Image processing apparatus and method, and medium therefor

[0002] 1. Field of the Invention

[0003] The present invention relates to a method of and an apparatus for processing image, as well as to a medium which provided a program implementing the image processing method. More specifically, the present invention pertains to an image processing apparatus and method, as well as a medium, which permit easy retouching such as three-dimensional editorial modification on the image of a three-dimensional object that is contained in a two-dimensional displayed image.

[0004] 2. Description of the Related Art

[0005] Various methods have been proposed that implement various processes on a two-dimensional image and extract information necessary for the processes from the two-dimensional image. Documents describing the methods include James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes, “Computer Graphics, principles and practice”, ADDISON-WESLEY PUBLISHING COMPANY, 1996 (hereinafter referred to as “Document 1”), Paul E. Debevec, Camillo J. Taylor, and Jitendra Malik, “Modeling and Rendering Architecture from Photographs: A hybrid geometry-and-image-based approach”, proceedings of SIGGRAPH 96, pp. 11-20 (hereinafter referred to as “Document 2”), Oliver Faugeras, “Three-dimensional computer version”, The MIT press (hereinafter referred to as “Document 3”), Kenneth P. Fishin, and Brian A. Barsky, “Family of New Algorithms for Soft Filling”, proceedings of SIGGRAPH 84, pp. 235-244 (hereinafter referred to as “Document 4”), Pat Hanrahan and Paul Haeberli, “Direct WYSIWYG Painting and Texuturing on 3D Shapes”, proceedings of SIGGRAPH 90, pp. 215-233 (hereinafter referred to as “Document 5”), Youichi Horry, Ken-ichi Anjyo, and Kiyoshi Arai, “Tour Into the Picture: Using a Spidery Mesh Interface to Make Animation from a Single Image”, proceedings of SIGGRAPH 97, pp. 225-232 (hereinafter referred to as “Document 6”), and Michael Gleicher, “Image Snapping”, proceedings of SIGGRAPH 95, pp. 183-190 (hereinafter referred to as “Document 7”).

[0006] In Document 1, image processing called “two-dimensional paint” is described in which a computer is used to perform processing in the same way as a designer or the like draws a picture on paper using a paintbrush or airbrushing (a technique that draws a picture by spraying paints on paper).

[0007] In this type of conventional two-dimensional paint, even when a three-dimensional object is displayed in an image, the image itself is treated on a two-dimensional plane. Accordingly, when characters are rendered irrespective of the direction of the three-dimensional object in a three-dimensional space which is displayed on the two-dimensional plane, or a figure is added, the image looks unnatural.

[0008] For example, in the case where a house-shaped three-dimensional object, as shown in FIG. 1A, is displayed, and characters are rendered on walls of the house without the direction of the characters, the characters do not look as if they are written on the walls, as shown in FIG. 1B. For adding a parallelepiped room to a wall of the house, in the case where a rectangle is rendered on the wall without the direction of the rectangle, the image looks unnatural, as shown in FIG. 1C. In the case where a cylinder is displayed in a two-dimensional image as shown in FIG. 2A, when characters are rendered on the side surface without ignoring the curvature of the side surface, the characters do not look as if they are written on the side surface, as shown in FIG. 2B.

[0009] Accordingly, in order that an image may not look unnatural in two-dimensional paint, it is required that a character or figure be rendered being transformed so as to match the direction of a three-dimensional object displayed in the two-dimensional image. Performing operations for the rendering requires a degree of experience.

[0010] Therefore, there is a method in which a user uses a ten-key pad or graphical user interface (GUI) to input an angle of inclination of the three-dimensional object displayed on the two-dimensional image so that a computer uses the input to transform a new character or figure to be rendered. In this method, the user must adjust the angle of inclination of the three-dimensional object to be input to the computer while viewing the rendered character or figure so that the rendered character or figure does not look unnatural. The adjustment also requires a degree of experience.

[0011] As described above, when the user instructs the rendering of a character or figure ignoring the direction of a three-dimensional object displayed on a two-dimensional image, the computer cannot display the character or figure as it looks natural, in other words, an image looking as if the character or figure was originally positioned cannot be obtained. This is due to lack of information on the position of the three-dimensional object displayed in the two-dimensional image and information (a position at which a landscape or the like was observed in the case where the two-dimensional image was obtained by performing image capture on a picture) on the position of image capture for the two-dimensional image.

[0012] Accordingly, there is a method that uses computer vision to find, from a two-dimensional image, the position of a three-dimensional object in a three-dimensional space displayed in the two-dimensional image, and the image capture position of the three-dimensional object.

[0013] In other words, in Document 2, a method is disclosed in which a three-dimensional shape such as a parallelepiped is correlated using a GUI with a structure in a plurality of photographs, and the size of the structure and the photographing position are found. In Document 3, other various methods for finding the position of a three-dimensional object and an image capture position on the object are disclosed.

[0014] In the above-described methods using the computer vision to compute a three-dimensional object position and an image capture position on the object, the principles of triangulation are used. This requires a plurality of images obtained by performing image capture on the same three-dimensional object from plurality of image capture positions. However, when two-dimensional paint is performed, the images are not always prepared, and when the two-dimensional images are photographs of a picture, the plurality of images is not used in general.

[0015] If the plurality of images obtained by performing image capture on the same three-dimensional object from the plurality of image capture positions can be prepared, the corresponding positions (e.g., vertices of the roof, etc., of a structure as a three-dimensional object) of the same three-dimensional object displayed on the images must be designated in order for the computer to compute the position of the three-dimensional object and the image capture position. The operation of designation for the images is complicated, and requires a time. In addition, in the case where based on the position (in the three-dimensional space) of the three-dimensional object displayed in the two-dimensional image and the image capture position, three-dimensionally natural rendering is performed, three-dimensional data, such as three-dimensional coordinates generated when the three-dimensional object is viewed from the image capture position, must be processed, which requires a great amount of operations for the processing.

[0016] In Document 4, a method for color processing for two-dimensional paint is disclosed.

[0017] In two-dimensional paint, the user uses a GUI to select a color for rendering and to perform rendering using the selected color. The color and brightness of a three-dimensional object displayed in the two-dimensional image vary depending on the positional relationship between the direction of the object surface and illumination. For example, painting an entire surface of the three-dimensional object in the same color (the same RGB levels) causes an unnatural rendering result. Accordingly, for obtaining a natural rendering result, a painting color must be gradationally changed considering the positional relationship between the direction of a surface to be painted and illumination. In particular, in the case where a surface to be painted is curved, pixel levels needs to be sequentially changed so that the rendering result is natural. Thus, a painting color must be selected for, for example, several pixels, which requires a great amount of operations.

[0018] Accordingly, in Document 4, a method called “tint fill” is disclosed in which in the case where a pixel color is a mixture of two colors and the two colors are known, either color can be changed into another color. The use of tint fill can efficiently cope with an area in which a foreground and a background are mixed, but cannot cope with object shading and light reflected by the object.

[0019] In Document 5, a method that transforms and pastes a character or handwritten figure input by operating a tablet or mouse along the surface of a three-dimensional object displayed in a two-dimensional image is disclosed.

[0020] Since the method disclosed in Document 5 uses three-dimensional information (e.g., the position of the three-dimensional object in a three-dimensional space, an image capture position, size, inclination, etc.) on the three-dimensional object displayed in the two-dimensional image, the method in Document 5 needs to recognize the three-dimensional information by using computer vision. However, when computer vision is used to recognize the three-dimensional information, a plurality of images are required as described above, and an amount of operations greatly increases.

[0021] Although the method in Document 5 can also use a three-dimensional model of a three-dimensional object generated in the computer, it must process three-dimensional data and requires a great amount of operations.

[0022] Regarding three-dimensional model generating, Document 2 discloses a technique that includes the steps of using computer vision technology to extract an outline of the shape of a three-dimensional object displayed in the two-dimensional image from the two-dimensional image, pasting a captured image onto the shape surface (performing texture mapping), and using these steps to generate a two-dimensional image viewed from a free viewpoint (captured from a free image capture position). Similar to the above-described cases, this technique requires a plurality of images because it uses computer vision technology to extract, from a two-dimensional image, an outline of the shape of a three-dimensional object displayed in the two-dimensional image.

[0023] In Document 6, a method is disclosed in which based on a single two-dimensional image, an image viewed from a viewpoint excluding the image capture position of the two-dimensional image is generated. This method only generates an image viewed from a viewpoint excluding the image capture position, and it is based on the assumption that a three-dimensional object displayed in a two-dimensional image or another adjacent object has a very simple shape of a plate. Accordingly, almost no serious problem occurs when an image viewed from another viewpoint is approximately generated. However, when a character or figure is added to the two-dimensional image, it is impossible to make the rendering result natural. For example, in the case where a cylinder is displayed in a two-dimensional image, a natural image formed by adding a character or figure on the side surface of the cylinder cannot be obtained.

[0024] In addition, it is required in many cases that when a two-dimensional image is processed, the user designates predetermined characteristic points (e.g., vertices) of a three-dimensional object in the two-dimensional image. The characteristic points are designated by operating a mouse or input pen, but it is difficult to accurately designate vertices of the three-dimensional object in the two-dimensional image. Accordingly, Document 7 discloses a method that includes the steps of extracting an edge (a portion whose brightness and color greatly change) in the two-dimensional image, and moving a cursor on the edge when the cursor is in the vicinity of the edge. Since all edges in the two-dimensional image are detected in the method in Document 7, sensitive response to noise or a texture in the two-dimensional image occurs, and the cursor may be moved to an unnecessary position.

[0025] Although many image processing techniques have been proposed heretofore as described above, these techniques require certain levels of skill and experience, as well as time, or necessitates a plurality of images of an object as viewed in different directions. These techniques therefore cannot suitably be used for the purpose of effecting three-dimensional edition or other retouching on the images of characters and patterns which are contained in a two-dimensional displayed image.

[0026] Thus, no image processing technique has heretofore been proposed which would enable easy three-dimensional edition or retouching on characters or patterns in a two-dimensional displayed image.

[0027] Accordingly, it is a primary object of the present invention to provide an image processing technique which permits a user to easily effect three-dimensional editorial work or other retouching on a two-dimensional displayed image.

[0028] To this end, according to one aspect of the present invention, there is provided an image processing apparatus for processing a two-dimensional moving picture, comprising: operating means operable to enable designation of shape information concerning the shape of a surface that forms part of a three-dimensional object displayed in the moving picture and designation of a characteristic point which is located on the moving picture; computing means for determining, based on the shape information and the characteristic point, a correlation equation expressing the correlation between a projected image which is obtained by projecting the surface constituting part of the three-dimensional object and an expanded image which is obtained by developing the surface into two-dimensional surface; first transforming means for transforming, based on the correlation equation, the projected image of the surface constituting part of the three-dimensional object in the moving picture into the expanded image; storing means for storing the expanded image; retouching means for effecting retouching on the expanded image stored in the storing means; and second transforming means for transforming, based on the correlation equation, the retouched expanded image back to the projected image in the moving picture.

[0029] According to another aspect of the present invention, there is provided an image processing method for processing a two-dimensional moving picture, comprising the steps of: preparing operating means operable to enable designation of shape information concerning the shape of a surface that forms part of a three-dimensional object displayed in the moving picture and designation of a characteristic point which is located on the moving picture; computing, in response to the designation of the shape information and the characteristic point by the operating means, a correlation equation expressing the correlation between a projected image which is obtained by projecting the surface constituting part of the three-dimensional object and an expanded image which is obtained by developing the surface into two-dimensional surface, based on the shape information and the characteristic point; executing first transformation for transforming, based on the correlation equation, the projected image of the surface constituting part of the three-dimensional object in the moving picture into the expanded image; storing the expanded image; retouching the expanded image stored in the storing step; and executing a second transformation for transforming, based on the correlation equation, the retouched expanded image back to the projected image in the moving picture.

[0030] According to still another aspect of the present invention, there is provided a medium for enabling a computer to execute a computer program for conducting image processing on a two-dimensional moving picture, the program comprising the steps of: computing, in response to designation of shape information concerning the shape of a surface that forms part of a three-dimensional object displayed in the moving picture and designation of a characteristic point which is located on the moving picture, the designation being made through operation of operating means that enables designation of the shape information and the characteristic point, a correlation equation expressing the correlation between a projected image which is obtained by projecting the surface constituting part of the three-dimensional object and an expanded image which is obtained by developing the surface into two-dimensional surface, based on the shape information and the characteristic point; executing first transformation for transforming, based on the correlation equation, the projected image of the surface constituting part of the three-dimensional object in the moving picture into the expanded image; storing the expanded image; retouching the expanded image stored in the storing step; and executing a second transformation for transforming, based on the correlation equation, the retouched expanded image back to the projected image in the moving picture.

[0031] The above and other objects, features and advantages of the present invention will become clear from the following description of the preferred embodiment when the same is read in conjunction with the accompanying drawings.

[0032] FIGS. 1A, 1B, and 1C are illustrations of rendering results using a conventional paint tool.

[0033] FIGS. 2A and 2B are illustrations of rendering results using a conventional paint tool.

[0034] FIG. 3 is a block diagram showing an image processing apparatus according to an embodiment of the present invention.

[0035] FIG. 4 is a block diagram showing the functions of the image processing apparatus shown in FIG. 3.

[0036] FIGS. 5A, 5B, and 5C are drawings showing the results of processing a two-dimensional image by the image processing apparatus shown in FIG. 4.

[0037] FIG. 6 is a drawing showing the results of processing a two-dimensional image by the image processing apparatus shown in FIG. 4.

[0038] FIG. 7 is a flowchart illustrating a process performed by the image processing apparatus shown in FIG. 4.

[0039] FIG. 8 is an illustration of a main window displayed when the image processing apparatus is activated.

[0040] FIGS. 9A and 9B are drawings illustrating copying.

[0041] FIGS. 10A, 10B, and 10C are drawings illustrating pasting.

[0042] FIG. 11 consists of drawings (A), (B), (C), and (D) illustrating texture copying by using a combination of copying and pasting.

[0043] FIG. 12 is an illustration of characteristic points designated by a user.

[0044] FIG. 13 is a flowchart illustrating the details of automatic correction in step S7 shown in FIG. 7.

[0045] FIG. 14 is a drawing illustrating step S22 shown in FIG. 13.

[0046] FIG. 15 is a drawing illustrating a method of extracting the contour line of three-dimensional object displayed in a two-dimensional image.

[0047] FIG. 16 is a flowchart illustrating the details of step S7 shown in FIG. 7.

[0048] FIG. 17 is a drawing illustrating correction of characteristic points.

[0049] FIGS. 18A, 18B, and 18C are drawings illustrating computation of an inverse transform expression and a right transform expression.

[0050] FIG. 19 is a drawing showing the relationship among a three-dimensional object in a three-dimensional space, an expanded image, and a projected image.

[0051] FIGS. 20A and 20B are drawings illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a parallelepiped.

[0052] FIGS. 21A, 21B, and 21C are drawings illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a cylinder.

[0053] FIG. 22 is a drawing illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a cylinder.

[0054] FIG. 23 is a drawing illustrating angle ν.

[0055] FIG. 24 is a flowchart illustrating computation of angle θ in the computation illustrated in FIG. 21.

[0056] FIGS. 25A, 25B, and 25C are drawing illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a sphere.

[0057] FIGS. 26A and 26B are drawings illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a sphere.

[0058] FIG. 27 is a drawing illustrating computation of μ from λ₁and λ₂shown in FIGS. 26A and 26B.

[0059] FIG. 28 is a drawing illustrating computation of μ from λ₁and λ₂shown in FIGS. 26A and 26B.

[0060] FIG. 29 is a drawing illustrating computation of μ from λ₁and λ₂shown in FIGS. 26A and 26B.

[0061] FIG. 30 is a drawing illustrating computation of μ from λ₁and λ₂shown in FIGS. 26A and 26B.

[0062] FIGS. 31A, 31B, and 31C are drawings illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a cone.

[0063] FIG. 32 is a drawing illustrating computation of an inverse transform expression and a right transform expression in the case where a three-dimensional object is a cone.

[0064] FIG. 33 is a drawing illustrating angle ν.

[0065] FIGS. 34A and 34B are drawings showing examples of characteristic points designated for a cone.

[0066] FIGS. 35A, 35B, 35C, and 35D are drawings illustrating matte processing.

[0067] FIG. 36 is a flowchart illustrating the object property/illumination change process in step S16 shown in FIG. 7.

[0068] FIG. 37 is a drawing illustrating step S53 shown in FIG. 36.

[0069] FIGS. 38A and 38B are drawings illustrating the results of the color distribution of an object.

[0070] FIG. 39 consists of drawings (A), (B), (C), and (D) illustrating erasing.

[0071] FIGS. 40A and 40B are illustrations of copying processing.

[0072] FIG. 41 is an illustration of a copy and paste processing performed on a still picture.

[0073] FIG. 42 is an illustration of a copy and paste processing performed on a dynamic image.

[0074] FIG. 43 is an illustration of a copy and paste processing executed on a moving picture composed of a plurality of frames showing a three-dimensional object, where the portion visible in one frame is also visible in other frames.

[0075] FIG. 44 is a detailed illustration of the copy and paste processing executed on a moving picture composed of a plurality of frames showing a three-dimensional object, where the portion visible in one frame is also visible in other frames.

[0076] FIG. 45 is an illustration of a copy and paste processing executed on a moving picture composed of a plurality of frames showing a three-dimensional object, where a portion visible in one frame is invisible in other frames.

[0077] FIGS. 46A to 46C are illustrations of points that are to be designated as being characteristic points.

[0078] FIG. 47 is a detailed illustration of the copy and paste processing executed on a moving picture composed of a plurality of frames showing a three-dimensional object, where a portion visible in one frame is invisible in other frames.

[0079] FIG. 48 is an illustration of a method for determining amounts of offsets.

[0080] FIG. 49 is an illustration showing the detail of the method for determining amounts of offsets.

[0081] FIG. 50 is a flowchart showing a copy and paste processing effected on a moving picture.

[0082] FIG. 51 is an illustration of the copy and paste processing effected on a moving picture.

[0083] FIG. 52 is an illustration of various types of medium usable as the medium in accordance with the present invention.

[0084] FIG. 3 shows an image processing apparatus according to an embodiment of the present invention. The image processing apparatus has a computer as a base, and can easily perform three-dimensional editing, etc., on a two-dimensional image.

[0085] An arithmetic circuit 1 consists of a central processing unit (CPU) or the like, and executes an application program stored in a program memory 2 under the control of an operating system (OS) loaded into the program memory 2, whereby performing various processes (described below) on a two-dimensional image. The program memory 2 consists of a random access memory (RAM), and temporarily stores the OS and the application program stored (recorded) in an external storage unit 7. A data memory 3 consists of a RAM, and temporarily stores data necessary for processing in the arithmetic circuit 1. A frame memory 4 consists of a RAM, and stores image data to be displayed on an image display unit 5. The image display unit 5 consists of, for example, a cathode-ray tube or liquid crystal display, and displays the image data stored in the frame memory 4. An input unit 6 consists of, for example, a combination of a mouse and a tablet, or a keyboard. The input unit 6 is operated to input a necessary command or data or to designate a predetermined position on the screen of the image display unit 5. The external storage unit 7 consists of, for example, a hard disk, a floppy disk, a compact disk ROM, or a magnetooptical disk, and stores the OS and the application program. The external storage unit 7 also stores data necessary for the operation of the arithmetic circuit and two-dimensional images (digital data) to be processed. An external interface 8 functions as an interface for receiving externally supplied data such as two-dimensional images captured by a camera 9 and data transmitted via a communication link (not shown).

[0086] The arithmetic circuit 1, the program memory 2, the data memory 3, the frame memory 4, the external storage unit 7, and the external interface 8 are connected by a bus, and programs and data are transmitted and received via the bus.

[0087] When main power is supplied to the image processing apparatus, the OS stored in the external storage unit 7 is read and loaded into the program memory 2, and it is executed by the arithmetic circuit 1. By operating the input unit 6 so that the application program is executed, the OS controls the arithmetic circuit 1 to read the application program from the external storage unit 7, and the read application program is executed after being loaded into the program memory 2. Thereby, for two-dimensional images that were captured by the camera 9 and stored in the external storage unit 7, various processes are described.

[0088] FIG. 4 shows the functions of the image processing apparatus (shown in FIG. 3) that are realized such that the arithmetic circuit 1 executes the application program.

[0089] A GUI event is supplied from the OS to an input event processor 11. The input event processor 11 analyzes the GUI event, and activates, in accordance with the analysis, a copy processor 12, a paste processor 13, a paste buffer selector 14, an erase processor 15, a matte processor 16, an object property processor 17, and a paint processor 18 so that processing is performed.

[0090] The types of the GUI event include, for example, a movement of a mouse cursor, mouse clicking (mouse button pressing and unpressing), selection from a menu on the GUI, and button operation. The GUI event is defined in a window system such as “X-Window”.

[0091] The copy processor 12 performs a copy operation for storing an image in a paste buffer (described below). The paste processor 13 performs a paste operation for pasting the image stored in the paste buffer on a two-dimensional image stored in an image buffer (described below). A paste buffer selector 14 performs a paste buffer selecting operation for selecting, from the image buffer, a buffer used as the paste buffer, and storing its buffer identification (ID) in buffer ID storage 21.

[0092] In the data memory 3, regions in which image pixel levels are stored in units of screens are reserved as image buffers. The image buffers are provided with buffer IDs so that each image buffer is identified. Among the image buffers, an image buffer whose ID is stored in the buffer storage 21 is particularly called a “paste buffer”. In each image buffer, the pixel levels (e.g., RGB values) of a two-dimensional image can be stored, and also a matte on an object displayed in the two-dimensional image can be stored. The matte is described below.

[0093] The erase processor 15 performs an erase operation for deleting part of an image stored in the paste buffer. The matte processor 16 performs for generating a matte for a three-dimensional object displayed in the two-dimensional image stored in the image buffer. The object property operator 17 performs object property processing for changing so-called “material properties” such as the color and material of the three-dimensional object displayed in the two-dimensional image stored in the image buffer, and illumination processing for changing illumination (changing the condition of illumination to the three-dimensional object in the two-dimensional image stored in the image buffer). The paint processor 18 performs a paint operation for painting two-dimensional paint on the image stored in the paste buffer.

[0094] A designated transform processor 19 corrects, in accordance with instructions from the copy processor 12 and the paste processor 13, the positions of characteristic points on the two-dimensional image stored in the image buffer, which are designated by the user by operating the input unit 6. The designated transform processor 19 sends the position-corrected characteristic points back to the copy processor 12 and the paste processor 13. An image transform processor 20 transforms, in accordance with instructions from the copy processor 12 and the paste processor 13, the image stored in the image buffer or the paste buffer. The buffer ID storage 21 stores the buffer ID of the image buffer selected as the paste buffer by the paste buffer selector. The buffer ID stored in the buffer ID storage 21 is supplied to the erase processor 15, the paint processor 18, and the image transform processor 20. Accordingly, the erase processor 15, the paint processor 18, and the image transform processor 20 recognize the image buffer as the paste buffer.

[0095] The display processor 22 performs processing necessary for displaying the two-dimensional image stored in the image buffer on the image display unit 5. Image data output from the display processor 22 are sent to the display processing by the OS, whereby the image data are written in the frame memory 4 so as to be displayed on the image display unit 5.

[0096] In FIG. 4, the blocks excluding the data memory 3 and the buffer ID storage 21 are realized such that the arithmetic circuit 1 executes the application program. Program components corresponding to the blocks are formed as modules.

[0097] According to the above-described image processing apparatus, for example, in the case where a two-dimensional image on which a house-shaped three-dimensional object is displayed is stored in the image buffer as shown in FIG. 5A, when characters are written ignoring the direction of walls of the house, a two-dimensional image that looks as if the characters were written along the walls can be obtained as shown in FIG. 5B. When a rectangle is drawn ignoring the direction of the walls of the house, a two-dimensional image in which a parallelepiped room is added to the walls can be obtained as shown in FIG. 5C. In addition, in the case where a two-dimensional image on which a cylinder as shown in FIG. 6A is displayed is stored in the image buffer, even when characters are written ignoring the direction of the side surface of the cylinder, a two-dimensional image that looks as if the characters were written along the side surface can be obtained as shown in FIG. 6B.

[0098] In other words, when the image of a character-written three-dimensional object is not actually captured by the camera 9, by implementing three-dimensional operations on the two-dimensional image captured by the camera 9, a two-dimensional image that looks as if the image of the character-written three-dimensional object was actually captured can be easily obtained.

[0099] Processing by the image processing apparatus shown in FIG. 4 is described with reference to the flowchart shown in FIG. 7.

[0100] When the application program is executed by the arithmetic circuit 1 (shown in FIG. 3), the image display unit 5 displays, for example, a main window as shown in FIG. 8. Command buttons 31 and Shape buttons 32 are displayed to the right of the main window, and three windows (hereinafter referred to as “buffer windows”) displaying two-dimensional images stored in the image buffer are displayed to the left of the window.

[0101] Each Command button is operated when a command is input. In the main window shown in FIG. 8, the Command buttons 31 consist of a Paste Select button that is operated when a paste buffer selection is performed, a Copy button that is operated when copying is performed, a Paste button that is operated when pasting is performed, an Erase button that is operated when erasing is performed, a Matte button that is operated when a matte operation is performed, a Material button that is operated when object property processing is performed, a Light button that is operated when illumination changing is performed, and a Paint button that is operated when painting is performed.

[0102] Each Shape button 32 is operated when the shape (shape information) of the surface of a three-dimensional object is input. In the main window shown in FIG. 8, the Shape buttons 32 consist of a Rectangle button, a Cylinder button, a Sphere button, and a Cone button that designate a rectangle, a cylindrical side surface, a sphere, and a conical surface, respectively.

[0103] In the main window shown in FIG. 8, the three buffer windows respectively display contents stored in image buffers #1, #2, and #3 among the image buffers (where the suffix i of image buffer #i represents, for example, a buffer ID). When a plurality of buffer windows are displayed, among the plurality of buffer windows, a buffer window displaying the content of a buffer of interest (described below) is displayed as a foreground (set to be active).

[0104] The operation of the main window is basically unified into a user interface in which an image buffer for processing is selected and a type of processing is subsequently selected. Accordingly, the user initially operates the input unit 6 to click any one of the buffer windows displayed in the main window, whereby an image buffer for processing is selected.

[0105] Specifically, when the user clicks any one of the buffer windows, in step S1, the input event processor 11 selects, as a buffer of interest, an image buffer whose storage content is displayed on the buffer window.

[0106] After that, the user clicks any one of the Command buttons 31. At this time, in step S2, the input event processor 11 recognizes which of the Command buttons 11 is clicked. The input event processor 11 proceeds to step S3, and determines which type of processing (operation) is designated, based on the recognition. In step S3, when the input event processor 11 has determined that paste buffer selection has been designated, in other words, when the Paste Selection button among the Command buttons 31 has been operated (clicked), the input event processor 11 proceeds to step S4, and activates the paste buffer selector 14 to perform paste buffer selection. In other words, the paste buffer selector 14 overwrites the buffer ID storage 21 with the buffer ID of the buffer of interest. Thereby, the image buffer corresponding to the present buffer of interest is used as a paste buffer, and the input event processor 11 proceeds back to step S1.

[0107] If the input event processor 11 has determined in step S3 that either copying or pasting has been instructed, in other words, when the Copy button or the Paste button among the Command buttons 31 has been operated, the input event processor 11 proceeds to step S5 at the time of the operation of the Shape button. Copying is a process that consists of generating an expanded image formed by expanding, on a two-dimensional plane, planes constituting a three-dimensional object displayed in a two-dimensional image stored in the buffer of interest, and storing the expanded image in the paste buffer. Pasting is a process that consists of pasting the image on the two-dimensional plane, which is stored in the paste buffer, on the planes constituting the three-dimensional object so that a projected image on the screen is generated, and pasting the projected image on the two-dimensional image stored in the buffer of interest. In either process, the planes constituting the three-dimensional object are processed. Accordingly, when the user operates the Copy button or the Paste button, the user designates (inputs) shape information on the planes constituting the three-dimensional object by successively operating any one of the Shape buttons 32.

[0108] In step S5, based on the operation of any one of the Shape buttons 32, the shape (shape information) of the planes constituting the three-dimensional object, which are copied or pasted, is recognized. When characteristic points are input, the input event processor 11 proceeds to step S6. In other words, in copying, it is required that portions (planes) generating the expanded image be specified by designating some of the points constituting the planes of the three-dimensional object displayed in the two-dimensional image stored in the buffer of interest. In pasting, in order that the projected image of the image stored in the paste buffer may be pasted on the planes constituting the three-dimensional object displayed on, for example, the two-dimensional image stored in the buffer of interest, the projected image needs to be specified by designating some of the points constituting the projected image. Accordingly, the user operates the Copy button or the Paste button, and operates any of the Shape buttons 32 to successively designate the shape of the three-dimensional object to be copied or pasted. After that, the user operates the input unit 6 to designate, as characteristic points, some of the points constituting the planes (to be processed) of the three-dimensional object in the two-dimensional image stored in the buffer of interest.

[0109] In step S6, the coordinates of the characteristic points designated as described above are recognized.

[0110] In the above-described case, the characteristic points are designated after designating the shape of the three-dimensional object. However, the shape of the three-dimensional object may be designated after designating the characteristic points. When copying is instructed, the copy processor 12 performs steps S5 and S6, and when pasting is instructed, the paste processor 13 performs steps S5 and S6.

[0111] In step S6, the copy processor 12 or the paste processor 13, which has recognized the characteristic points designated by the user, supplies the characteristic points to the designated transform processor 19. When the designated transform processor 19 receives the characteristic points, in step S7, it corrects their positions, and supplies the corrected points to the copy processor 12 or the paste processor 13, which supplied the characteristic points.

[0112] In step S8, the input event processor 11 determines whether either the Copy button or the Paste button has been operated. If the input event processor 11 has determined in step 8 that the Copy button has been operated, the copy processor 12 performs copying. At this time, the input event processor 11 proceeds from step S8 to S9, and the copy processor 12 computes a transform expression (hereinafter referred to as a “inverse transform expression”) for transforming, based on the shape information and the characteristic points, the planes of the three-dimensional object in the two-dimensional image stored in the buffer of interest into an expanded image. The copy processor 12 supplies it to the image transform processor 20. In step S10, the image transform processor 20 transforms, based on the inverse transform expression, the planes (specified by the characteristic points) of the three-dimensional object in the two-dimensional image stored in the buffer of interest into an expanded image. In step S11, the image transform processor 20 copies (performs overwriting with) the expanded image in the image buffer whose buffer ID is stored in the buffer ID storage 21, that is, in the paste buffer. After that, the input event processor 11 returns to step S1.

[0113] According to the above-described copying, when a two-dimensional image on which a house-shaped three-dimensional object V is displayed as shown in FIG. 9A is stored in the buffer of interest, by designating, as shape information, a rectangle that is the shape of a wall S with vertical lines, and designating, as characteristic points, four vertices P₁, P₂, P₃, and P₄on the wall S, the two-dimensional image (expanded image) of the wall S with vertical lines, obtained by expanding the three-dimensional object V on the two-dimensional plane, is generated as shown in FIG. 9B and stored in the paste buffer.

[0114] Therefore, when the two-dimensional image (as shown in FIG. 9A) on which the three-dimensional object V is displayed is stored in the image buffer, in order that an expanded image of the wall S with vertical lines may be obtained, the user needs to perform the following operations. First, the user sets, as a buffer of interest, the image buffer storing the two-dimensional image on which the three-dimensional object V is displayed. Second, the user operates the Copy button instructing copying. Third, the user operates the Rectangle button designating the shape of the wall S. Fourth, the user designates, as characteristic points, four vertices P₁, P₂, P₃, and P₄of the rectangle forming the wall S with vertical lines in the two-dimensional image.

[0115] In step S8, if the input event processor 11 has determined that the Paste button has been operated, the paste processor 13 performs pasting. In this case, the input event processor 11 proceeds from step S8 to S12, and the paste processor 13 computes a transform expression for transforming, based on the shape information and the characteristic points, the image on the two-dimensional plane stored in the paste buffer into a projected image. The paste processor 13 supplies it to the image transform processor 20. In step S13, the image transform processor 20 transforms, based on the transform expression, the image stored in the paste buffer into a projected image. In step S14, the image transform processor 20 pastes the projected image on the plane (specified by characteristic points) of the three-dimensional object displayed in the two-dimensional image stored in the buffer of interest. After that, the input event processor 11 returns to step S1.

[0116] According to the above-described pasting, when the image on the two-dimensional plane (shown in FIG. 10A) is stored in the paste buffer by the copying described with reference to FIGS. 9A and 9B, and the two-dimensional image on which the house-shaped three-dimensional object V is stored in the buffer of interest, by designating, as shape information, a rectangle that is the shape of an unfigured wall S′, and designating, as characteristic points, four vertices P₁′, P₂′, P₃′, and P₄′ of the unfigured wall S′, an expanded image of the wall S′, formed in the case where the three-dimensional object V is projected on the screen when the image stored in the paste buffer is pasted on the wall S′ of the three-dimensional object V, is generated and pasted on the wall S′ specified by the characteristic points P₁′, P₂′, P₃′ , and P₄′ of the three-dimensional object V displayed on the two-dimensional image stored in the buffer of interest.

[0117] Therefore, when the image shown in FIG. 10A is stored in the paste buffer, and the two-dimensional image (shown in FIG. 10B) on which the house-shaped three-dimensional object V is displayed is stored, in order that a two-dimensional image in which the image stored in the paste buffer is pasted on the wall S′ of the three-dimensional object V may be obtained, the user needs to perform the following operations. First, the user sets, as a buffer of interest, the image buffer storing the two-dimensional image on which the three-dimensional object V is displayed. Second, the user operates the Paste button for instructing pasting. Third, the user operates the Rectangle button for designating the shape of the wall S′. Fourth, the user designates, as characteristic points, four vertices P₁, P₂, P₃, and P₄of the rectangle forming the wall S′ in the two-dimensional image.

[0118] As is clear from the above-described descriptions, by performing the copying described with FIGS. 9A and 9B while using, as a buffer of interest, an image buffer storing a two-dimensional image on which the three-dimensional object V is displayed, and performing the pasting described with FIGS. 10A, 10B, and 10C, the texture of the wall S can be copied on another wall S′. In other words, the user can copy the texture of the wall S in the two-dimensional image on another wall S by only performing the above-described copying and pasting without paying attention to the directions of the walls S and S′.

[0119] In addition, by using conventional two-dimensional paint, or the like, to write the characters “ABC” or “DEF” in the paste buffer, similar to the writing of characters on a plane, and using pasting to paste the characters on the wall of the house shown in FIG. 5A, a natural two-dimensional image as shown in FIG. 5B can be obtained. In other words, it is not necessary to write the characters along the wall.

[0120] By performing copying so that an expanded image is stored in the paste buffer, or selecting, as the paste buffer, an image buffer storing a texture, and performing pasting three times after designating, as characteristic points, vertices p₁, p₂, p₃, and p₄constituting the plane S₁(shown in FIG. 5C), vertices p₃, p₄, p₅, and p₆constituting the plane S₂(shown in FIG. 5C), and vertices P₂, P₃, P₆, and p₇constituting the plane S₃(shown in FIG. 5C), a two-dimensional image in which a parallelepiped room is added to the wall of the house can be easily created (In this case, rectangles are designated for pieces of shape information of planes S₁, S₂, and S₃).

[0121] As is understood from FIGS. 5A to 5C, a three-dimensional object for pasting may be displayed or may not be displayed in the two-dimensional image stored in the buffer of interest. In other words, FIG. 5B shows that a three-dimensional object for pasting (three-dimensional object on which characters are pasted) is displayed on the original two-dimensional image (the two-dimensional image shown in FIG. 5A. FIG. 5C shows that a three-dimensional object (parallelepiped defined by the planes S₁, S₂, and S₃shown in FIG. 5C) for pasting is not displayed on the original two-dimensional image. In the case where a three-dimensional object for pasting is not on the original two-dimensional image, correction of characteristic points designated by the user, which are points of the three-dimensional object, that is, step S7 shown in FIG. 7, is skipped. This is because user-designated characteristic points are corrected to be positioned on the contour line of the three-dimensional object in step 7 as described below, but when the three-dimensional object is not displayed on the original two-dimensional image, its contour line does not exist.

[0122] An expanded image obtained by copying based on a two-dimensional image may be pasted on a two-dimensional image excluding the original two-dimensional image. The plane of a three-dimensional object on which the expanded image is pasted may not have the same shape as the plane of the original three-dimensional object for generating the expanded image. For example, by performing copying on a latticed wall of a three-dimensional object displayed in a two-dimensional image as shown in FIG. 11A, storing the expanded image in the paste buffer as shown in FIG. 11B, and performing pasting on the side plane of a cylinder displayed in a two-dimensional image as shown in FIG. 11C, the latticed expanded image can be pasted along the side plane of the cylinder. In other words, the texture of a three-dimensional object can be pasted as the texture of another three-dimensional object.

[0123] In the paste buffer, not only images obtained by copying, but also arbitrary images can be stored. According to pasting, an arbitrary image can be pasted on a three-dimensional object displayed in a two-dimensional image (As described above, the three-dimensional object does not need to be displayed on the two-dimensional image).

[0124] According to copying and pasting, in addition to the above-described operations, various three-dimensional operations on a two-dimensional image can be performed. For example, in the case where the plane of a three-dimensional object displayed in the two-dimensional image is irregular, by storing an expanded image of the plane in the paste buffer, deleting the irregularity in the paste buffer, and using pasting to paste the irregularity-deleted image on the original three-dimensional object, a two-dimensional image on which the irregularity-deleted three-dimensional object is displayed can be obtained. In addition, for example, when copying and pasting on the same three-dimensional object is performed, by replacing characteristic points to be designated for pasting by characteristic points to be designated for pasting, movement, enlargement, reduction, and other non-linear transformation (e.g., transformation producing virtual effects of image capture using a lens having optical characteristics, such as a so-called “fish-eye lens”, etc.) on a three-dimensional object can be performed.

[0125] Referring back to FIG. 7, if the input event processor 11 has determined in step S3 that the matte operation has been instructed, in other words, when among the Command buttons 31, the Matte button has been operated, the input event processor 11 proceeds to step S15, and activates the matte processor 16 to perform the matte operation. At this time, the mate processor 16 generates and stores, in the buffer of interest, the matte of the three-dimensional object displayed in the two-dimensional image stored in the buffer of interest, and returns to step S1. The details of the matte operation are described below.

[0126] If the input event processor 11 has determined in step S3 that object property processing or illumination change has been instructed, in other words, among the Command buttons 31, the Material button or the Light button has been operated, the input event processor 11 proceeds to step S16. In step S16, the input event processor 11 activates the object property processor 17 to perform the object property processing or illumination change (object property/illumination change processing), and returns to step S1. The details of the object property processing or illumination change are described below.

[0127] If the input event processor 11 has determined in step S3 that erasing has been instructed, in other words, when among the Command buttons 31, the Erase button has been operated, the input event processor 11 proceeds to step S17. In step 17, the input event processor 11 activates the erase processor 17 to perform erasing, and returns to step S1. The details of erasing are described below.

[0128] If the input event processor 11 has determined in step S3 that painting has been instructed, in other words, when among the Command buttons 31, the Paint button has been operated, the input event processor 11 proceeds to step S18. In step S18, the input event processor 11 activates the paint processor 18 to perform painting, and return to step S1. In painting, by using, for example, a paint tool performing two-dimensional paint similar to conventional one, part of an image stored in the paste buffer is erased or another image is added in accordance with user operations from the input unit 6.

[0129] As described above, in the case where the plane of a three-dimensional object displayed in the two-dimensional image is irregular, by storing an expanded image of the plane in the paste buffer, deleting the irregularity in the paste buffer, and using pasting to paste the irregularity-deleted image on the original three-dimensional object, a two-dimensional image on which the irregularity-deleted three-dimensional object is displayed can be obtained.

[0130] By performing copying on a two-dimensional image on which the three-dimensional object is displayed as shown in FIG. 5A, storing an expanded image of the walls in the paste buffer, using painting to write the characters “ABC” or “DEF” on the image stored in the paste buffer, and using pasting to paste the character-written image on the original walls, a natural two-dimensional image can be obtained as shown in FIG. 5B. In other words, two-dimensional paint enables facile three-dimensional rendering.

[0131] When copying described with reference to FIGS. 9A and 9B, or pasting described with reference to FIGS. 10A to 10C, is performed, it is difficult for an operation-experienced user to accurately designate the positions of vertices P₁to P₄or vertices P₁′ to P₄′ as characteristic points. In other words, when copying described with reference to FIGS. 9A and 9B is performed, the user-designated characteristic points P₁to P₄shift from the positions of vertices to be essentially designated, as shown in FIG. 12. When the two-dimensional image defocuses or is blurred, it is impossible for even the experienced user to accurately designate, as characteristic points, the positions of vertices to be essentially designated.

[0132] Accordingly, in step S7 in FIG. 7, automatic correction is performed so that the positions of the characteristic points designated by the user are corrected to the positions to be essentially designated.

[0133] In other words, in step S7, the copy processor 12 or the paste processor 13 activates the designated transform processor 19 to supply shape information and characteristic points designated from the input unit 6 operated by the user.

[0134] When the designated transform processor 19 receives the shape information and the characteristic points, it performs the automatic correction for correcting the characteristic points.

[0135] The points to be designated are basically points on the contour line (borderline) of planes of a three-dimensional shape used for copying or pasting, as described below. Accordingly, the designated transform processor 19 extracts the contour line of the three-dimensional shape (plane) used for copying or pasting, and corrects the characteristic points so as to be positioned on the contour line.

[0136] FIG. 13 shows a flowchart describing automatic correction performed by the designated transform processor 19. To make description simpler, an example is given hereinafter wherein a plane of a three-dimensional object displayed on a two-dimensional image, which is used for the copying or pasting, is a quadrangle as shown in FIG. 12. Therefore, as illustrated in FIG. 12, four vertices of the quadrangle should be designated as characteristic points.

[0137] In the designated transform processor 19, firstly in step 21, a combination of two adjacent points is selected from characteristic points inputted (designated) by the user. Specifically, points that should be designated as characteristic points are, as described above, basically on a contour line of a plane of the three-dimensional object displayed on the two-dimensional image, which is used for the copying or pasting. Also it is determined in advance, according to a shape of the three-dimensional object, which points on the contour line the user should designate as characteristic points, as described in the following. Since two adjacent characteristic points on the contour line can be recognized from the characteristic points inputted by the user based on the shape information, the designated transform processor 19 selects one of the combinations of two characteristic points recognized in such a manner.

[0138] In step S22, a region to be used for edge detection, performed in step S23 described below, (hereinafter referred to as an edge candidate region) is detected. Specifically, in step S22, for example, a line segment connecting two characteristic points selected in step S21 (hereinafter referred to as selected characteristic points) is obtained, and pixels within a range of a predetermined distance (a distance of five-pixels, for example) from the line segment are detected. Such detected pixels form an edge candidate region.

[0139] Specifically, given that the characteristic points P₁to P₄are designated, for example, as shown in FIG. 12, when the characteristic points P₁and P₂are selected as the selected characteristic points, a shaded region shown in FIG. 14 is detected as the edge candidate region.

[0140] As well as allowing the user to input a desired distance by operating the input unit 6, it is possible to preset a distance range from the line segment connecting two selected characteristic points, wherein pixels are detected.

[0141] After the edge candidate region is detected, the process proceeds to step S23 wherein edge detection is performed using the edge candidate region.

[0142] Various types of conventional methods are proposed heretofore as edge detection methods, all of which are applicable here. In this embodiment, the following method, for example, is taken.

[0143] Specifically, pixels within the edge candidate region are filtered by an edge detection filter called a Sobel operator, and a gradient (gradient vector) in each pixel is obtained. It is conventionally known in the art to detect pixels having a greater gradient. In this embodiment, however, an inner product of a unit vector meeting at right angles with the line segment connecting the two selected characteristic points, namely, a normal vector, and a gradient in each pixel within the edge candidate region is computed, and a pixel having an inner product not smaller than a predetermined threshold level is detected as a pixel forming an edge (hereinafter referred to as an edge pixel).

[0144] As indicated in FIG. 12, the characteristic points are designated such that they do not exactly coincide with, but are in the vicinity of, the vertices of the quadrangle which form the plane of the three-dimensional object used for the copying or pasting. Therefore, the line segment connecting two selected characteristic points approximately coincides with a contour line of the quadrangle which forms the plane of the three-dimensional object used for the copying or pasting. As a result, an inner product of a normal vector and a gradient represents a gradient's component in the normal vector's direction, or in the direction approximately perpendicular to the contour line. Hence, edge detection using values of the inner products reduces adverse effects, compared with edge detection using values of gradients. In other words, it is possible to perform edge detection unaffected by noise or texture such as a moire pattern of a two-dimensional image.

[0145] After the edge pixels are detected, the process proceeds to step S24 wherein a straight line crossing the edge pixels is obtained, using, for example, the least squares method. Specifically, in step S24, a line minimizing the sum of the squares of the distances from the edge pixels detected in step S23 (hereinafter referred to as an edge straight line) is obtained.

[0146] In step S25, it is determined whether edge straight lines are obtained for all combinations of two adjacent characteristic points inputted by the user. If the result is negative, the process returns to step S21, and two adjacent characteristic points for which an edge straight line is not yet obtained are selected as selected characteristic points and a similar processing is repeated.

[0147] If it is determined that edge straight lines are obtained for all combinations of two adjacent points, i.e., edge straight lines are obtained respectively for a combination of characteristic points P₁and P₂, a combination of P₂and P₃, a combination of P₃and P₄, and a combination of P₄and P₁as in FIG. 12, intersection points of four edge straight lines obtained for the four combinations are computed in step S26. Specifically, an intersection point of the edge straight line obtained for the combination of the characteristic points P₄and P₁and the edge straight line obtained for the combination of the characteristic points P₁and P₂(hereinafter referred to as a first intersection point), an intersection point of the edge straight line obtained for the combination of the characteristic points P₁and P₂and the edge straight line obtained for the combination of the characteristic points P₂and P₃(hereinafter referred to as a second intersection point), an intersection point of the edge straight line obtained for the combination of the characteristic points P₂and P₃and the edge straight line obtained for the combination of the characteristic points P₃and P₄(hereinafter referred to as a third intersection point), and an intersection point of the edge straight line obtained for the combination of the characteristic points P₃and P₄and the edge straight line obtained for the combination of the characteristic points P₄and P₁(hereinafter referred to as a fourth intersection point) are computed. Further in step S26, the characteristic point closest to each intersection point is corrected and adjusted to the intersection point, i.e. the characteristic points P₁to P₄are corrected to the positions of first to fourth intersection points, and the process returns.

[0148] The above described automatic correction enables the characteristic points P₁to P₄designated by the user to be moved to the vertices of the quadrangle, freeing the user from designating accurate positions of characteristic points. Specifically, the user should only designate points in the vicinity of the positions that should be designated as characteristic points.

[0149] In the process shown in FIG. 7, the automatic correction of characteristic points is set to operate every time. However it is possible to set the automatic correction such that it operates only when the user gives an instruction. Also it is possible to allow the user to further correct characteristic points after being corrected by the automatic correction.

[0150] Next, the automatic correction can be operated using the edge pixel detection method illustrated in FIG. 13, for example, as described below.

[0151] Specifically, it is assumed that a two-dimensional image, such as that shown in FIG. 15, is stored in a buffer of interest. A shaded part in FIG. 15 indicates a three-dimensional object displayed on a two-dimensional image (hereinafter referred to as a projected image hereinafter, since it is the three-dimensional object in three-dimensional space projected onto a screen).

[0152] In FIG. 15, points P_Aand P_Bof the two-dimensional image are designated as characteristic points of the projected image of the three-dimensional object. The points P_Aand P_Bare adjacent to each other on a contour line of the projected image of the three-dimensional object.

[0153] For the two-dimensional image stored in the buffer of interest, a two-dimensional coordinate system having a horizontal x_paxis and a vertical y_paxis is defined as shown in FIG. 15. The form of the three-dimensional object displayed on the two-dimensional image is recognizable from shape information. Once the shape of the three-dimensional object is recognized, a contour line between the characteristic points P_Aand P_Bof the projected image (since a contour line of the projected image basically forms a closed loop, there are two contour lines between the characteristic points P_Aand P_B; here, a contour line having no other characteristic point (indicated by a bold line in FIG. 15) is selected from the two contour lines (indicated by bold and thin lines in FIG. 15)) can generally be defined as a function f of variables x_pand y_p, f=0, expressed with a plurality of parameters. (The function f has a different type (a package in the field of programming) according to the form of the three-dimensional object.)

[0154] Specifically, expressing the parameters by C₁, C₂, . . . , C_N(N is an integer not smaller than 1), a contour line between the characteristic points P_Aand P_Bis defined as:

[0155] f(x_p, y_p, C₁, C₂, . . . , C_N)=0  (1)

[0156] In the meantime, pixels forming the true contour line generally have a component of the pixel level which suddenly varies in the direction perpendicular to the contour line. Hence, when the parameters C₁, C₂, . . . , C_Nthat maximize the component in the direction of the normal vector (perpendicular to the contour line) in the pixel, out of a differential value of the pixel level of pixels on the screen coordinate system expressed by the expression (1), are computed, a line defined by the expression (1) using the parameters represents the true contour line.

[0157] Then, energy E_fof a line corresponding to the contour line between the characteristic points P_Aand P_Bdefined by the expression (1) (as described above, there are two lines between the characteristic points P_Aand P_Bwhich correspond to the contour line of the projected image; here, it is the line having no other characteristic point) is defined as: $\begin{array}{cc}{E}_f=\frac{\sum _{\{\left(x_p,y_p\right)|f\left(x_p,y_p,P_1,...,P_N\right)=0}\}\left|\nabla B\left(x_p,y_p\right)·n_f\left(x_p,y_p\right)\right|}{K}& \left(2\right)\end{array}$

[0158] In the expression (2), Σ indicates summation for all pixels on the line expressed by the expression (1). Also ∇ represents the nabla operator and B(x_p, y_P) represents a pixel level of a pixel at a position (x_P, y_P) on the screen coordinate system. A vector n_f(x_P, y_P) represents a normal vector at a pixel at a position (x_P, y_P) (a unit vector at a position (x_P, y_P) of a vector (here, e.g., an outward vector in a region enclosed by the contour line) perpendicular to the line expressed by the expression (1)). Also |x| represents a norm of a vector x and · represents an inner product. In addition, K represents the total of pixels on the line expressed by the expression (1).

[0159] ∇B(x_P, y_P) in the expression (2) represents a differential value of a pixel level of a pixel on the line expressed by the expression (1), namely, a gradient, which can be computed using the Sobel operator. Specifically, ∇B(x_P, y_P) can be computed using the following expression:

[0160] ∇B(x_p, y_p)=(Δ_xB(x_p, y_p), Δ_yB(x_p, y_p))

[0161] $\begin{array}{cc}\begin{array}{c}{\Delta }_{x}B\left(x_p,y_p\right)=B\left(x_p+1,y_p-1\right)+2B\left(x_p+1,y_p\right)+\\ B\left(x_p+1,y_p+1\right)-(B\left(x_p-1,y_p-1\right)+\\ 2B\left(x_p-1,y_p\right)+B\left(x_p-1,y_p+1\right))\end{array}\begin{array}{c}{\Delta }_{x}B\left(x_p,y_p\right)=B\left(x_p-1,y_p+1\right)+2B\left(x_p,y_p+1\right)+\\ B\left(x_p+1,y_p+1\right)-(B\left(x_p-1,y_p-1\right)+\\ 2B\left(x_p,y_p-1\right)+B\left(x_p+1,y_p-1\right))\end{array}& \left(3\right)\end{array}$

[0162] In addition, a parameter C_n(n=1, 2, . . . , N) maximizing energy E_fcan be obtained as follows. Expressing the initial value of the parameter C_nby C_n′ for example, the expression (2) is computed using the parameter C_nbeing varied within a range of from C_n′−Ε_nto C_n′+ε_nto obtain the parameter C_nwhen the value is a maximum. (ε_nn is a predetermined infinitesimal value.) Specifically, given that the contour line between the characteristic points P_Aand P_Bis a line segment L, for example, the expression (2) can be expressed as: $\begin{array}{cc}{E}_f=\frac{\sum _{\left(x_p,y_p\right)\in L}\left|\nabla B\left(x_p,y_p\right)·n_L\right|}{K}& \left(4\right)\end{array}$

[0163] Here, Σ represents, as described above, the line expressed by the expression (1), namely, summation of all pixels on the line segment L. Also a vector n₁corresponds to the vector n_f(x_P, y_P) in the expression (2), which is a normal vector of the line segment L. In the expression (4), K represents the total of pixels on the line segment L.

[0164] Meanwhile, if the contour line between the characteristic points P_Aand P_Bis a line segment, the expression (1) can be expressed using three parameters C₁, C₂, and C₃, for example, as:

[0165] f(x_P, y_P, C₁, C₂, C₃)=C₁x_P+C₂y_P+C₃  (5)

[0166] Given that the coordinates of the characteristic points P_Aand P_Bare (x₁, y₁) and (x₂, y₂), respectively, the parameters C₁, C₂, and C₃in the expression (5) can be expressed as:

[0167] C₁=y₁−y₂, C₂=−x₁+x₂, C₃=x₁y₂−x₂y₁  (6)

[0168] Therefore in this case, the values expressed by the expression (6) are used as the initial values C₁′, C₂′, and C₃′ of the parameters C₁, C₂, and C₃for computing energy E_fexpressed by the expression (4) while varying each value, and the parameters C₁, C₂, and C₃that maximize the energy E_fare obtained. The expression (5) using these parameters C₁, C₂, and C₃represents the true contour line between the characteristic points P_Aand P_B. Thus, moving the characteristic points P_Aand P_Bonto the line expressed by the expression (5) means that they are moved onto the true contour line.

[0169] Provided that the contour line between the characteristic points P_Aand P_Bis a line segment, varying the parameters C₁, C₂, and C₃corresponds to varying the coordinates of each characteristic point P_Aand P_B, being (x₁, y₁) and (x₂, y₂), respectively. Accordingly, the coordinates x₁, y₁, x₂, and y₂are the parameters of the function f.

[0170] When the contour line between the characteristic points P_Aand P_Bis a line segment, the initial values are provided for three parameters of the function f in the expression (5) by the characteristic points P_Aand P_B. Meanwhile, when the contour line between the characteristic points P_Aand P_Bis a curve, the initial values are not necessarily provided for all the parameters of the function f. In the latter case, a predetermined value may be given as an initial value, or the user may input a desired initial value.

[0171] In the above description, only two adjacent characteristic points are used so that the description is simpler to understand. However, a problem such as in the following may occur in actual processing. Given that three characteristic points, namely, a characteristic point #1, a characteristic point #2 adjacent to #1, and a characteristic point #3 adjacent to #2, are designated on a contour line, when the above automatic correction first using the characteristic points #1 and #2, and then using the characteristic points #2 and #3 are processed, the position of the characteristic point #2 after being corrected by the automatic correction using the characteristic points #1 and #2, and the position of the characteristic point #2 after being corrected by the automatic correction using the characteristic points #2 and #3 may not meet each other.

[0172] Therefore in the actual processing, parameters of the expression (1) defining the contour line between the characteristic points #1 and #2 (hereinafter referred to as first parameters) and parameters of the expression (1) defining the contour line between the characteristic points #2 and #3 (hereinafter referred to as second parameters) may preferably be altered so that such a deviation of corrected positions of the characteristic point #2 does not occur. In this case, the first and second parameters are not independent; their relationship is restricted by a condition that the corrected positions of the characteristic point #2 do not deviate from each other. Consequently, the relationship between the first parameters maximizing the energy E_fof the expression (2) using the characteristic points #1 and #2, and the second parameters maximizing the energy E_fof the expression (2) using the characteristic points #2 and #3 may not satisfy the condition that the corrected positions of the characteristic point #2 do not deviate from each other. In such a case, the first and second parameters that maximize the sum of the energy E_fof the expression (2) using the characteristic points #1 and #2, and the energy E_fof the expression (2) using the characteristic points #2 and #3 can be obtained.

[0173] To make the description simpler, an example is considered in which the deviation of one characteristic point #2 occurs. However, such a deviation may occur in all the characteristic points on the same contour line. Thus, it is preferred to compute parameters of the expression (1) defining a contour line between a combination of adjacent points for all the combinations using the above method. Specifically, it is preferred to compute the total energies E_fof combinations of adjacent characteristic points and obtain parameters that maximize the total.

[0174] Now referring to a flowchart in FIG. 16 and to FIG. 17, the above automatic correction processed in step S7 in FIG. 7 is described.

[0175] To make the description simpler, it is assumed that a projected image is a quadrangle and four vertices of the quadrangle are designated as characteristic points P₁to P₄.

[0176] Also in this case, energy E_fexpressed by the expression (4) is computed by varying three parameters C₁, C₂, and C₃in the expression (5), i.e. x-coordinates x₁and x₂and y-coordinates y₁and y₂of two characteristic points in the expression (6), and x₁, x₂, y₁, and y₂maximizing the energy E_fare obtained. Since such computation is quite complex, a simplified method is used here to reduce the computational complexity.

[0177] Specifically at first in step S31, one of four characteristic points P₁to P₄is selected as a characteristic point of interest. In step S32, energies E_fof two sides of the quadrangle crossing the characteristic point of interest (connecting the characteristic point of interest and other characteristic points) are computed. For example, given that the points P₁to P₄are designated as characteristic points of a shaded projected image, as in FIG. 17, and the characteristic point P₄is selected as a characteristic point of interest, energies E_fof sides P₄P₁and P₃P₄, which cross the characteristic point of interest P₄, among four sides of a quadrangle P₁P₂P₃P₄, are obtained in step S32.

[0178] In step S33, one of the two sides, for which the energies E_fare computed in step S32, having a greater energy E_fis selected as a line closer to a contour line of the projected image. The characteristic point of interest is shifted along a straight line extending that selected side (occasionally referred also as an edge straight line). Then energies E_fof two sides formed by the shifted characteristic point of interest and two characteristic points adjacent to the shifted characteristic point of interest are computed. Specifically, given that the characteristic point P₄is selected as the characteristic point of interest, as in FIG. 17, when energy E_fof the side P₃P₄is greater than energy E_fof the side P₄P₁, the characteristic point of interest P₄is shifted along the straight line P₃P₄as represented by an arrow D in the drawing. Then energy E_fof the side P₃P₄formed by the shifted characteristic point of interest P₄and one characteristic point P₃adjacent to P₄, and energy E_fof the side P₄P₁formed by the shifted characteristic point of interest P₄and the other characteristic point P₁adjacent to P₄are computed.

[0179] After that, it is determined in step S34 whether the sum of the energies of two sides computed in step S33 is a maximum. If the result is negative, the process returns to step S33 and the characteristic point of interest is shifted again along the edge straight line, and a similar processing is repeated.

[0180] On the other hand, when it is determined that the sum of the energies of the two sides computed in step S33 is a maximum, or as described above, when, by shifting the characteristic point P₄along the straight line P₃P₄, a normal vector n₁₄of the side P₄P₁approximately meets at right angles with a side connecting two vertices that should originally be designated by the characteristic points P₁and P₄, the process proceeds to step S35. In step S35 it is determined whether all characteristic points P₁to P₄have been used as characteristic points of interest. If the result is negative, the process returns to step S31 and the characteristic point that has not yet been used as a characteristic point of interest is selected as a characteristic point of interest, and step S32 and those that follow are repeated.

[0181] If it is determined in step S35 that all characteristic points P₁to P₄have been used as characteristic points of interest, the process proceeds to step S36. In step S36, it is determined whether movements of the characteristic points P₁to P₄in the processing of the previous step S33 exceed a predetermined threshold level ε. If the result is affirmative, the process returns to step S31 and repeats similar processing. Specifically, each of the characteristic points P₁to P₄is sequentially selected as the characteristic point of interest, of which position is shifted so that the sum of energies of the two sides formed by each characteristic point is a maximum.

[0182] In the meantime, when it is determined in step S36 that the movements of the characteristic points P₁to P₄in step S33 do not exceed the predetermined threshold level ε, or as described above, when there is almost no movement of the characteristic points P₁to P₄in the processing of the previous step S33, the process returns.

[0183] Next in the copy processor 12 or paste processor 13 shown in FIG. 4, the corrected characteristic points and shape information are used to compute the inverse transform expression or transform expression, as described below.

[0184] For example, when the camera 9 (FIG. 3) captures the image of a three-dimensional object in three-dimensional space shown in FIG. 18A, a two-dimensional image outputted by the camera 9 displays a projected image (projected picture), as shown in FIG. 18B, which is the three-dimensional object being projected onto a screen. Therefore, a two-dimensional image on which such a projected image is displayed is stored in an image buffer (except for a paste buffer).

[0185] Meanwhile, an expanded image, which is the surface of the three-dimensional object in the three-dimensional space (FIG. 18A) expanded onto a two-dimensional plane, is stored in the paste buffer, as shown in FIG. 18C. Specifically, if the surface of the three-dimensional object is, for example, a plane, a two-dimensional image seeing the plane from the front is stored in the paste buffer. If a plane of the three-dimensional object is a side surface of a cylinder, a two-dimensional image which is the side surface expanded onto a two-dimensional plane as a rectangular shape is stored in the paste buffer. Also if a plane of the three-dimensional object is a sphere, a two-dimensional image which is the sphere expanded onto a two-dimensional plane, using a method such as equidistant cylindrical projection used in cartography or the like, is stored in the paste buffer. If a plane of the three-dimensional object is a three-dimensional curved surface, such as Bezier curved surface expressed by two parameters s and t, a two-dimensional image which is the curved surface expanded onto a two-dimensional plane, using the parameter s and t as a horizontal axis and vertical axis, is stored in the paste buffer. In addition, for example, if the three-dimensional object is a body of revolution, a two-dimensional image which is the body of revolution expanded onto a two-dimensional plane, using the circumference direction or rotation axis as a horizontal axis or vertical axis, is stored in the paste buffer.

[0186] In the copying, as shown in FIG. 18B, by designating a few characteristic points on the projected image of (the plane of) the three-dimensional object displayed on the two-dimensional image stored in the buffer of interest (indicated by ▪ in FIG. 18B), and shape information of (the plane of) the three-dimensional object, the inverse transform expression is computed. Using the inverse transform expression, the expanded image shown in FIG. 18C is generated from the projected image shown in FIG. 18B, which is stored in the paste buffer. In the pasting, as shown in FIG. 18B, by designating a few characteristic points on a region on which the projected image of (the plane of) the three-dimensional object is pasted (indicated by ▪ in FIG. 18B), and shape information (of the plane) of the three-dimensional object, the transform expression is computed. Using the transform expression, the projected image shown in FIG. 18B is generated from the image (expanded image) shown in FIG. 18C, which is pasted on the image buffer.

[0187] Therefore, inter-relationship among the three-dimensional object in three-dimensional space, its projected image, and its expanded image becomes important in computing the inverse transform expression and transform expression.

[0188] Now referring to FIG. 19, a two-dimensional coordinate system for the expanded image, a three-dimensional coordinate system for the three-dimensional object, and a two-dimensional coordinate system for the projected image are considered. The x and y coordinates in the two-dimensional coordinate system for the expanded image are expressed by x_eand y_e; the x, y, and z coordinates in the three-dimensional coordinate system for the three-dimensional object are expressed by x₀, y₀, and z₀; and the x and y coordinates in the two-dimensional coordinate system for the projected image are expressed by x_pand y_p. Also the pixel level of the projected image at the point (x_p, y_p) is expressed by B(x_p, y_p), and the pixel level of the expanded image at the point (x_e, y_e) is expressed by C(x_e, y_e). Finally the mapping, wherein the expanded image is pasted onto the surface of the three-dimensional object, is expressed by ψ, and the mapping, wherein the three-dimensional object is transformed into the projected image, is expressed by M.

[0189] Here, the two-dimensional coordinate system for the expanded image, the three-dimensional coordinate system for the three-dimensional object, and the two-dimensional coordinate system for the projected image can be selected in an arbitrary manner. Specifically, the three-dimensional coordinate system for the three-dimensional object, for example, is selected suitably for expressing the three-dimensional object. Taking a concrete example, when the three-dimensional object is a cylinder, let the origin (0, 0) be the center of a circle of either of two bottom surfaces, and a rotation axis (a straight line connecting the centers of circles of two bottom surfaces) be coincident with the y₀axis. When the three-dimensional object is a rectangle, for example, let the origin (0, 0, 0) be the lower left vertex of four vertices of the rectangle, and a horizontal axis or vertical axis be coincident with either the x₀axis or y₀axis.

[0190] The point (x₀, y₀, z₀) on the three-dimensional object has a one-to-one correspondence to the point (x_e, y_e) on the expanded image. The relationship between them, namely, the mapping ψ (or ψ⁻¹), is obtainable by geometric computation. Accordingly, when the relationship between the point (x₀, y₀, z₀) on the three-dimensional object and the point (x_e, y_e) on the expanded image, namely, the mapping M (or M¹), is given, the mapping M which transforms the point (x_e, y_e) on the expanded image into the point (x_p, y_p) can be computed.

[0191] Nevertheless, when only the three-dimensional object and the projected image are considered, the mapping M cannot be computed unless it is known which point (x₀, y₀, z₀) on the three-dimensional object is projected onto which point (x_e, y_e) on the projected image. Computing from the projected image on which point (x_e, y_e) the point (x₀, y₀, z₀) on the three-dimensional object is projected basically needs preparation of a plurality of two-dimensional images for using a triangulation principle, as described above. Also expressing a three-dimensional model of the three-dimensional object requires a memory for storing three-dimensional data as temporary data, which sees the point (x₀, y₀, z₀) on the three-dimensional object from a viewpoint (the origin of camera's coordinates as described below), and computation for working with such three-dimensional data.

[0192] Therefore, the mapping ψ is obtained here based on the shape information inputted by the user, and the mapping M is obtained based on the characteristic points inputted by the user. Also the relationship between the expanded image and the projected image, namely, the mapping Mψ which transforms the expanded image into the projected image, is obtained. The mapping Mψ is a transform expression, and the mapping which performs the inverse of the transformation corresponding to the mapping Mψ, i.e., (Mψ)⁻¹, is an inverse transform expression. Hence, the inverse transform expression is obtainable by computing the inverse matrix of the mapping Mψ, using, for example, the Gaussian elimination method.

[0193] Since the inverse transform expression (Mψ)⁻¹is easily obtained once the transform expression Mψ is computed, the computation of the transform expression is described hereunder.

[0194] As described above, the point (x₀, y₀, z₀) on the three-dimensional object has a one-to-one correspondence to the point (x_e, y_e) on the expanded image. Accordingly, the point (x₀, y₀, z₀) is expressed in an expression (7) using the mapping ψ and the point (x_e, y_e) as: $\begin{array}{cc}\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)=\psi \left(x_e,y_e\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)& \left(7\right)\end{array}$

[0195] When the camera 9 captures the image of the three-dimensional object in the three-dimensional space to output the projected image, three-dimensional coordinates for expressing the position of the camera 9 (x_c, y_c, z_c) (hereinafter referred to as camera's coordinates) may be given. The camera's coordinates (x_c, y_c, z_c) of the point (x₀, y₀, z₀) are expressed in the following expression using nine elements r₁₁, r₁₂, r₁₃, r₂₁, r₂₂, r₂₃, r₃₁, r₃₂, and r₃₃for representing a rotation in the three-dimensional space, and three elements t₁, t₂, and t₃for representing a translation as: $\begin{array}{cc}\left(\begin{array}{c}{x}_c\\ y_c\\ z_c\end{array}\right)=\left(\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{21}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right)\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right)& \left(8\right)\end{array}$

[0196] The point (x_p, y_p) on the projected image is the point (x_c, y_c, z_c) on the three-dimensional object in the camera's coordinates projected onto the screen. The transformation is perspective transformation (projection transformation) as described in Document 1. Therefore it can be expressed by a rational formula, wherein both denominator and numerator are a linear expression of one of the x-coordinate, y-coordinate, and z-coordinate (a linear term and constant term of one of x, y, and z). Specifically, letting w be an arbitrary number expressing a homogeneous coordinate, the point (x_p, y_p) of the projected image on the screen is expressed in an expression (9) using the homogeneous coordinate as: $\begin{array}{cc}\left(\begin{array}{c}{\mathrm{wx}}_p\\ {\mathrm{wy}}_p\\ w\end{array}\right)=\left(\begin{array}{ccc}{f}_h& 0& 0\\ 0& f_v& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{x}_c\\ y_c\\ z_c\end{array}\right)& \left(9\right)\end{array}$

[0197] Let f be a focal distance of the camera 9, and h and v be a horizontal length and vertical length, respectively, of one pixel of the screen in the camera's coordinate system. Then, f_hand f_vin the expression (9) represent f×h and f×v, respectively.

[0198] It is clear from the expression (9) that the value in which the x-coordinate wx_por y-coordinate wy_pof the homogeneous coordinate divided by the z-coordinate w is the x-coordinate x_por y-coordinate y_pof the projected image.

[0199] From the expressions (7) to (9), an expression (10) is derived as: $\begin{array}{cc}\left(\begin{array}{cccc}{w}_1{x}_{\mathrm{p1}}& w_2{x}_{\mathrm{p2}}& w_3{x}_{\mathrm{p3}}& w_4{x}_{\mathrm{p4}}\\ w_1{y}_{\mathrm{p1}}& w_2{y}_{\mathrm{p2}}& w_3{y}_{\mathrm{p3}}& w_4{y}_{\mathrm{p4}}\\ w_1& w_2& w_3& w_4\end{array}\right)=M\left(\begin{array}{cccc}{\psi }_1\left(0,0\right)& {\psi }_1\left(L_W,0\right)& {\psi }_1\left(L_W,L_H\right)& {\psi }_1\left(0,L_H\right)\\ {\psi }_2\left(0,0\right)& {\psi }_2\left(L_W,0\right)& {\psi }_2\left(L_W,L_H\right)& {\psi }_2\left(0,L_H\right)\\ 1& 1& 1& 1\end{array}\right)=M\left(\begin{array}{cccc}0& L_W& L_W& 0\\ 0& 0& L_H& L_H\\ 1& 1& 1& 1\end{array}\right)& \left(10\right)\end{array}$

[0200] In the expression (10), as is evident from FIG. 19, M is a matrix which transforms ψ(x_e, y_e), i.e., (x₀, y₀, z₀) into (x_p, y_p), which can be expressed in the following expression as: $M=\left(\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& m_{14}\\ m_{21}& m_{22}& m_{23}& m_{24}\\ m_{31}& m_{32}& m_{33}& 1\end{array}\right)=\left(\begin{array}{cccc}{r}_{11}{f}_h/t_3& r_{12}{f}_h/t_3& r_{13}{f}_h/t_3& t_1{f}_h/t_3\\ r_{21}{f}_v/t_3& r_{22}{f}_v/t_3& r_{23}{f}_v/t_3& t_2{f}_v/t_3\\ r_{31}/t_3& r_{32}/t_3& r_{33}/t_3& 1\end{array}\right)$

[0201] ψ(ψ₁, ψ₂, ψ₃) in the expression (10) is obtainable by geometric computation based on the shape information inputted by the user. The matrix M in the expression (10) is also computable based on the characteristic points inputted by the user. The matrix M generally has m₁₁, m₁₂, m₁₃, m₁₄, m₂₁, m₂₂, m₂₃, m₂₄, m₃₁, m₃₂, and m₃₃as unknowns, as shown in the expression (11). Accordingly, when a total of six points on the projected image, i.e., five points with 2-degrees of freedom and a point with 1-degree of freedom, are designated as characteristic points, m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to m₃₃are computable. When more than six points are designated, eleven elements of m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to m₃₃can be computed using, for example, the least squares method.

[0202] As described above, the matrix M is computed here based on the characteristic points inputted by the user. Therefore it is not necessary to work with the three-dimensional data of the three-dimensional object in the camera's coordinates, nor computation using the three-dimensional data. It is also unnecessary to use a plurality of two-dimensional images of the same three-dimensional object shot from a plurality of positions. The matrix M is computable using a two-dimensional image of the three-dimensional object shot from one position. As a result, the mapping Mψ (and (Mψ)⁻¹) can be obtained without computation using the three-dimensional data in the camera's coordinates, and without using a plurality of two-dimensional images.

[0203] Next, computation of ψ and M for three-dimensional objects having primitive shapes is described below.

[0204] If the three-dimensional object is, for example, a rectangular parallelepiped, ψ in the expression (7), that is, the relationship between a point (x₀, y₀, z₀) on a plane (rectangle) of the rectangular parallelepiped in three-dimensional space and a point (x_e, y_e) of an expanded image can be given, for example, as follows. $\begin{array}{cc}\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)=\left(\begin{array}{c}{x}_e\\ y_e\\ 0\end{array}\right)& \left(12\right)\end{array}$

[0205] where, as the two dimensional coordinates of a projected image (hereinafter referred to as “screen coordinates”), the coordinates of a plane with z₀=0 in the three-dimensional coordinate system on the three-dimensional object (hereinafter referred to as “three-dimensional coordinates”), where each x coordinate of the screen coordinates is equal to each x coordinate of the three-dimensional coordinates, and each y coordinate of the screen coordinates is equal to each y coordinate of the three-dimensional coordinates (x_e=x₀, y_e=y₀), that is, the coordinates of the x-y plane in the three-dimensional coordinates are used. Further, as a three-dimensional coordinate system, the coordinates, where one of the vertices of a plane of the rectangular parallelepiped corresponds to the origin and the plane is included in the x-y plane, are used.

[0206] Since z₀=0 by the expression (12), the need for the elements m₁₃, m₂₃, m₃₃of the third row of the matrix M of the expression (11) is eliminated. In this case, the expression (10) can be given as follows: $\begin{array}{cc}\left(\begin{array}{c}{\mathrm{wx}}_p\\ {\mathrm{wy}}_p\\ w\end{array}\right)=M\left(\begin{array}{c}{x}_e\\ y_e\\ 0\\ 1\end{array}\right)=\left(\begin{array}{ccc}{m}_{11}& m_{12}& m_{14}\\ m_{21}& m_{22}& m_{24}\\ m_{31}& m_{32}& 1\end{array}\right)\left(\begin{array}{c}{x}_e\\ y_e\\ 1\end{array}\right)& \left(13\right)\end{array}$

[0207] Therefore, since the number of the elements of the matrix M to be computed is the eight elements m₁₁, m₁₂, m₁₄, m₂₁, m₂₂, m₂₄, m₃₁, and m₃₂, these eight elements can be found by designating, as characteristic points, four points on the projected image corresponding to four points on the expanded image (given the information of eight coordinates in total, that is, t_pand y_pof four characteristic points).

[0208] When the three-dimensional object is a rectangular parallelepiped, the projected image of the plane (rectangle) is a quadrangle such as shown in FIG. 20A. Here, suppose that the four vertices of the projected image of the quadrangle are designated as characteristic points by a user. Let the vertices of the projected image of the quadrangle be vertices P₁, P₂, P₃, and P₄clockwise from the top-left vertex, such as shown in FIG. 20A, while letting the vertices of a rectangular expanded image stored in a paste buffer be vertices E₁, E₂, E₃and E₄clockwise from the top-left vertex, such as shown in FIG. 20B so that P₁, P₂, P₃, and P₄correspond to E₁, E₂, E₃and E₄, respectively.

[0209] Moreover, let the coordinates of points E₁, E₂, E₃and E₄of the two dimensional coordinates of the expanded image (hereinafter referred to as “paste-buffer coordinates”) be (0, 0), (L_W, 0), (L_W, L_H), and (0, L_H), respectively, where L_Windicates the width of the rectangular expanded image and L_Hindicates the height of the rectangular expanded image. The user can input any desired values to those lengths as far as the memory sizes of the paste buffer can hold. Other than the user setting the desired values of L_Wand L_H, it is also possible that the size of the expanded image is set to a predetermined proportion, for example, 80 percent, of the size (area) of the paste buffer.

[0210] By letting the screen coordinates of the points P₁, P₂, P₃, and P₄, as characteristic points designated by the user, be (X_p1, Y_p1), (X_p2, Y_p2), (X_p3, Y_p3), and (X_p4, Y_p4), respectively, the next expression (14) is obtained from the expression (10). $\begin{array}{cc}\left(\begin{array}{cccc}{w}_1{x}_{\mathrm{p1}}& w_2{x}_{\mathrm{p2}}& w_3{x}_{\mathrm{p3}}& w_4{x}_{\mathrm{p4}}\\ w_1{y}_{\mathrm{p1}}& w_2{y}_{\mathrm{p2}}& w_3{y}_{\mathrm{p3}}& w_4{y}_{\mathrm{p4}}\\ w_1& w_2& w_3& w_4\end{array}\right)=M\left(\begin{array}{cccc}0& L_w& L_w& 0\\ 0& 0& L_H& L_H\\ 1& 1& 1& 1\end{array}\right)& \left(14\right)\end{array}$

[0211] where w₁, w₂, w₃, and w₄of the expression (14) are arbitrary numbers indicating homogeneous coordinates.

[0212] The matrix M of the expression (14) leads the eight elements m₁₁, m₁₂, m₁₄, m₂₁, m₂₂, m₂₄, m₃₁, and m₃₂of the expression (13) to be unknowns. Since eight equations can be derived based on the expression (14), the eight unknowns m₁₁, m₂₁, m₁₄, m₂₁, m₂₂, m₂₄, m₃₁, and m₃₂, that is, the matrix M, can be solved. Incidentally the characteristic points designated by the user and automatically corrected as described above are basically used as the characteristic points for solving the matrix M.

[0213] When the three-dimensional object is a rectangular parallelepiped and then either a copying or a pasting is performed on the plane (rectangle), ψ is computed by the expression (12), along with M by the expression (14). The mapping Mψ is directly used as a right transform expression in case of the pasting. For the copying, the mapping obtained by computing the inverse matrix of the mapping Mψ is used as an inverse transform expression. Irrespective of the case where the three-dimensional object is a rectangular parallelepiped, the expression used in the pasting is called the “right transform expression” because the mapping Mψ is directly used, while the expression used in the copying is called the “inverse transform expression” because the inverse mapping of the mapping Mψ is used.

[0214] When the copying is performed, in the image transform processor 20, the pixels within the region of the quadrangle P₁P₂P₃P₄as the projected image in the buffer of interest, as shown in FIG. 20A, are written into, in accordance with the inverse transform expression (Mψ)⁻¹, the corresponding positions of the quadrangle in the rectangle E₁E₂E₃E₄of the paste buffer, as shown in FIG. 20B.

[0215] If one pixel in the buffer of interest corresponds to a plurality of pixels in the paste buffer, the level of the one pixel may be written into all the pixels in the paste buffer. Interpolated values obtained by performing interpolation on the pixels in the buffer of interest may be written into the paste buffer. For example, if a pixel A of the buffer of interest corresponds to two adjacent pixels a₁and a₂of the paste buffer, the level of the pixel A may be written into the pixel a₁, and the mean value of the levels of the pixel A and its adjacent pixel in the buffer of interest may be written into the pixel a₂.

[0216] On the contrary, if a plurality of pixels of the buffer of interest correspond to a single pixel of the paste buffer, for example, the level of any one of the pixels may be written into the paste buffer, or weighted addition value obtained by filtering the pixels may be written into the paste buffer.

[0217] The writing of the pixel levels into the paste buffer may be performed by computing the location of a pixel of the paste buffer transformed from a pixel of the projected image in the buffer of interest in accordance with the inverse transform (Mψ)⁻¹as described above, or computing the location of a pixel (of the projected image) transforming into a pixel of the expanded image.

[0218] When the pasting is performed, in the image transform processor 20, the pixels within the region of the rectangle E₁E₂E₃E₄as the expanded image (shown in FIG. 20B) in the paste buffer are written into, in accordance with the right transform expression Mψ, the corresponding positions in the quadrangle P₁P₂P₃P₄of the projected image (shown in FIG. 20A, in the buffer of interest, whereby the projected image is pasted onto the two dimensional image stored in the buffer of interest.

[0219] If one pixel in the paste buffer corresponds to a plurality of pixels in the buffer of interest, the level of one pixel may be written into all the pixels in the buffer of interest, similarly to the copying as described above. Also, interpolated data obtained by performing interpolation on a plurality of pixels in the paste buffer may be written into the plurality of pixels in the buffer of interest.

[0220] On the contrary, if a plurality of pixels in the paste buffer corresponds to a single pixel in the buffer of interest, for example, the level of any one of the pixels may be written into the buffer of interest, or weighted addition value obtained by filtering the plurality of pixels may be written into the buffer of interest.

[0221] The writing of pixel levels into the buffer of interest may be performed by computing the location of a pixel in the buffer of interest transformed from a pixel of the projected image in the paste buffer in accordance with the right transform Mψ as described above, or computing the location of a pixel (of the expanded image) transforming into each pixel of the projected image.

[0222] Next, if the three-dimensional object is, for example, a cylinder, when the copying for creating and storing an expanded image of its side into the paste buffer, or the pasting for pasting the image stored in the paste buffer onto its side is performed, the inverse transform expression or the right transform expression is respectively computed as follows.

[0223] A three-dimensional coordinate system for representing the location of the cylinder in a three-dimensional space is assumed as shown in FIG. 21A. In the three-dimensional coordinate system, the center of either base (circle), for example, the upper base is used as the origin (0, 0, 0), and the direction from the upper base to the lower base and the direction from O₅(described below) to the center of the upper base are respectively used as the y₀axis and the z₀axis. A direction in which a right hand screw advances when being turned from the y₀axis to the z₀axis is used as the x₀axis.

[0224] It is assumed that a projected image, obtained by capturing the image of the cylinder in FIG. 21A from an upper front position with respect to a point O₅, is displayed in the two dimensional image in the buffer of interest. The projected image of the cylinder is displayed as, for example, the shaded part in FIG. 21B.

[0225] In FIG. 21B, points P₁and P₂or points P₃and P₄indicate the endpoints of the intersections (which are shown as the bold lines in FIG. 21B) where the side of the cylinder on the projected image intersects its upper base or its lower base. Specifically, the point P₁indicates an intersecting point where the left of a contour line of the side of the cylinder in the projected image intersects its upper base. The point P₂indicates an intersecting point where the right of the contour line of the projected image intersects the upper base. The point P₃indicates an intersecting point where the right of the contour line of the projected image intersects the lower base. The point P₄indicates an intersecting point where the left of the contour line of the projected image intersects the lower base.

[0226] Points O₁, O₂, O₃, and O₄in FIG. 21A correspond to points P₁, P₂, P₃, and P₄in FIG. 21B, respectively.

[0227] The point O₅in FIG. 21A bisects an arc O₁O₂(viewed as a projected image) where the point O₅corresponds to the point P₅in FIG. 21B. Also, a point O₆in FIG. 21A bisects an arc O₃O₄(viewed as a projected image). The point O₆corresponds to the point P₆in FIG. 21B.

[0228] As described above, since the z axis is directed from the point O₅to the center of the upper base, and it also goes through the origin at the center of the upper base, the point O₅is included in the y₀-z₀plane with x₀=0. Therefore the point O₆is included in the y₀-z₀plane as well.

[0229] In FIG. 21B, a point P₇is a vanishing point on the cylinder, which is a point where lines P₁P₄and P₂P₃meet. Thus, the point P₆is on a straight line between points P₅and P₇. Accordingly, the point P₆has one degree of freedom of the point P₆in the direction of a line P₅P₇.

[0230] An angle θ (radian) in FIG. 21A is formed between the point O₁as the center (the origin of the three-dimensional coordinates) of the upper base and the point O₂(thus, the angle θ is also formed between the point O₃as the center of the lower base and the point O₄). Here, let “r” be the radius of the base of the cylinder. If the expanded image corresponding to the projected image of the side of the cylinder is stored in the paste buffer, the relationship between the radius r and the angle θ is given as the following expression (assuming that the circumferential direction of the base of the cylinder corresponds to the horizontal direction of the expanded image for example).

[0231] r=L_w/θ  (15)

[0232] where L_windicates the width (horizontal length) of the expanded image as described above.

[0233] Let the angle φ be as follows

[0234] φ=θ(x_e−L_w/2)/L_w  (16)

[0235] where x_eindicates an x coordinate (horizontal coordinate) of the paste buffer coordinate system. It is hereinafter assumed that the top-left vertex of a rectangular expanded image is used as the origin of the coordinates, and the direction from the top to the base, and the direction from the left to the right are respectively used as the x_eaxis and the y_eaxis.

[0236] If the angle φ is expressed as the expression (16) regarding the cylinder, ψ of the expression (7), that is, the relationship between the point (x₀, y₀, z₀) of its side of three-dimensional space and the point (x_e, y_e) of the expanded image is expressed by the following expression, based on consideration of the expanded image of the cylinder in FIG. 21A. $\begin{array}{cc}\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)=\left(\begin{array}{c}r\sin \phi \\ y_e\\ -r\cos \phi \end{array}\right)& \left(17\right)\end{array}$

[0237] Unlike the case described with reference to FIGS. 20A and 20B, there is no unnecessary element of the matrix M. Thus, eleven elements m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to M₃₃in the expression (11) must be found. The eleven elements can be computed as described above by designating, as characteristic points, a total of six points (of the projected image) consisting for five predetermined points and a point having one degree of freedom.

[0238] Accordingly, it is assumed that the user designates as characteristic points six points P₁to P₆shown in FIG. 21B. Then, let the vertices of a rectangular expanded image stored in a paste-buffer be vertices E₁, E₂, E₃, and E₄, clockwise from the top-left vertex, as shown in FIG. 21C, and let the midpoints of lines E₁E₂and E₃E₄be points E₅and E₆so that the points P₁, P₂, P₃, P₄, P₅, and P₆correspond to the points E₁, E₂, E₃, E₄, E₅, and E₆, respectively.

[0239] In addition, let the coordinates of the points E₁, E₂, E₃, E₄, E₅, and E₆(in the paste buffer coordinate system) be (0, 0), (L_W, 0), (L_W, L_H), (0, L_H), (L_W/2, 0) and (L_W/2, L_H), respectively.

[0240] By establishing and solving equations for the eleven elements m₁₁to m₁₄, m₂₁to m₂₄and m₃₁to m₃₃in accordance with the expression (10) using the relationships described above, these eleven elements, that is, the matrix M can be found.

[0241] In this case, the characteristic points P₁through P₆for computing the matrix M are automatically corrected. Either characteristic point P₅or P₆is further automatically corrected so as to be positioned on a straight line between the other auto-corrected characteristic point and the vanishing point P₇. For example, among the characteristic points P₅and P₆, one point P₆is automatically corrected, and is further automatically corrected so as to be positioned on the same straight line between the other automatically corrected characteristic point P₅and the vanishing point P₇. This is because the characteristic point P₆is one degree of freedom, as described above.

[0242] If the height of the cylinder in FIG. 21A is equal to the height L_Hof the expanded image in FIG. 21C, the coordinates of the points O₁, O₂, O₃, O₄, O₅and O₆are (−r sin(θ/2), 0, −r cos(θ/2)), (r sin(θ/2),0,−r cos(θ/2)), (r sin(θ/2),L_H, −r cos(θ/2)), (−r sin(θ/2), L_H, −r cos(θ/2)), (0, 0, −r) and (0, L_H, −r), respectively.

[0243] The matrix M found regarding the cylinder, as described above, is a function of the angle θ in FIG. 21A. Thus, the angle θ must be found. The two dimensional image displays the projected image of the cylinder in which the straight line connecting the characteristic point P₅and the vanishing point P₇in parallel to the axis y_pin FIG. 21B, but in general, the projected image of the cylinder is likely to be formed as shown in FIG. 22 so that the straight line connecting the characteristic point P₅and the vanishing point P₇is inclined with the axis y_p.

[0244] Accordingly, when an angle α is formed between the straight line connecting the characteristic point P₅and the vanishing point P₇and the axis x_p, a method for finding the angle is described below.

[0245] From consideration of the x_p′ axis and the y_p′ axis obtained by counterclockwise rotating the x_paxis and the y_paxis of the screen coordinate system by the angle α, the relationship between these axes and the projected image of the cylinder corresponds to that shown in FIG. 21B. Hence, by assuming a coordinate system defined by the x_p′ axis and the y_p′ axis obtained by counterclockwise rotating the x_paxis and the yp axis by the angle α, the angle θ is found.

[0246] In this case, the relationship between the point (x_p, y_p) and the point (x_p′, y_p′) can be expressed by the following expression, using homogeneous coordinates. $\begin{array}{cc}\left(\begin{array}{c}{\mathrm{wx}}_p^{\prime }\\ {\mathrm{wy}}_p^{\prime }\\ w\end{array}\right)=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{\mathrm{wx}}_p\\ {\mathrm{wy}}_p\\ w\end{array}\right)& \left(18\right)\end{array}$

[0247] The matrix M′ be defined by, for example, the following expression. $\begin{array}{cc}{M}^{\prime }=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)M=\left(\begin{array}{cccc}{m}_{11}^{\prime }& m_{12}^{\prime }& m_{13}^{\prime }& m_{14}^{\prime }\\ m_{21}^{\prime }& m_{22}^{\prime }& m_{23}^{\prime }& m_{24}^{\prime }\\ m_{31}^{\prime }& m_{32}^{\prime }& m_{33}^{\prime }& 1\end{array}\right)& \left(19\right)\end{array}$

[0248] where the matrix M of the expression (19) is given in the expression (11).

[0249] Based on the expression (10), and the expressions (15) to (19), the next expressions (20) are derived.

[0250] x_p′=(m₁₁′rsin φ+m₁₂′y_e−m₁₃′rcos φ+m₁₄′)/(m₃₁′rsin φ+m₃₂′y_e−m₃₃′rcos φ+1)

[0251] _p′=(m₂₁′rsin φ+m₂₂′y_e−m₂₃′rcos φ+m₂₄′)/(m₃₁′rsin φ+m₃₂′y_e−m₃₃′rcos φ+1)  (20)

[0252] In other words, regarding the cylinder shown in FIG. 21A as the three-dimensional object, let a point O be on an arc O₁O₅O₂displayed as the projected image, then the point O is projected onto a point P on an arc P₁P₅P₂on the projected image in FIG. 21B. If the point P corresponds to the point E in the paste buffer coordinate system as shown in FIG. 21C, the x coordinate corresponding to the point E can be expressed as x_e−L_w/2 with respect to E₅as a reference.

[0253] The points E₅and E in the paste buffer coordinate system in FIG. 21C correspond to the points O₅and O of the cylinder in the three-dimensional space in FIG. 21A. Thus, the x_ecoordinate x_e−L_w/2 (with respect to the position as a reference for the point E) corresponds to the position of the point O on the circumference of the upper base of the cylinder in FIG. 21A.

[0254] That is, φ defined in the expression (16) above indicates the angle formed by the point O₅, the origin and the point O, and x_p′ and y_p′ of the expression (20) indicate the coordinates of the point P corresponding to the point O in the projected image.

[0255] Moving the point O with respect to O₅as a reference along the circumference of the upper base of the cylinder, that is, varying the value of the angle φ from 0 radian causes the point P corresponding to the position O in the projected image to move along an arc P₅P₂or an arc P₅P₁with respect to the point P₅as a reference. An angle formed by the point P, the vanishing point P₇, and the point P₅is represented by ν as shown in FIG. 23.

[0256] The coordinates of the point P are given by (x_p′, y_p′), found by the expression (20). Therefore, the angle ν can be found by the following expression when the coordinates of the point P₇are represented as (x_p7′, y_p7′) as shown in FIG. 23.

[0257] ν=tan⁻¹(|y_p′−y_p7′|/|x_p′−x_p7′|)  (21)

[0258] The angle ν is a function of the angle φ because as the point P moves, that is, the angle φ varies, the angle ν varies.

[0259] In FIG. 23, assuming that the direction of right-handed rotation with respect to the line P₅P₇as a reference line is the positive direction of the angle ν, the angle ν is maximum when the point P corresponds to the point P₂, while the angle ν is minimum when the point P corresponds to the point P₁.

[0260] The point P corresponds to either the point P₁or P₂when the expression |φ|=θ/2 holds (the point P corresponds to the point P₁when φ=−θ/2, while the point P corresponds to the point P₂when φ=θ/2). Thus, the angle ν takes its extremes when the expression |φ|=θ/2 holds.

[0261] Accordingly, let a value obtained by partially differentiating the angle ν of the expression (21) using φ, be 0, as follows: $\begin{array}{cc}\frac{\partial v}{\partial \phi }=0& \left(22\right)\end{array}$

[0262] The expression (22) holds when |φ|=θ/2 where θ is the angle to be found (=2φ) (an angle formed by the point O₁, the center of the upper base of the cylinder, and the point O₂where the points P₁and P₂correspond to the points O₁and O₂in FIG. 21).

[0263] Since the left-hand side of the expression (22), obtained by partially differentiating ν of the expression (22) using φ, is a function which has two variables φand θ, computation of φ and θ satisfying the expression (22) becomes complicated. Therefore, the angle θ satisfying the expression (22) is found by using approximate calculation.

[0264] FIG. 24 shows a flowchart illustrating a process for finding the value of θ satisfying the expression (22) by performing approximate calculation (hereinafter referred to as the “θ computation process”). The θ computation process is performed, for example, by the copy processor 12 in the copying, or by the paste processor 13 in the pasting.

[0265] In the θ computation process, the θ is set to π (radian) in step S41. In step S42, the matrix M′ satisfying the expression (22) is found based on the angle θ and the characteristic points. In step S43, ν of the expression (21) is found based on the matrix M′ obtained in step S42. In step S44, φ obtained when ν is an extreme value, that is, φ satisfying the expression (22) is found. The process then proceeds to step S45.

[0266] Two values of φ satisfying the expression (21) are found, and the two values are represented as φ₁and φ₂. In step S45, θ is newly set to |φ₁|+|φ₂|, and the process proceeds to step S46. In step S46, the process determines whether the angle θ has converged, in other words, the process determines whether the absolute value of the difference between the new θ and the previous θ is not more than a predetermined threshold value. If the process has determined in step S46 that the angle θ has not converges, which means that the absolute value of the difference between the new θ and the previous θ is more than the predetermined threshold value. The process returns to step S42 from which the angle θ is used to repeatedly perform the steps subsequent to step S42.

[0267] If the process has determined in step S46 that the angle θ has converged, which means that the absolute value of the difference between the new θ and the previous θ is not more than the predetermined threshold value, the new θ is determined as the value to be found, whereby the θ computation process is completed.

[0268] It has been described using taking an example that there is the vanishing point P₇of the projected image in the cylinder (shown in FIGS. 21B and 23). If there is no vanishing point of the projected image, which means that lines P₁P₄and P₂P₃are parallel in the projected image, y_p′ of the expression (20) takes its extreme values when the expression |φ|=θ/2 holds. Let the value of the partial differential of y_p′ of the expression (20) using φ be 0, as shown in the expression (23). By performing the θ computation process in FIG. 24, based on the expressions (20) and (23) instead of the expressions (21) and (22) as described above, the computation of the values of θ when y_p′ takes its extreme value can be obtained. $\begin{array}{cc}\frac{\partial y_p^{\prime }}{\partial \phi }=0& \left(23\right)\end{array}$

[0269] Next, if the three-dimensional object is, for example, a sphere, when the copying for creating and storing an expanded image of the sphere into the paste buffer, or the pasting for pasting the image stored in the paste buffer onto the sphere, is performed, the inverse transform expression or the right transform expression is computed respectively as described below. Here, the expanded image of the sphere is created by, for example, the equidistant cylindrical projection.

[0270] In this case, assuming that there is a three-dimensional coordinate system for representing the location of the sphere in the three-dimensional space as shown in FIG. 25A, where the center of the sphere is the origin (0, 0, 0) and the x₀axis, the y₀axis, and the z₀axis are located so as to be perpendicular to one another (the x₀axis is a direction in which a right hand screw advances when being turned from the y₀axis to the z₀axis.

[0271] Hereinafter, an intersecting point (on the positive side of the y₀coordinate) where the sphere and the y₀axis intersect is referred to as the “north pole”, while an intersection (circle) where the sphere and the x₀-z₀plane with y₀=0 intersect is referred to as the “equator”.

[0272] For example, a projected image obtained by capturing the image of the sphere in FIG. 25A from a predetermined position is displayed in the two dimensional image stored in the buffer of interest, as shown in FIG. 25B, where the straight line P₅P₆connecting the points P₅and P₆(described below) is inclined at the angle α with respect to the x_paxis.

[0273] Points P₁and P₃in FIG. 25B indicate two endpoints of the intersection (circle) of the sphere and the y₀-z₀plane (with x₀=0) displayed in the projected image, where the point P₁indicates the endpoint on the same side of the north pole while the point P₃indicates the other endpoint at the opposite side of the north pole with respect to the x₀-z₀plane as the standard. Points P₂and P₄indicate the two endpoints of the equator as the intersection of the sphere and the x₀-z₀plane displayed in the projected image where the point P₂indicates the endpoint at the positive side of the x₀coordinate while the point P₄indicates the other endpoint at the negative side of the x₀coordinate.

[0274] Points O₁, O₂, O₃, and O₄in FIG. 25A correspond to points P₁, P₂, P₃, and P₄in FIG. 25B, respectively.

[0275] Points O₅, O₆and O₇in FIG. 25A indicate the center of the sphere, the north pole, and the bisecting point of an arc O₂O₄(viewed as a projected image) along the equator, respectively, where points P₅, P₆and P₇in FIG. 25B correspond to points O₅, O₆and O₇, respectively.

[0276] For example, if the expanded image of the entire sphere in FIG. 25A is stored in the paste buffer in accordance with the equidistant cylindrical projection, the radius R of the sphere can be given by the following expression where the equator corresponds to the horizontal direction of the expanded image.

[0277] R=L_w/(2π)  (24)

[0278] Regarding the sphere, ψ in the expression (7), by considering the expanded image of the sphere in FIG. 25A in accordance with the equidistant cylindrical projection, the relationship between a point (x₀, y₀, z₀) of the sphere in three-dimensional space and a point (x_e, y_e) of the expanded image can be given as follows: $\begin{array}{cc}\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)=\left(\begin{array}{c}R\sin \left(\pi y_e/L_H\right)\cos \left(2\pi x_e/L_w\right)\\ R\cos \left(\pi y_e/L_H\right)\\ R\sin \left(\pi y_e/L_H\right)\sin \left(2\pi x_e/L_w\right)\end{array}\right)& \left(25\right)\end{array}$

[0279] Also, there is no unnecessary element of the matrix M in this case. Thus, the eleven elements m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to m₃₃, described in the expression (11), must be found.

[0280] At this time, for example, P₁to P₇in FIG. 25B are designated as characteristic points. Initially, for example, points P₂, P₄, P₅and P₆having two degrees of freedom are designated as characteristic points. Points P₁, P₃, and P₇are designated so as to be on the straight line P₅P₆connecting the point P₅and the point P₆. Hence, since points P₁, P₃and P₇are on the straight line P₅P₆, they have one degree of freedom.

[0281] Since points P₂, P₄, P₅and P₆have two degrees of freedom, while points P₁, P₃and P₇have one degree of freedom, finding the values of these seven points P₁to P₇enables the eleven elements m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to m₃₃to be found.

[0282] The entire sphere in FIG. 25A, in accordance with the equidistant cylindrical projection, is stored in the paste buffer, and its expanded image is shown in FIG. 25C.

[0283] The region with the diagonal lines in FIG. 25C indicates the expanded image of the projected image in FIG. 25B, while the region without the diagonal lines indicates the expanded image of the sphere which is not displayed in the projected image in FIG. 25B. Thus, regarding the sphere in FIG. 25A, a curved line L₁corresponds to the boundary between the visible part and the invisible part of the sphere in the projected image in FIG. 25B. A line L₂bisecting the vertical direction of the rectangular expanded image corresponds to the equator.

[0284] Points E₁, E₂, E₃and E₄on the curved line L₁correspond to characteristic points P₁, P₂, P₃and P₄of the projected image in FIG. 25B, respectively. The y_ecoordinate of the point E₁on the curved line L₁takes the minimum value while the y_ecoordinate of the point E₃on the curved line L₁takes the maximum value. Points E₂and E₄indicate two intersecting points of the curved line L₁and the line L₂corresponding to the equator, where the x_ecoordinate of the point E₂is greater than that of the point E₄.

[0285] A line E₆connecting the origin (0, 0) and a point (L_w, 0) in the paste buffer coordinate system in FIG. 25C corresponds to the point P₆in FIG. 25B. A point E₇in FIG. 25C bisecting a line E₂E₄corresponds to a point P₇in FIG. 25B.

[0286] Next, let predetermined angles of the sphere in FIG. 25A be μ, λ₁, and λ₂as shown in FIG. 26. In other words, let the angle ∠O₂O₅O₄formed between the two endpoints P₁and P₂and the origin of the sphere O₅be μ as shown in FIG. 26(A), where points P₁and P₂are the endpoints of the intersection, that is, the circumference (or the equator), which the sphere and the x₀-z₀plane intersect as displayed in the projected image. Let the endpoints of the intersection, that is, the circumference, of the sphere and the y₀-z₀plane, as displayed in the projected image, be points O₁and O₃where the endpoint O₁is on the same side of the north pole while the endpoint O₃is on the other side of the north pole. Then let angle ∠O₃O₅O₇formed by points O₃, O₇, and O₅, and angle ∠O₁O₅O₇formed by point O₁, O₇and O₅, be λ₁and λ₂, respectively, as show in FIG. 26(B). Here the angle λ₁+λ₂indicates the angle ∠O₁O₅O₃formed by points O₁, O₃, and the center of the sphere O₅.

[0287] The coordinates of points O₁, O₂, O₃, O₄, O₅, O₆, and O₇in FIG. 25A are (0, R sin λ₂, R cos λ₂), (R sin(μ/2), 0, R cos(μ/2)), (0, −R sin λ₁, R cos λ₁), (−R sin(μ/2), 0, R cos(μ/2)), (0, 0, 0), (0, R, 0) and (0, 0, R), respectively.

[0288] The matrix M to be found regarding the sphere is a function of the angles μ, λ₁and λ₂in FIG. 26. Thus, the values of these angles must be found. Also, in FIG. 25B, the values of these angles must be found, while considering the fact that a line P₅P₆between the point P₅corresponding to the center of the sphere and the point P₆corresponding to the north pole, and the x_paxis form the angle α in FIG. 25B.

[0289] Then the values of angles μ, λ₁and λ₂are obtained by considering the x_p′ axis and the y_p′ axis formed by counterclockwise rotating the xp axis and yp axis by the angle α.

[0290] Also, in this case, the relationship between the points (x_p, y_p) and (x_p′, y_p′) can be given by the expression (18) by using homogeneous coordinates.

[0291] In FIG. 25A, let the coordinates of the intersection (circle) of the sphere and the y₀-z₀plane with x₀=0 be (0, R sin β, R cos β), and let the coordinates of the intersection (the points on the equator) of the sphere and the x₀-z₀plane with y₀=0 be (R sin γ, 0, R cos γ). Let x_p′ and y_p′ coordinates obtained by projecting of the point (0, R sin β, R cos β) onto the screen coordinate system be x_P1and y_p1. Let the x_p′ and y_p′ coordinates obtained by projecting the point (R sin γ, 0, R cos γ) onto the screen coordinate system be x_P2and y_p2. The coordinates x_P1and y_p2can be given by the expressions (26) and (27), respectively, using the elements of the matrix M′ defined in the expression (19).

[0292] x_p1=(m₁₂′Rsin β+m₁₃′Rcos β+m₁₄′)/(m₃₂′Rsin β+m₃₃′Rcos β+1)  (26)

[0293] y_p2=(m₂₁′Rsin γ+m₂₂′Rcos γ+m₂₄′)/(m₃₁′Rsin γ+m₃₃′Rcos γ+1)  (27)

[0294] Regarding the sphere in FIG. 25A, x_P1of the expression (26) takes an extreme when the expression β=−λ₁or β=λ₂holds, while y_P2of the expression (27) takes an extreme when the expression γ=μ/2 or γ=−μ/2 holds. Let each value obtained by partially differentiating x_P1of the expression (26) and y_P2of the expression (27) using β and γ, respectively be 0 as follows: $\begin{array}{cc}\frac{\partial x_{\mathrm{p1}}}{\partial \beta }=0& \left(28\right)\\ \frac{\partial y_{\mathrm{p2}}}{\partial \gamma }=0& \left(29\right)\end{array}$

[0295] The expression (28) holds when β=−λ₁or β=λ₂, in which λ₁and λ₂are values to be solved, while the expression (29) holds when γ=μ/2 or γ=−μ/2, in which μ is a value to be solved.

[0296] Since the elements of the matrix M′ are functions of μ, λ₁and λ₂, a value obtained by partially differentiating x_p1of the expression (28) using β becomes a function with four variables μ, λ₁, λ₂and β, and a value obtained by partially differentiating y_p2of the expression (29) with respect to γ becomes a function with four variables μ, λ₁, λ₂and γ. Hence, computing the values of μ, λ₁and λ₂which satisfy the expressions (28) and (29) becomes complicated. However, it becomes comparatively easier to obtain the values of μ, λ₁and λ₂by using an approximate calculation in the same manner as the θ computation process as described above, where the values of μ, λ₁and λ₂are initialized with predetermined values and then the elements of the matrix M′ are computed thereby computing the values of β and γ with the values of the elements, which leads the values of μ, λ₁and λ₂to be updated.

[0297] This example uses the equidistant cylindrical projection for creating the expanded image of the sphere. However, a technique of creating the expanded image of the sphere is not limited to the above-described example. That is, it is also possible to create the expanded image of the sphere by using Mercator's projection or Lambert conformal projection.

[0298] Regarding the sphere, the expanded image of the entire sphere is stored in the paste buffer as shown in FIG. 25C. Since the projected image corresponds only to the region with the diagonal lines in FIG. 25C, the region with the diagonal lines in FIG. 25C is displayed in the two dimensional image when the projected image transformed from the expanded image is pasted onto the two dimensional image stored in the buffer of interest. Since the whole image stored in the paste buffer is not displayed as the projected image in the two dimensional image, while considering above, two-dimensional paint operations, such as rendering a character and the like, must be performed over the expanded image pasted on the sphere. If the characters are pasted onto the projected image of the sphere in FIG. 25B, the pasting must be performed by rendering the characters in the region with the diagonal lines in FIG. 25C. If some of the characters are rendered outside the region, those characters are not pasted on the projected image in FIG. 25B.

[0299] Next, when the three-dimensional object is a sphere, μ, λ₁, and λ₂in FIG. 26 are respectively obtainable in a similar manner to the θ calculation, as described using, 24. The angle μ can be obtained by the following relationship after obtaining λ₁and λ₂.

[0300] That is, suppose there are points O₅′ and O₇′, as shown in FIG. 27, besides the points O₁through O₇shown in FIG. 25A. The point O₅′ is an intersection of a line segment O₁O₃and a perpendicular line extended from the point O₅to the line segment O₁O₃, and the point O₇l′ is an intersection of a line segment O₂O₄and a line segment O₅O₇.

[0301] When φO₅′O₅O₇′ is symbolized by λ3, λ3 is given as the following expression by employing λ₁, or ∠O₃O₅O₇, and λ₂, or ∠O₁O₅O₇as shown in FIG. 26B. $\begin{array}{cc}\begin{array}{c}{\lambda }_3=\left({\lambda }_1+{\lambda }_2\right)/2-{\lambda }_1\\ =\left({\lambda }_2-{\lambda }_1\right)/2\end{array}& \left(30\right)\end{array}$

[0302] The intersectional plane of the sphere in FIG. 27 and the plane O₁O₆O₇O₃(y₀z₀plane) forms a circle, as shown in FIG. 28, with a center at O₅and with an equal radius to the sphere in FIG. 27. In FIG. 28, D₁is given by the following expression, when the length of line segment O₃O₅′ is D₁,

[0303] D₁=Rsin((λ₁+λ₂)/2)  (31)

[0304] where R represents a radius of the sphere in three-dimensional space as shown in the expression (24).

[0305] Moreover, in FIG. 28, D₂is also given by the following expression, when the length of the line segment O₅′O₇′ is D₂.

[0306] D_2=Rcos((λ₁+λ₂)/2)tan λ₃  (32)

[0307] The intersectional plane of the plane of the points O₁, O₂, O₃, and O₄, and the sphere in FIG. 27 forms a circle with a center at O₅′ and with a radius of the line segment O₃O₅′, in which the line segments O₅′O₃and O₅′O₂are equal to each other since they are the radius of the circle with the center at O₅′. Accordingly, the length of the line segment O₅′O₂is equal to the length of line segment O₅′O₃, or D₁given by the expression (31). Thus, in FIG. 29, D₃can be obtained from the Pythagorean proposition, where the length of the line segment O₂O₇′ is D₃. $\begin{array}{cc}\begin{array}{c}{D}_3=\sqrt{\left(D_1^2-D_2^2\right)}\\ =R\sqrt{\left({\sin }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right)-{\cos }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right){\tan }^2{\lambda }_3\right)}\end{array}& \left(33\right)\end{array}$

[0308] Further, the intersectional plane of the sphere in FIG. 27 and the plane of the points O₅, O₂, O₇, and O₄forms the circle with a center at O₅and with a radius R as shown in FIG. 30, similarly to FIG. 26A. Since sine of angle μ/2 is obtained from FIG. 30 by computing D₃/R, the following expression holds for μ, λ₁and λ₂. $\begin{array}{cc}\begin{array}{c}\sin \left(\mu /2\right)=D_3/R\\ =\sqrt{\left({\sin }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right)-{\cos }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right){\tan }^2{\lambda }_3\right)}\\ =\sqrt{({\sin }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right)-{\cos }^2\left(\left({\lambda }_1+{\lambda }_2\right)/2\right){\tan }^2\left(\left({\lambda }_2-{\lambda }_1\right)/2\right)}\end{array}& \left(34\right)\end{array}$

[0309] As described above, λ₁and λ₂are obtained in a similar manner to the θ calculation described using, 24, and μ is obtainable by substituting λ₁and λ₂into the expression (34).

[0310] In the θ calculation, steps from S42 through S45 are necessarily repeated until θ converges. Thus, it is not required to repeat steps when μ is obtained from λ₁and λ₂Hence, μ is calculated faster and with less operations.

[0311] In the case that the three-dimensional object is, for example, a cone, when copying for storing a generated expanded side image of the cone in the paste buffer, or pasting an image stored in the paste buffer to the side of the cone, inverse transformation or transformation is computed in the following manner.

[0312] Namely, in this case, as shown in FIG. 31A, suppose that three-dimensional coordinates in three-dimensional space indicate the position of the cone. In these coordinates, suppose a y₀axis from a vertex designated as the origin (0, 0, 0), which corresponds to the center of the fan formed by expanding the side of the cone, is directed to the center of the base surface O₅, and further suppose a z₀axis from a below mentioned point O₄is directed to the center of the base surface O₅, where a x₀axis coincides with the direction of a right-hand screw when the right-hand screw is twisted from the y₀axis to the z₀axis.

[0313] When, for example, the projected image of the cone in FIG. 31A, whose image was captured from a relatively higher position, is displayed in the two-dimensional image of the buffer of interest, the projected image of the cone appears as an shaded image in FIG. 31B.

[0314] In FIG. 31B, points P₁and P₅correspond to the points O₁and O₅in FIG. 31A, respectively. Moreover, the points P₂and P₃are end points of the intersectional line of the projected image of the side of the cone (the projected image formed when only the side plane is projected) and the projected image of the base of the cone (the projected image formed when only the base surface is projected). The right and left ends of the intersectional line are designated as the points P₂and P₃, respectively. The points P₂and P₃correspond to the points O₂and O₃in FIG. 31A, respectively.

[0315] In FIG. 31A, points O₄and O₆are the points bisect the front and the back of the base circle, between the points O₂and O₃, and correspond to the points P₄and P₆in FIG. 31B, respectively.

[0316] Here, as described above, the z₀axis is directed from the point O₄to the center of the base circle, or point O₅, and the point O₅is on the y₀axis. Further, as described, the points O₄, O₅, and O₆are on the same straight line, hence, these points from O₄to O₆are all on the y₀z₀plane (the plane at x₀=0). In consequence, in FIG. 31B, the points P₄, P₅, and P₆, respectively corresponding to the points O₄, O₅and O₆, are also on the same straight line. That is, when two of the points from P₄to P₆are designated, for example the points P₄and P₆, the remaining point, the point P₅, is necessarily on the straight line P₄P₆. That means one degree of freedom.

[0317] In the case where the expanded image corresponding to the projected image of the side of the cone is stored in the paste buffer, the relationship between the angle θ in FIG. 31A representing ∠O₂O₅O₃, and the radius of the base of the cone designated as r, is given by the expression (15), similarly to the case for the cylinder described using, 21, where, for example, the direction of the base circle of the cone coincides with the lateral direction of the expanded image.

[0318] Further, in the case when the angle φ is given by the expression (16), in a similar manner as the cylinder described using, 21, ψ of the expression (7) on the side of the cone, or the relationship of the point on the side of the cone between (x₀,y₀,z₀) in three-dimension space, and (x_e, y_e) on the expanded image, is represented by the following expression through taking account of the expanded image of the cone in FIG. 31A. $\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)=\left(\begin{array}{c}\frac{y_e}{L_H}r\sin \phi \\ y_e\\ -\frac{y_e}{L_H}r\cos \phi \end{array}\right)$

[0319] In this event, the eleven elements m₁₁to m₁₄, m₂₁to m₂₄and m₃₁to m₃₃of the matrix, as described by the expression (11), which are necessarily obtained, can be calculated in the manner as heretofore described by designating six points in total on the projected image, composed of five points having two degrees of freedom and one point having one degree of freedom.

[0320] At this point, the user is requested to designate six characteristic points, for example, the point P₁through the point P₆in FIG. 31B. Namely, five points having two degrees of freedom, for example, the point P₁through the point P₄and the point P₆, are designated, and one point having one degree of freedom, for example, the point P₅, is designated. When the points P₄and P₆are designated, the point P₅is on the straight line P₄P₆as described, so that the point P₅has one degree of freedom.

[0321] Further, as shown in FIG. 31C, in the paste buffer coordinate system of the expanded image stored as a rectangle in the paste buffer, the line segment from an origin (0,0) to point (L_w, 0), coordinates (L_w, L_H), (0, L_H) and (L_w/2, L_H) are designated as E₁, E₂, E₃, and E₄, respectively, where P₁corresponds to E₁, P₂to E₂, P₃to E₃, and P₄to E₄.

[0322] According to the expression (10) in conjunction with the above relationship, the eleven elements m₁₁to m₁₄, m₂₁to m₂₄, and m₃₁to m₃₃, that is, matrix M, are given by solving expressions for the eleven elements m₁₁to m₁₄, m₂₁to m₂₄and m₃₁to m₃₃.

[0323] In the example in FIG. 31C, a rectangle is used as the expanded image of the side of the cone, other expanded images of the cone also may be employed, for example, such as a fan shape which is merely an expanded image of the side of the cone in FIG. 31A.

[0324] In the case of using the fan as the expanded image of the side of the cone, it is required to render characters along the circle of the expanded image of the fan when characters are to be pasted in parallel with the base surface on the projected image of the side of the cone. On the contrary, in the case of using the rectangle as the expanded image of the side of the cone as shown in FIG. 31C, rendering characters in the expanded image in parallel with the x_eaxis and pasting, enables characters to be pasted in parallel with the base surface on the projected image of the side of the cone.

[0325] When the height of the cone in FIG. 31A is equal to the longitudinal length L_Hof the expanded image in FIG. 31C, coordinates of points O₁, O₂, O₃, O₄, O₅, and O₆are (0, 0, 0), (r sin(θ/2), L_H, −r cos(θ/2)), (−r sin(θ/2), L_H, −r cos(θ/2)), (0, L_H, −r), (0, L_H, 0) and (0,L_H,r), respectively. Matrix M for the cone obtained as described above is a function of the angle θ similar to the case of cylinder described using. 21, hence this angle θ needs to be calculated. The projected image of the cone in FIG. 31B is displayed on two-dimensional image in a manner that the straight line between the characteristic points P₄and P₅is in parallel with the y_paxis, however it is generally common, for example as shown in FIG. 32, that the straight line between the characteristic points P₄and P₅on the projected image of the cone are inclining about the y_paxis.

[0326] The method for computing θ is described, where the angle made by the straight line between the characteristic points P₄and P₅′ and the x_paxis is α.

[0327] When y₀=y_e=L_Hin the expression (35), the expression (35) is equal to the expression (17) representing ψ on the cylinder. Accordingly, in this case x_pand y_pon the cone are equal to x_pand y_pon the cylinder, respectively. When the angle formed by the straight line between two characteristic points P₄and P₅, and the x_paxis is α, as described, in the coordinate system defined by the axes of x_p′ and y_p′, where the x_p′ and y_p′ axes obtained by rotating counterclockwise the x_pand y_paxes by α, the point P on the projected image (in FIG. 31), corresponding to the point O on the circle of the base of the cone, is represented similarly to the case of the cylinder.

[0328] That is, regarding the cone, the coordinates of x_p′ and y_p′ of point P on the projected image, corresponding to the point O on the circle of the base of the cone, are given by the expression (20). When an angle formed by the point P corresponding to the point O, P₁, and P₄on the projected image is designated as ν, as shown in FIG. 33, and the coordinates of P₁is represented as (x_p′, y_p′), the angle ν is represented as the following expression.

[0329] ν=tan⁻¹(|y_p′−y_p1′|/|x_p′−x_p1′|)  (36)

[0330] The angle ν of the expression (36) is a function of the angle φ (=∠O₄O₅O) (in FIG. 31A) formed by the point O, O₅and O₄, and is extreme when |φ|=θ/2 is satisfied.

[0331] Hence, instead of the expression (21), θ of the cone in FIG. 31A is given by using the expression (36), similarly to the case of the cylinder.

[0332] When the vertex P₁is inside the projected image in FIG. 31B or FIG. 33, θ is 2π (360 degrees) since the entire side is to be seen.

[0333] The foregoing description is about the method for computing ψ and M regarding some primitive shapes as three-dimensional objects, and computation of ψ and M of three-dimensional objects having different shapes from the above-described shapes can be performed if the shapes are represented by functions.

[0334] The characteristic points are not limited to the points described above. For example, in FIG. 20, the vertices of the quadrangle, or the projected image of the rectangle, are designated as the characteristic points, while each midpoint of the sides of the quadrangle is instead designated as the characteristic points of the quadrangle. Regarding the sphere, points from P₁through P₇corresponding to the points O₁to O₇, respectively, shown in FIG. 25A, are designated as the characteristic points. However, for example, the projected points on the two-dimensional image corresponding to the points O₁to O₆, and a point O₈may be designated as the characteristic points of the sphere, when the point O₈is the other end of the intersection than the point O₇, of the straight line of points O₅and O₇, and the sphere shown in FIG. 25A.

[0335] Regarding the cone, for example, points as shown in FIG. 34 may be designated as the characteristic points. That is, as shown in FIG. 34A, the midpoint of the line segment between the vertex O₁of the cone and the center of the base is represented as O₆₀. The points of intersection of the circle of the base and the x₀y₀plane are represented as O₃₀and O₅₀, and the points of intersection of the circle of the base and the y₀z₀plane are represented as O₂₀and O₄₀. Here, the coordinates of the points O₂₀, O₃₀, O₄₀, O₅₀and O₆₀correspond to (r, L_H, O), (0, L_H, −r), (−r, L_H, O), (0, L_H, r) and (0, L_H/2, O), respectively. When points on the projected image corresponding to the points O₁, O₂₀, O₃₀, O₄₀, O₅₀and O₆₀, are represented as points P₁, P₂₀, P₃₀, P₄₀, P₅₀and P₆₀, respectively, as shown in FIG. 34B, these points may be designated as the characteristic points.

[0336] It is apparent from the foregoing description that a displayed point (a visible point) and a non-displayed point (an invisible point) on the projected image both can be designated as the characteristic points.

[0337] The correspondence of the designated point as the characteristic point of the projected image to the position of the expanded image stored in the paste buffer is predetermined.

[0338] In the pasting, basically as described above, the image stored in the paste buffer is pasted on the surface of the three-dimensional object, which is specified by the characteristic points designated by the user, on the two-dimensional image stored in the buffer of interest. That is, the pixel levels of the image stored in the paste buffer are used to overwrite the two-dimensional image in the buffer of interest. However, for example, in the case when a new three-dimensional object is added to the two-dimensional image stored in the buffer of interest by using the pasting, simply overwriting with the pixel levels of the image stored in the paste buffer on the buffer of interest enables the new three-dimensional object to be added in front of the three-dimensional object already displayed in the two-dimensional image stored in the buffer of interest, but not at the back thereof.

[0339] Hence, in this embodiment, by using a matte, the new three-dimensional object can be added at the back of the three-dimensional object already displayed in the two-dimensional image stored in the buffer of interest.

[0340] The matte is a so-called “gray scale image” representing a shape of an object displayed in the two-dimensional image, and is also called an “α-image” since the gray scale value represents a contribution rate (hereinafter referred to as “α”) to the object at each pixel composing the two-dimensional image. Here, the matte of the two-dimensional image stored in the buffer of interest is adapted to be generated by the matte processing in step 15 in FIG. 7. As described above, the matte of two-dimensional image is stored in the image buffer storing the two-dimensional image.

[0341] Each pixel level is between, for example, zero and one. The level of each pixel composing a foreground is one. The level of each pixel composing a background is zero. The level of each pixel composing a boundary region between the foreground and the background is proportional to the ratio of the foreground to the background included in each pixel (As the ratio of the foreground is greater, the pixel level is closer to one, and as the ratio of the background is greater, the pixel level is closer to zero).

[0342] Accordingly, for example, in the case when a two-dimensional image displaying a house-shaped three-dimensional object, as shown in FIG. 35A, is stored in the buffer of interest, when performing the matte operation using the house as a foreground and the other part as a background, a matte is generated in the matte processor 16, as shown in FIG. 35B, and stored in the buffer of interest. In FIG. 35B, each pixel level of the shaded portion is one, and each pixel level of the blank portion is zero.

[0343] The following disclosed methods can be applied to the matte generation. They are, for example, “Intelligent Scissors for Image Composition”, Proceedings of SIGGRAPH 95, pp. 191-198, by Eric N. Mortensen and William A. Barrett, and the inventions in Japanese Unexamined Patent Publication Nos. 8-331455 and 10-143654 (both disclosed by Sony Corporation in Japan).

[0344] When the matte, together with the two-dimensional image, is stored in the buffer of interest, operation of the input unit 6 enables the matte operation to be valid or not.

[0345] For example, when the two-dimensional image or the matte are stored in the buffer of interest, as shown in FIG. 35A or FIG. 35B, and when the pasting of, for example, a cylinder on the two-dimensional image of the buffer of interest is instructed, the projected image of the cylinder simply overwrites the two-dimensional image, so that a two-dimensional image having a cylinder in front of the house is generated, as shown in FIG. 35C.

[0346] When the matte is invalid, the two-dimensional image C by the pasting is generated based on the expression C=αA+(1−α)B, where the level of pixels composing the two-dimensional image stored in the buffer of interest is A, the level of pixels composing the projected image of the cylinder is B, and the level of pixels composing the matte is α.

[0347] Accordingly, in this case, a two-dimensional image having the cylinder behind the house is generated, as shown in FIG. 35D.

[0348] The pasting using the matte enables a so-called occlusion to occur, which causes the hidden part of the three-dimensional object displayed in the two-dimensional image to be invisible. Hence, in this case the matte is functioning as a flag which determines whether overwriting using the projected image is allowed.

[0349] It is possible for the matte to be generated by the user.

[0350] Referring to the flowchart in FIG. 36, the object property/illumination change process in step 16 in FIG. 7 is described. In this case, it is assumed that the matte of the two-dimensional image has already been generated and stored in the buffer of interest.

[0351] In the object property/illumination change, in step S51, the color distribution of the region of the object displayed in the two-dimensional image stored in the buffer of interest is computed, which distribution is represented by the matte also stored in the buffer of interest. In step S51, the region of the object displayed in the two-dimensional image stored in the buffer of interest is recognized based on the matte stored in the buffer of interest, and the distribution (color distribution) (in the RGB space) of the RGB value of pixels composing the region is computed.

[0352] In step S52, following the computation of the color distribution, among pixels composing the region of the object, a pixel (hereinafter referred to as “maximum chromatic pixel”) having a maximum chroma is detected based on the color distribution computed in step S51.

[0353] FIG. 37 shows an example of the color distribution of the object in RGB space (the portion circumscribed with the solid line). FIG. 37 shows the color distribution in which each of the maximum R, G, and B values is normalized to one. Therefore, in the RGB space, a point in which (R, G, B)=(0, 0, 0) represents black, and a point in which (R, G, B)=(1, 1, 1) represents white.

[0354] In the RGB space, the chroma is larger farther from the straight line of the points of black and white, in other words, the straight line of the points (0, 0, 0) and (1, 1, 1) in FIG. 37. Accordingly, in step S52, a pixel (in the color distribution obtained in step S51) having the color located at the point being farthest from the straight line of the point (0, 0, 0) and (1, 1, 1) is detected. In the color distribution in FIG. 37, the pixel of color indicated by C in the same Fig. is computed as the maximum chromatic pixel.

[0355] Here, the color C of the maximum chromatic pixel (RGB level) is considered as the original color of the object displayed in the two-dimensional image stored in the buffer of interest.

[0356] The process proceeds to step S53, and the color distribution computed in step S51 is transformed, then the pixel levels (RGB levels) in accordance with the transformed color distribution are written in the corresponding pixels composing the two-dimensional image stored in the buffer of interest, and the process returns. In step S53, the color C of the maximum chromatic pixel is converted into predetermined color D, then the other colors of pixels composing the region of the object are converted in accordance with a predetermined rule.

[0357] Specifically, when the object property process for changing material attributes of the object, such as color or tone, is commanded, in step S53, the color C of the maximum chromatic pixel is converted into the predetermined color D, and in a similar manner, the other colors of pixels composing the region of the object are linearly converted.

[0358] That is, as shown in FIG. 37, when the color C of the maximum chromatic pixel is converted, the colors (in the color distribution) between the color C and the color at the point (1, 1, 1) representing white are converted into the color between the color D and the color at the point (1, 1, 1) representing white. Moreover, the colors (in the color distribution) between the color C and the color at the point (0, 0, 0) representing black are linearly converted into the color between the color D and the color of the point (0, 0, 0) representing black. In consequence, the color distribution of the object indicated by the solid line in FIG. 37 is changed to that indicated by the dotted line. In this case, as the colors are linearly converted, the brightest color Q in the original color distribution is converted into the brightest color T in the transformed color distribution. Similarly, the darkest color P in the original color distribution is converted into the darkest color S in the transformed color distribution.

[0359] The colors between the color C and the color at the point (1, 1, 1) representing white are specular reflective elements from the object by specular reflection of rays from an illuminant, and the colors between the color C and the color at the point (0, 0, 0) representing black are diffused reflective elements from the object by diffused reflection of rays from the illuminant. Hence, as described above, the colors of the projected image of the object displayed in the two-dimensional image stored in the buffer of interest can be converted while maintaining the original shade by lineally transforming specular reflective elements and diffused reflective elements of the color distribution of the object separately. That is, in the case described above, the color of the object displayed in the two-dimensional image is changeable from the color C to the color D.

[0360] The color change is based on the theory of the dichromatic reflection model.

[0361] In the above-described case, respective specular reflective elements or diffused reflective elements of the color distribution of the object are linearly transformed, meanwhile specular reflective elements or diffused reflective elements can non-linearly be transformed in step S53.

[0362] For example, in FIG. 37, by transforming almost all colors between the color C and Q of specular reflective elements to the region around the color D, and transforming the color Q and colors around the color Q to the region around the color T, the object can be provided with strong specular reflection on a limited part of the object. In this case, the entirely bright object having rich diffused reflection (as shown in FIG. 38A) can be changed to the object with strong specular reflection at a part thereof as shown in FIG. 38B.

[0363] As described above, non-linear transformation of diffused reflective elements or specular reflective elements enables the region of specular reflection on the object to be narrowed or broadened, or enables the intensity of specular reflection to be changed, hence the material tone of the object can be changed.

[0364] The generation of the strong reflection at a part of an object, as illustrated above, is generally due to a small surface roughness of the object. Accordingly, the surface of the object can be changed to a smoother surface by changing the color distribution.

[0365] When the illumination change process is commanded, in step S53 in FIG. 36, the colors between the color C and Q of specular reflective elements in FIG. 37 are converted so as to be distributed along from the color C to the color of a new illuminant.

[0366] Since the brightest color in the color distribution of the object is considered as a color of the illuminant, an apparent effect in which the illuminant looks to be changed to a new one can be obtained by transforming the colors distributed from the color C to the brightest color Q, to the colors distributed from the color C to the color of the new illuminant.

[0367] Referring to step S17 in FIG. 7, the erasing is described. As described above, since the erasing is to eliminate a part of the two-dimensional image stored in the image buffer, the erasing can basically be performed by the following steps. The first is the copying of the image buffer as the buffer of interest, which stores the two-dimensional image to be erased, to store the expanded image in the paste buffer, the second is the erasing for eliminating a part of the expanded image stored in the paste buffer, and the last is the pasting to paste the projected image, corresponding to the expanded image, on the original position in the buffer of interest.

[0368] However, in the case of erasing the cylinder buried in a grid pattern wall in the two-dimensional image as shown in FIG. 39A, sequential processing from the copying to store the wall of the expanded image in the paste buffer as shown in FIG. 39B to the erasing of simply eliminating the part of the cylinder followed by the pasting, results in an unnatural two-dimensional image without any design at the part where the cylinder originally existed.

[0369] Therefore, the erasing can eliminate a part of the expanded image, and can reproduce the eliminated part of background.

[0370] That is, for example, in the case of eliminating the cylinder buried in the grid pattern wall in the two-dimensional image, shown in FIG. 39A, when the expanded image of the wall is stored in the paste buffer, as shown in FIG. 39B, by performing the copying, the erasing can eliminate the cylinder part, and can render the background corresponding to the eliminated part. Hence, the grid pattern is entirely drawn on the expanded image of the paste buffer, as shown in FIG. 39C.

[0371] Accordingly, the cylinder is eliminated as shown in FIG. 39D by the pasting with the expanded image as described above, and further the natural two-dimensional image with grid pattern is reproduced at a part where the cylinder originally existed.

[0372] As described above, the methods of eliminating a part of the image, and reproducing a background having the eliminated part, which are disclosed in, for example, Japanese Unexamined Patent Publication No. 9-128529 (corresponding to European Unexamined Patent Publication No. 772,157) and 10-105700 (corresponding to U.S. Pat. No. 5,892,853) both being of the same assignee as this application, may be employed.

[0373] In the embodiment described heretofore, copying processing is performed such that the expanded image is formed only of a single surface of the three-dimensional object contained in the two-dimensional displayed image stored in the buffer of interest, and such a sole expanded image is stored in the paste buffer. This, however, is only illustrative and the copying processing may be executed such that expanded images are formed of two or more adjacent surfaces on the three-dimensional object contained in the two-dimensional displayed image stored in the buffer of interest and these expanded images are collectively stored in the paste buffer.

[0374] FIG. 40A shows, by way of example, a process in which only a single surface of a three-dimensional object is developed into expanded image. Referring to this Figure, it is assumed that the buffer of interest stores a two-dimensional image containing an image of a house as a three-dimensional object. The house has a wall on which a lattice-like pattern is rendered and an adjacent plain wall. When consecutive characters or a continuous pattern is to be rendered on these two walls, the copying processing is executed first on the wall having the lattice pattern. Namely, the expanded image of this wall is stored in the paste buffer and retouching such as rendering characters or pattern is effected on the stored expanded image. The pasting processing is then executed to paste the lattice-pattern wall having the characters or pattern rendered thereon. The described operation is repeatedly conducted for the plain wall. Namely, expanded image of the plain wall is formed through the copying processing and stored in the paste buffer. Characters or a pattern is rendered on the stored expanded image of the plain wall, and the image of the plain wall now carrying characters or the pattern is set back to the house image by the pasting processing. It will be seen that this operation is quite troublesome because the surfaces defining the three-dimensional objects are processed, i.e., developed, retouched and pasted, in a one-by-one fashion. In addition, it is not easy to render a character string or continuous pattern that extends over two ore more surfaces defining the three-dimensional object.

[0375] Referring now to FIG. 40B, expanded images are formed of both the wall having the lattice pattern and the adjacent plain wall, and these expanded images are stored in the paste buffer. In this case, character string or pattern can easily be drawn on the consecutive two walls, i.e., the wall having the lattice pattern and the plain wall.

[0376] Pasting operation is also possible such that the expanded images of two or more consecutive surfaces stored in the paste buffer are simultaneously set back to the corresponding positions on the three-dimensional object.

[0377] Thus, according to the present invention, an inverse transformation expression for developing the image of a surface constituting part of three-dimensional object in a two-dimensional displayed image into a two-dimensional expanded image is determined based on the shape information and the characteristic point designated by the user. Thus, the image of a surface defining part of the three-dimensional object is transformed into an expanded image in accordance with this inverse transformation expression. The invention also determines, in accordance with the shape information and the characteristic point as designated by the user, a forward transformation expression for transforming the expanded image into a projected image. The projected image formed through the processing in accordance with the forward transformation expression is pasted to an area which is identified by the characteristic point on the two-dimensional displayed image. It is therefore easy to effect three-dimensional editorial work or other types of retouching on the image of a three-dimensional object contained in a two-dimensional displayed image.

[0378] Therefore, when a picture of an object taken by the camera 9 is available, the image processing apparatus of FIG. 3 enables the position, shape or texture of the object without requiring taking the picture of the object again and again. This image processing apparatus also can be used by a designer when the designer wishes to effect trimming or retouching on the image taken up through a camera. Further, the image processing apparatus can be employed in, for example, preparation of virtual-reality space which is now finding spreading use, as well as for various editorial purposes. Obviously, the image processing apparatus of this embodiment can process not only the image of an actual objet taken up by a camera, but also images obtained through photographing paintings or pictures, as well as images formed by using computer graphics techniques.

[0379] Thus, a series of operations are executed including the copying processing effected on a surface of a three-dimensional object contained in a two-dimensional displayed image stored in a buffer of interest, storing the result of the copying processing in a paste buffer, retouching the image thus stored in the paste buffer, and pasting the retouched image to the original surface of the three-dimensional object. In this series of operations, the pasting processing is performed by using the same shape information and characteristic point as those used in the copying processing. This is because in the process of the kind described the shape information and the characteristic point used in the pasting processing are naturally identical to those employed in the copying processing. It is laborious and wasteful for the user to input again the same shape information and characteristic point when pasting the retouched image back to the original surface of the three-dimensional object. Further, there is a risk that the user when pasting may designate a characteristic point which is not identical to that used in the copying processing, even if an automatic correcting processing is executed. In such a case, the region in which the retouched projected image is pasted is not registered with the region in the two-dimensional image from which the expanded image is derived, with the result that an unnatural two-dimensional image is formed as a result of the pasting.

[0380] It is also to be understood that the image processing apparatus shown in FIG. 3 can equally be used both for still and moving two-dimensional images.

[0381] A description will be given first as to the case where the two-dimensional image to be obtained is a still image. It is assumed here that the two-dimensional image to be processed is a still image (projected image) of a three-dimensional object such as, for example, a cylindrical body as shown in Segment (A) of FIG. 41. In this case, the copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function Mψ of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the cylindrical body displayed in the still image and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B (x_p, y_p) of a point (x_p, y_p) on the projected image in the still picture is written in the paste buffer as the pixel value C (x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of the side or peripheral surface of the cylindrical body developed into two-dimensional image is obtained as shown in Segment (B) of FIG. 41. The projected image of the cylindrical body shown in Segment (A) of FIG. 41 is an image which is obtained when the cylindrical body, having the peripheral surface hatched as illustrated, is viewed from a level slightly above the top face of the cylindrical body. The expanded image of the peripheral surface of the cylindrical body, therefore, is patterned by hatching as shown in Segment (B) of FIG. 41.

[0382] For the purpose of the writing a string of characters “ABCD” on the peripheral surface of the cylindrical body shown in Segment (A) of FIG. 41, characters “A”, “B”, “C” and “D” are written on the expanded image of the peripheral surface shown in Segment (B) of FIG. 41, in the same way as writing these characters on a two-dimensional surface as shown in Segment (C) of FIG. 41, whereby an expanded image carrying the characters “A”, “B”, “C” and “D” is obtained as shown in Segment (D) of FIG. 41. The pixel value of a point, i.e., the coordinates of a pixel (x_e, y_e) on the expanded image of the peripheral surface of the cylindrical body, carrying the characters “A”, “B”, “C” and “D”, is expressed by C′(x_e, y_e). Then, the pasting processing is executed by writing into the buffer of interest, the pixel value C′(x_e, y_e) of the point on the expanded image as the pixel value B′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image, based on the same function Mψ as that used in the copying processing, whereby a two-dimensional image containing a natural projected image, equivalent to an image obtained by photographing the cylindrical body and carrying the character string “ABCD” on and along the peripheral surface, is obtained as shown in Segment (E) of FIG. 41.

[0383] It will be seen that the pasting processing is executed by using the shape information and the characteristic point which are the same as those used in the copying processing.

[0384] A description will now be given of a case in which the two-dimensional image to be processed is a moving picture which is formed of first to third frames (or fields) as shown in Segments (A), (B) and (C) of FIG. 42. In this case, the series of processings including the copying processing for transforming the projected image in the two-dimensional displayed image into an expanded image, retouching the expanded image and pasting the retouched image back to the projected image are executed in the following manner. This series of processings will be referred to as “copy and paste processing”, hereinafter. The first to third frames shown in Segments (A) to (C) of FIG. 42 illustrate images of a cylindrical body which is moving leftward while rotating clockwise when viewed from the top, as seen from a level slightly above the top of the cylindrical body. This is equivalent to images that are obtained when the viewer moves counterclockwise around the cylindrical body while the latter is moving leftward.

[0385] In this case, the copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function My of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the cylindrical body displayed in the first frame (Segment (A) of FIG. 42) and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B₁(x_p, y_p) of a point (x_p, y_p) on the projected image in the first frame is written in the paste buffer as the pixel value C₁(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of the side or peripheral surface of the cylindrical body shown in the first frame developed into two-dimensional image is obtained.

[0386] For the purpose of the writing a string of characters “A”, “B”, “C” and “D” on the peripheral surface of the cylindrical body in the same way as that explained before with reference to Segment (A) of FIG. 41, characters “A”, “B”, “C” and “D” are written on the expanded image of the peripheral surface, whereby an expanded image carrying the characters “A”, “B”, “C” and “D” is obtained as shown in Segment (D) of FIG. 42. The pixel value of a point, i.e., the coordinates of a pixel (x_e, y_e)n on the peripheral surface of the cylindrical body shown in the first frame, now carrying the characters “A”, “B”, “C” and “D”, is expressed by C₁′(x_e, y_e). Then, the pasting processing is executed by writing into the buffer of interest, the pixel value C₁′(x_e, y_e) of the point on the expanded image as the pixel value B₁′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the first frame, based on the same function Mψ as that used in the copying processing, whereby a first frame image containing a natural projected image, equivalent to an image obtained by photographing the cylindrical body carrying the character string “ABCD” on and along the peripheral surface, is obtained as shown in Segment (G) of FIG. 42.

[0387] Then, a copying processing is executed on the image contained in the second frame. The copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function Mψ of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the cylindrical body displayed in the second frame (Segment (B) of FIG. 42) and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B₂(x_p, y_p) of a point (x_p, y_p) on the projected image in the second frame is written in the paste buffer as the pixel value C₂(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of the side or peripheral surface of the cylindrical body of the second frame is obtained.

[0388] The purpose of the process which is being described is to obtain a projected image of the cylindrical body carrying the character string “ABCD” on and along the peripheral surface thereof. It is recalled here that the image of the cylindrical body is presented as a moving picture such that the cylindrical body rotates clockwise. Accordingly, even though all the characters ABC and D are visible in the first frame image shown in Segment (G) of FIG. 42, the leftmost character “A” becomes invisible in the final projected image of the second frame. Therefore, characters “B”, “C” and “D”, except for the invisible character “A”, are rendered on the second frame expanded image of the peripheral surface of the cylindrical body, as shown in Segment (E) of FIG. 42, such as to leave a grater space between the character “D” and the edge of the expanded image than in Segment (D) of this Figure. The pixel value of a point, i.e., the coordinates of a pixel (x_e, y_e), on the expanded image of the peripheral surface of the cylindrical body image of the second frame is expressed by C₂′(x_e, y_e). Then, the pasting processing is executed by writing into the buffer of interest, the pixel value C₂′(x_e, y_e) of the point on the expanded image as the pixel value B₂′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the second frame, based on the same function Mψ as that used in the copying processing, whereby a second frame image containing a natural projected image, equivalent to an image obtained by photographing the cylindrical image which has been rotated clockwise to hide the character “A”, is obtained as shown in Segment (H) of FIG. 42.

[0389] Then, a copying processing is executed on the image contained in the third frame. The copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function Mψ of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the cylindrical body displayed in the third frame (Segment (C) of FIG. 42) and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B₃(x_p, y_p) of a point (x_p, y_p) on the projected image in the third frame is written in the paste buffer as the pixel value C₃(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of the side or peripheral surface of the cylindrical body of the third frame is obtained.

[0390] The image of the cylindrical body is presented as a moving picture such that the cylindrical body rotates clockwise. Accordingly, even though the characters “B”, “C” and “D” are visible in the second frame image shown in Segment (H) of FIG. 42, the leftmost character “B” becomes invisible in the final projected image of the third frame. More specifically, the character “A” which is visible in the first frame projected image becomes invisible in the second frame projected image and further the character “B” disappears in the final projected image of the third frame. Therefore, only the characters “C” and “D”, except for the invisible character “A” and “B”, are rendered on the third frame expanded image of the peripheral surface of the cylindrical body, as shown in Segment (F) of FIG. 42, such as to leave a further grater space between the character “D” and the edge of the expanded image than in Segment (E) of this Figure. The pixel value of a point, i.e., the coordinates of a pixel (x_e, y_e), on the expanded image of the peripheral surface of the cylindrical body image of the third frame is expressed by C₃′(x_e, y_e). Then, the pasting processing is executed by writing into the buffer of interest, the pixel value C₃′(x_e, y_e) of the point on the expanded image as the pixel value B₃′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the third frame, based on the same function Mψ as that used in the copying processing, whereby a third frame image containing a natural projected image, equivalent to an image obtained by photographing the cylindrical image which has been further rotated clockwise to hide the characters “A” and “B”, is obtained as shown in Segment (H) of FIG. 42.

[0391] It will be seen from the foregoing description that three-dimensional editorial work or other retouching operations can easily be effected on a two-dimensional image regardless of whether the image is moving or still.

[0392] The above-described technique, however, is laborious particularly when the two-dimensional image to be processed is a moving picture image, because the retouching operation such as rendering characters has to be executed on the expanded image of a three-dimensional object in each of a plurality of frames.

[0393] In addition, in the case of an object of the type described in connection with FIG. 42, the user is obliged to conduct different types of retouching work on the expanded images of different frames, such that all the characters “A” to “D” are written for the first frame, characters “B”, “C” and “D” except for “A” are written for the second frame and only the characters “C” and “D” are written for the third frame. This also heavily burdens the user.

[0394] A description will now be given of an embodiment which enables easier copy and paste processing on a moving two-dimensional picture image.

[0395] A description will first given of a case where no part of the image of the three-dimensional object becomes invisible despite the motion of the picture.

[0396] Segments (A), (B) and (C) of FIG. 43 show images carried by first to third frames of a moving picture of a rectangle which is a surface of a rectangular parallelepiped body. It is intended here that this moving picture is processed into a moving picture composed of the first to third frames each having a character string “ABCD” as shown in Segments (D) to (F) of FIG. 43.

[0397] The frames of the moving picture shown in Segments (A) to (C) of FIG. 43 illustrate the rectangle as a surface of the rectangular parallelepiped body which is rotating clockwise when viewed from the upper side. This is equivalent to images obtained when a photographer moves counterclockwise around the rectangular parallelepiped body while photographing.

[0398] It will be understood that the rectangle as a plane is wholly-visible, i.e., the projected images show the entire portion of the rectangle, even when the rectangle is rotated or translated in a three-dimensional space, although the two-dimensional shape and size, i.e., the contour and size of the projected image, may vary, unless the angle between the visual axis and a line normal to the plane of the rectangle exceeds 90 degrees.

[0399] A copying processing is executed on the image contained in the third frame. The copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function Mψ of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the rectangle displayed in the first frame (Segment (A) of FIG. 43) and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B₁(x_p, y_p) of a point (x_p, y_p) on the projected image in the first frame is written in the paste buffer as the pixel value C₁(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of one surface of the rectangular parallelepiped body of the first frame is obtained.

[0400] In this case, the object to be processed is the image of a rectangle which is one of the surfaces presented by a parallelepiped body. The correlation between a point (x₀, y₀, Z₀) in the three-dimensional space and the corresponding point (x_e, y_e) on the expanded image is expressed by the aforesaid equation (12). The point (x_e, y_e) is correlated to the corresponding point (x_p, y_p) on the projected image of the rectangle by the aforesaid equation (13).

[0401] Coordinates of four apices of the tetragon providing the projected image of the rectangle are defined, clockwise starting from the left upper apex, as P₁(=(x_P1, y_p1), P₂(=(x_P2, y_p2), P₃(=(x_P3, y_p3), and P₄(=(x_P4, y_p4). At the same time, four apices of the expanded image of the rectangle are termed as E₁, E₂, E₃and E₄, clockwise starting from the left upper apex. Thus, the apices P₁, P₂, P₃and P₄of the projected image correspond to the apices E₁, E₂, E₃and E₄of the expanded image, respectively. The coordinates of the points E₁, E₂, E₃and E₄on the paste buffer coordinate system are defined as (0, 0), (L_W, 0), (L_W, L_H) and (0, L_H), respectively. As stated before, L_Wand L_Hrespectively represent the width and height of the expanded image of the rectangle. The equation (14) is derived from the equation (13) and the term M in the equation (13) is determined by solving the equation (14).

[0402] Therefore, the function Mψ in the equation (10), that determines the correlation between the point (x_p, y_p) on the projected image of the rectangle displayed in the first frame (Segment (A) of FIG. 43) and the coordinates (x_e, y_e) of the corresponding point on the expanded image, can be determined when the user designates, as the shape information, that the projected image to be processed is an image of a rectangle and designates, as the characteristic points, the coordinates of the four apices of the rectangle on the two-dimensional displayed image. More specifically, the term M can be determined based on the characteristic points and ψ can be determined based on the shape information.

[0403] Then, based on the function Mψ, the pixel value B₁(x_p, y_p) of a point (x_p, y_p) on the projected image in the first frame is written in the paste buffer as the pixel value C₁(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of one surface of the rectangular parallelepiped body of the first frame is obtained, as shown in Segment (A) of FIG. 44.

[0404] The described operation is performed also on the second and third frames. Namely, the function Mψ in the equation (10), that determines the correlation between the point (x_p, y_p) on the projected image of the rectangle displayed in each of the second and third frames and the coordinates (x_e, y_e) of the corresponding point on the expanded image, can be determined when the user designates the coordinates of the four apices of the rectangle on the two-dimensional image displayed in each frame. It will be appreciated that there is no need for designation of the shape information, because the shape information used in the processing of the first frame image can be used also for each of the second and third frames. Expanded images of one surface of the rectangular parallelepiped body contained in the second and third frames can be formed based on the function Mψ thus determined.

[0405] More specifically, for the image on the second frame, the pixel value B₂(x_p, y_p) of a point (x_p, y_p) on the projected image in the second frame is written in the paste buffer based on the function Mψ, as the pixel value C₂(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of one surface of the rectangular parallelepiped body of the second frame is obtained, as shown in Segment (B) of FIG. 44. Similarly, for the third frame, the pixel value B₃(x_p, y_p) of a point (x_p, y_p) on the projected image in the third frame is written in the paste buffer based on the function Mψ, as the pixel value C₃(x_e, y_e) of the corresponding point (x_ey_e) on the expanded image, whereby an expanded image of one surface of the rectangular parallelepiped body of the third frame is obtained, as shown in Segment (C) of FIG. 44.

[0406] The expanded images are thus formed of the rectangle for the respective frames of the moving picture, and are stored in the paste buffers and displayed on the display unit 5. In this case, therefore, three image buffers are used as the paste buffers. The projected images contained in the first to third frames are of an identical rectangle. The expanded images of the first to third frames, shown in Segments (A) to (C) of FIG. 44, therefore, have the same values of height L_Hand the width L_Wof the rectangle.

[0407] Then, in the image processing apparatus of FIG. 4, one of the image buffers which is not occupied is selected and used as a buffer which stores the expanded image of the whole of the rectangle to be processed. This image buffer will be referred to also as a “paint buffer”. A storage area that can store the expanded image of the whole of the rectangle to be processed is secured in the paint buffer. This storage area will be referred to also as a “overall expanded image storage area”. In this case, the width and height of the expanded image, formed by developing the whole or entire part of the rectangle to be processed, are equal to the width L_Wand height L_Hof the expanded images of the rectangle contained in the first to third frames. Therefore, an area having a width L_Wand height L_His secured as the overall expanded image storage area, as shown in Segment (D) of FIG. 44.

[0408] In the meantime, the user while monitoring the expanded image displayed on the display unit 5 operates the input unit 5, so as to render the characters “A”, “B”, “C” and “D” at desired addresses in the overall expanded image storage area. The thus rendered characters are written in the overall expanded image storage area of the paint buffer by the operation of the paint processor 18, as shown in Segment (D) of FIG. 44.

[0409] The position of a point in the overall expanded image storage area secured in the paint buffer is expressed in terms of a two-dimensional coordinate system having an X-axis and a Y-axis which extend, respectively, to the right and downward from an origin which is located on the left upper corner of the storage area. This coordinate system is the same as the paste buffer coordinate system. The pixel value (stored value) of a point (x_e, y_e) in the overall expanded image storage area is expressed as D(x_e, y_e).

[0410] Character images are thus rendered by the user and are stored in the overall expanded image storage area. The paint processor 18 then cuts at least part out of the overall expanded image storage area. This cut-out part will be referred to as a “cut-out region”, hereinafter. Retouching such as an editorial work is effected on the basis of the cut-out region, on the expanded image of rectangle which is contained in each of the first to third frames and which is stored in each paste buffer.

[0411] In the illustrated case, the paint processor 18 cuts out the entirety of the overall expanded image storage area. Thus, the cut-out region is the same as the entirety of the overall expanded image storage area. The paint processor 18 then writes the contents of the cut-out region onto the expanded images of rectangle displayed in the first to third frames.

[0412] This operation will be described in more detail. The pixel value (stored value) of a point (x_e, y_e) in the overall expanded image storage area is expressed as D(x_e, y_e). A function, which is “1” where the pixel value D(x_e, y_e) is not “0” and “0” where the pixel value D (x_e, y_e) is 0, is expressed by α(x_e, y_e). The results of writing of the content carried by the cut-out region onto the pixels (x_e, y_e) of the expanded images of the rectangle shown in the first, second and third frames are respectively represented by C₁′(c_e, y_e), C₂′(x_e, y_e) and C₃′(x_e, y_e). The paint processor 18 then determines, in accordance with the following equation (37), the expanded images C₁′(x_e, y_e), C₂′(x_e, y_e) and C₃′(x_e, y_e) of the rectangle in the first, second and third frames in which the contents of the respective cut-out regions have been written.

[0413] C₁′(x_e, y_e)=

[0414] α(x_e, y_e)D

[0415] (x_e, y_e)+(1−

[0416] α(x_e, y_e))C₁

[0417] (x_e, y_e)C₂

[0418] ′(x_e, y_e)=

[0419] α(x_e, y_e)D(x_e,

[0420] y_e)+(1−α(x_e,

[0421] y_e))C₂(x_e,

[0422] y_e)C₃(x_e,

[0423] y_e)=α(x_e, y_e)D

[0424] (x_e, y_e)+(1−

[0425] α(x_e, y_e))C₃

[0426] (x_e, y_e)  (37)

[0427] As a result of this processing, the expanded images of the rectangle contained in the first to third frames as shown in Segments (A) to (C) of FIG. 44 are transformed into expanded images carrying the character string “ABCD” as shown in Segments (E) to (G) of the same Figure.

[0428] The characters “A”, “B”, “C” and “D” to be added to each frame may be written in a manner as if these characters are rendered by a translucent paint. This can be achieved by setting the term α(x_e, y_e) to a value which is greater than 0 but less than 1.0, e.g., 0.5, in the equation (37) shown above.

[0429] Then, a pasting processing is executed by writing into the buffer of interest (image buffer in which the first frame data is stored) the pixel value C₁′(x_e, y_e) of the point (x_e, y_e) on the expanded image (Segment (E) of FIG. 44) of the rectangle contained in the first frame which now carries the character string “ABCD”, as the pixel value B₁′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the first frame, based on the same function Mψ as that used in the copying processing on the first frame. As a result, a first frame image containing a projected image of the rectangle carrying the character string “ABCD” is obtained as shown in Segment (D) of FIG. 43.

[0430] Similarly, a pasting processing is executed for the second frame. This is effected by writing into the buffer of interest (image buffer in which the second frame data is stored) the pixel value C₂′(x_e, y_e) of the point (x_e, y_e) on the expanded image (Segment (F) of FIG. 44) of the rectangle contained in the second frame which now carries the character string “ABCD”, as the pixel value B₂′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the second frame, based on the same function Mψ as that used in the copying processing on the second frame. As a result, a second frame containing the projected image of the rectangle carrying the character string “ABCD” is obtained, as shown in Segment (E) of FIG. 43.

[0431] Likewise, a pasting processing is executed for the third frame. This is effected by writing into the buffer of interest (image buffer in which the third frame data is stored) the pixel value C₃′(x_e, y_e) of the point (x_e, y_e) on the expanded image (Segment (G) of FIG. 44) of the rectangle contained in the third frame which now carries the character string “ABCD”, as the pixel value B₃′(x_p, y_p) of the corresponding point (x_p, y_p) on the projected image of the third frame, based on the same function Mψ as that used in the copying processing on the third frame. As a result, a third frame containing the projected image of the rectangle carrying the character string “ABCD” is obtained, as shown in Segment (F) of FIG. 43.

[0432] Thus, the content rendered by the user is written in the overall expanded image storage area of the paint buffer as shown in Segment (D) of FIG. 44, and the entirety of the overall expanded image storage area is extracted as the cut-out region. Editorial work, which is the rendering of the character string “ABCD” in this case, is effected on the expanded images of the rectangle displayed in the first to third frames of the moving picture. It is therefore possible to obtain the desired two-dimensional moving picture, without taking the labor of repeatedly effecting the editorial work such as rendering of the character string on the expanded image of the three-dimensional object separately for the individual frames. In other words, for obtaining the moving picture carrying the character string “ABCD”, constituted by projected images of the rectangle presented by three frames, it suffices to render the character string “ABCD” only once.

[0433] A description will now be given of a copy and paste processing which is used in a case where different frames of a moving picture provide different visible portions of a three-dimensional object.

[0434] Segments (A) to (C) of FIG. 45 show images on first to third frames of a moving picture. A cylindrical body is shown as a moving three-dimensional object, by way of example. It is intended here that the moving picture is transformed into a moving picture which shows the cylindrical body carrying a character string “ABCD” on and along its peripheral surface and which is composed of first to third frames as shown in Segments (D) to (F) of FIG. 45.

[0435] Segments (A) to (C) of FIG. 45 show projected images of the cylindrical body which is rotating clockwise and moving to the left when viewed from a level slightly above the top surface thereof, as in the case of the images shown in Segments (A) to (C) of FIG. 42.

[0436] Since the peripheral surface of the cylindrical body is a three-dimensional curved surface, appearance of the displayed image, as well as the visible portion, i.e., the portion which appears on the displayed two-dimensional image, vary in accordance with rotation or translation of the cylindrical body (or position of the viewer) in the three-dimensional space. For instance, a portion of the peripheral surface which is visible in the image on a frame may completely disappear in the image on another frame of the moving picture.

[0437] As in the process described before, the copying operation is conducted by determining, based on the shape information and the characteristic point that are designated by the user, the function Mψ of the equation (10) which expresses the correlation between coordinates (x_p, y_p) of a point on the projected image of the cylindrical body displayed in the first frame (Segment (A) of FIG. 45) and the coordinates (x_e, y_e) of the corresponding point on the expanded image. Then, based on the function Mψ, the pixel value B₁(x_p, y_p) of a point (x_p, y_p) on the projected image in the first frame is written in the paste buffer as the pixel value C₁(x_e, y_e) of the corresponding point (x_e, y_e) on the expanded image, whereby an expanded image of the side or peripheral surface of the cylindrical body, corresponding to the projected image contained in the first frame, is obtained.

[0438] The object to be processed in this case is the side or peripheral surface of a cylindrical body. For the reasons stated before in connection with FIGS. 21A to 21C, the correlation between the coordinates (x₀, y₀, z₀) of a point on the peripheral surface in the three-dimensional space and the coordinates (x_e, y_e) of the corresponding point are given by the following equation (38). $\begin{array}{cc}\left(\begin{array}{c}{x}_o\\ y_o\\ z_o\end{array}\right)=\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\end{array}\right)=\left(\begin{array}{c}\frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(x_e-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ y_e\\ \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(x_e-\frac{L_{\mathrm{W1}}}{2}\right)\right)\end{array}\right)& \left(38\right)\end{array}$

[0439] The equation (38) is equivalent to the equation (17), since it is derived from the equation (17) by substituting the equation (15) or (16) for the element “r” or “φ” of the equation (17) and substituting O₁and L_W1for θ and L_Wof the equation (17), respectively. As stated before, the portion of the curved peripheral surface of the cylindrical body which is visible on the projected image varies in accordance with rotation o translation of the cylindrical body within the three-dimensional space. Therefore, in the equation (17), the value of the factor θ which indicates the angle formed around the center of the top surface of the cylindrical portion by the visible section of the peripheral surface (see FIG. 21A), as well as the value of the factor L_W(length of the side E₁E₂or E₄E₃shown in FIG. 21C) which indicates the width of the expanded image corresponding to the portion of the peripheral surface visible on the projected image, varies according to the frame of the moving picture. In the following description, therefore, symbols θ₁,L_W1; θ₂,L_W2; θ₃,L_W3are used to indicates the values of the factors θ and L_Wobtained with the first, second and third frames, respectively.

[0440] It is to be understood that the first to third frames show projected images of an identical cylindrical body. Therefore, the ratios L_W1/θ₁, L_W2/θ₂and L_W3/θ₃, indicative of the radii of the top surface of the cylindrical bodies in the images of the respective frames, have an identical value.

[0441] In regard to the peripheral surface of a cylindrical body, the correlation between the coordinates (x_p, y_p) of a point on the projected image and the coordinates (x_e, y_e) of the corresponding point on the expanded image is expressed by the following equation (39) which is obtained by incorporating the equation (38) in the equation (10). $\begin{array}{cc}\left(\begin{array}{c}w{x}_p\\ w{y}_p\\ w\end{array}\right)=M\left(\begin{array}{c}{\psi }_1\left(x_e,y_e\right)\\ {\psi }_2\left(x_e,y_e\right)\\ {\psi }_3\left(x_e,y_e\right)\\ 1\end{array}\right)=M\left(\begin{array}{c}\frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(x_e-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ y_e\\ \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(x_e-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ 1\end{array}\right)& \left(39\right)\end{array}$

[0442] As will be seen from the equation (11), the matrix M in the equation (39) has eleven elements: m₁₁to m₁₄, m₂₁to m₂₄and m₃₁to m₃₃. As explained before with reference to FIGS. 21A to 21C, these elements can be determined based on 6 (six) characteristic points: five points having two degrees of freedom and one point having one degree of freedom, and corresponding points on the paste buffer coordinate system, provided that such six characteristic points are designated. Using the matrix M thus determined, it is possible to transform the projected image of the peripheral surface of the cylindrical body into a corresponding expanded image, in accordance with the equation (39).

[0443] To explain in more detail, the user designates six points marked by black solid circles (●) in FIG. 46A which shows the projected image of the cylindrical body contained in the first frame. These six points are the same as the points P₁to P₆explained before in connection with FIG. 21B. The expanded image corresponding to the peripheral surface of the cylindrical body sown in the first frame has a rectangular shape. As explained before in connection with FIG. 21C, the apices of the rectangle are indicated by E₁, E₂, E₃and E₄, as mentioned in the clockwise direction starting from the left upper apex. The bisector point of the line E₁E₂and the bisector point of the line E₄E₃are respectively represented by E₅and E₆. The points P₁, P₂, P₃, P₄, P₅and P₆are respectively correlated to the points E₁, E₂, E₃, E₄, E₅and E₆. The coordinates (x_e, y_e) of the points E₁, E₂, E₃, E₄, E₅and E₆are respectively given as (0, 0), (L_W1, 0), (L_W1, L_H), (0, L_H), (L_W1/2, 0) and (L_W1/2, L_H). At the same time, the coordinates (x_p, y_p) of the points P₁, P₂, P₃, P₄, P₅and P₆designated as the characteristic points are respectively expressed by (x_p1, y_p1),(x_p2, y_p2), (x_p3, y_p3), (x_p4, y_p4), (x_p5, y_p5) and (x_p6, y_p6) Using these expressions, the following equation (40) is derived from the equation (39). The matrix M in the equation (39) can be determined by solving the equation (40). $\begin{array}{cc}\begin{array}{c}\left(\begin{array}{cccccc}{w}_1{x}_{\mathrm{p1}}& w_2{x}_{\mathrm{p2}}& w_3{x}_{\mathrm{p3}}& w_4{x}_{\mathrm{p4}}& w_5{x}_{\mathrm{p5}}& w_6{x}_{\mathrm{p6}}\\ w_1{y}_{\mathrm{p1}}& w_2{y}_{\mathrm{p2}}& w_3{y}_{\mathrm{p3}}& w_4{y}_{\mathrm{p4}}& w_5{y}_{\mathrm{p5}}& w_6{y}_{\mathrm{p6}}\\ w_1& w_2& w_3& w_4& w_5& w_6\end{array}\right)=M\left(\begin{array}{cccccc}{\psi }_1\left(0,0\right)& {\psi }_1\left(L_{\mathrm{W1}},0\right)& {\psi }_1\left(L_{\mathrm{W1}},L_H\right)& {\psi }_1\left(0,L_H\right)& {\psi }_1\left(\frac{L_{\mathrm{W1}}}{2},0\right)& {\psi }_1\left(\frac{L_{\mathrm{W1}}}{2},L_H\right)\\ {\psi }_2\left(0,0\right)& {\psi }_2\left(L_{\mathrm{W1}},0\right)& {\psi }_2\left(L_{\mathrm{W1}},L_H\right)& {\psi }_2\left(0,L_H\right)& {\psi }_2\left(\frac{L_{\mathrm{W1}}}{2},0\right)& {\psi }_2\left(\frac{L_{\mathrm{W1}}}{2},L_H\right)\\ {\psi }_3\left(0,0\right)& {\psi }_3\left(L_{\mathrm{W1}},0\right)& {\psi }_3\left(L_{\mathrm{W1}},L_H\right)& {\psi }_3\left(0,L_H\right)& {\psi }_3\left(\frac{L_{\mathrm{W1}}}{2},0\right)& {\psi }_3\left(\frac{L_{\mathrm{W1}}}{2},L_H\right)\\ 1& 1& 1& 1& 1& 1\end{array}\right)\\ =M(\begin{array}{ccc}\frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(0-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(L_{\mathrm{W1}}-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(L_{\mathrm{W1}}-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ 0& 0& L_H\\ \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(0-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(L_{\mathrm{W1}}-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(L_{\mathrm{W1}}-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ 1& 1& 1\end{array}\\ \begin{array}{ccc}\frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(0-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(\frac{L_{\mathrm{W1}}}{2}-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\sin \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(\frac{L_{\mathrm{W1}}}{2}-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ L_H& 0& L_H\\ \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(0-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(\frac{L_{\mathrm{W1}}}{2}-\frac{L_{\mathrm{W1}}}{2}\right)\right)& \frac{L_{\mathrm{W1}}}{{\theta }_1}\cos \left(\frac{{\theta }_1}{L_{\mathrm{W1}}}\left(\frac{L_{\mathrm{W1}}}{2}-\frac{L_{\mathrm{W1}}}{2}\right)\right)\\ 1& 1& 1\end{array})\end{array}& \left(40\right)\end{array}$