Video hand image three-dimensional computer interface

[0002] I. Field of the Invention

[0003] This invention pertains to a method and apparatus for inputting commands to a computer using hand signals. More particularly, the present invention relates to a video gesture-based computer interface wherein images of hand gestures are used to control a computer and wherein motion of the user's hand or a portion thereof is tracked in a three-dimensional coordinate system with five degrees of freedom.

[0004] II. Description of the Related Art

[0005] Various types of computer control and interface devices exist for inputting commands to a computer. Such devices may for example take the form of a computer mouse, joystick or trackball, wherein a user manipulates the interface device to perform a particular operation such as to select a specific entry from a menu of options, perform a “click” or “point” function, etc. A significant problem associated with such interface devices is that a surface area is needed for placement of the device and, in the case of a mouse, to accommodate device movement and manipulation. In addition, such interface devices are generally connected by a cable to a computer CPU with the cable typically draped across the user's desk, causing obstruction of the user's work area. Moreover, because interface device manipulation for performing operations is not consistent with common communication movements, such as the use of a pointing finger hand gesture to select a menu entry, as opposed to maneuvering a mouse until the cursor rests on the desired menu entry, a user must become comfortable and familiar with the operation of the particular interface device before proficiency in use may be attained.

[0006] To address these drawbacks, a video interface system for enabling a user to utilize hand gestures to issue commands to a computer has been developed and is described in commonly assigned U.S. patent application of Segen entitled “Video Hand Image Computer Interface”, Ser No. 08/887,765 filed Jul. 3, 1997 (hereinafter “Segen”), which is hereby incorporated herein by reference in its entirety. The Segen system, by way of preferred example, utilizes a video camera or other video input device connected to an image processing computer, with the camera positioned to receive images of an object such as a user's hand. The image processing capabilities of the computer act upon predetermined recognized hand gestures as computer commands. Hand images from the camera are converted to a digital format and input to the computer for processing. The results of the processing and attempted recognition of each image are sent to an application or the like for performing various functions or operations.

[0007] However, the use of both traditional two-dimensional input devices and the Segen system is problematic in advanced computer-based three-dimensional object selection and manipulation applications. In such applications, a virtual three-dimensional environment is typically displayed to the user with one or more displayed virtual objects and command menus positioned within the virtual environment. The user may delete, move and otherwise change the objects in the virtual environment or create new objects. The user may also select various commands from the command menus. Other functions may be performed in the virtual environment such, for example, as drawing curves. Traditional input devices are extremely difficult to use in such a virtual environment because traditional devices control only two degrees of freedom and, thus, a combination of several input devices or, with a single device, more than one operation, is required to control three or more degrees of freedom as is necessary in three-dimensional applications. Such a combination control scheme is cumbersome, unintuitive and requires significant training on the user's part. The Segen system provides for three degrees of freedom which is more than adequate for issuing commands, but may not be sufficient for use in certain three-dimensional applications where interaction with three-dimensional objects is necessary.

[0008] It would thus be desirable to provide a computer interface that enables common and intuitive hand gestures and hand motions to be used for interacting with a three-dimensional virtual environment. It would further be desirable to provide a system and method for tracking hand gestures and hand motions in a three-dimensional coordinate system with five degrees of freedom.

[0009] This invention relates to the use of images input to a computer for effectuating computer control. The present invention provides a system and method for tracking the three-dimensional position and orientation of a finger of the user's hand with five degrees of freedom. The tracking of the user's finger by the inventive system enables the user to utilize and control computer applications requiring three-dimensional interaction.

[0010] The system includes a computer for controlling operation of the system, having image processing capabilities and a display for displaying a virtual environment to a user, and at least two video input devices, such as video cameras, connected to the computer for acquiring video images and for enabling the computer to operatively identify, interpret, and track motions and gestures of a user's hand made by the user within an identification zone monitored by the video input devices (hereinafter “cameras”). The identification zone is an area that is typically, although not necessarily, near the computer system towards which the first and second cameras are oriented. The configuration and functionality of the cameras and the computer are similar to the above-incorporated Segen system, except that while the Segen system employs only a single camera, the system of the present invention utilizes two or more cameras.

[0011] It should be noted that while the inventive system is largely based on the above-incorporated Segen system, the system of the present invention provides important additional inventive features that are essential for enabling the user to control applications requiring three-dimensional interaction. These additional features are possible because the inventive system uses two or more cameras to simultaneously capture images of the user's hand, while the Segen system employs only a single camera to capture a single image.

[0012] During operation of the system, hand images from the cameras are continually converted to a digital format and input to the computer for processing. As in the Segen system, the results of the processing and attempted recognition of each image are then sent to an application or the like for performing various functions or operations. However, when the inventive computer recognizes a hand gesture as a “point” gesture with one finger extended, the computer uses information derived from the images to track three-dimensional coordinates of the extended finger of the user's hand with five degrees of freedom. Essentially, the computer utilizes two-dimensional images obtained by each camera to derive three-dimensional position (in an x, y, z coordinate system) and orientation (azimuth and elevation angles) coordinates of the extended finger. This feature is not possible in the Segen system because at least two cameras are necessary to generate the information required for three-dimensional position and orientation tracking. Thus, the present invention provides an advantageous supplement to the functionality of the Segen system because, in addition to issuing commands to a computer using hand gestures (as described in detail in Segen), the inventive system enables user control of applications that require three-dimensional interaction.

[0013] The inventive method of tracking three-dimensional coordinates of the user's finger with five degrees of freedom is performed as follows. Initially, the two cameras are calibrated to derive a perspective projection matrix for each camera. The computer then simultaneously acquires a first image of the user's hand from the first camera and a second image of the user's hand from the second camera. The images are then classified by the computer to identify the class of each image. This classification process is performed in the same manner as in the Segen system. Once the class of each image is identified, the computer determines whether both images belong to the “point” class (where one of the user's fingers is extended) and, in the process of doing so, also identifies a portion of each image corresponding to the user's extended finger, such as the index finger. If both images belong to the “point” class, the computer determines pixel coordinates of the “tip” of an extended finger of the user's hand and the orientation angle of the extended finger's axis for each of the two images in accordance with the pixel coordinate and orientation angle determination technique used by the Segen system.

[0014] The computer first generates a first virtual line from the first image pixel coordinates of the tip of the extended finger of the user's hand and the orientation angle of the extended finger's axis, and similarly generates a second virtual line from the second image pixel coordinates and orientation angle. The computer then determines the three-dimensional coordinates of the tip of the extended finger by locating a virtual intersection of the first and second virtual lines that corresponds to the fingertip's three-dimensional position. Thus, the computer tracks the three-dimensional coordinates of the tip of the user's extended finger by generating the first and second virtual lines and determining, for each first and second image frame, the coordinates of the virtual intersection.

[0015] To determine the three-dimensional orientation of the extended finger, i.e. the azimuth and elevation angles, the computer generates, for the first image, a first parameter vector representative of a first three-dimensional virtual plane containing the axis of the extended finger as viewed from the position of the first camera. The computer also generates, for the second image, a second parameter vector representative of a second three-dimensional virtual plane containing the axis of the extended finger as viewed from the position of the second camera. The computer then generates the first and second virtual planes and determines a third virtual line corresponding to a line formed by intersection of the first and second virtual planes. Finally, the computer derives the azimuth and elevation angles of the extended finger from the parameters of the third virtual line. Thus, the computer tracks the three-dimensional orientation of the user's extended finger by generating the first and second virtual planes lines and determining, for each first and second image frame, the parameters of the third virtual line formed by the intersection of the first and second virtual planes.

[0016] Other objects and features of the present invention will become apparent from the following detailed description considered in conjunction with the accompanying drawings. It is to be understood, however, that the drawings are designed solely for purposes of illustration and not as a definition of the limits of the invention, for which reference should be made to the appended claims.

[0017] In the drawings, wherein like reference characters denote corresponding or similar elements throughout the various figures:

[0018] FIG. 1 is a block diagram of a video hand image three-dimensional computer interface system in accordance with the present invention;

[0019] FIG. 2 is a logic flow diagram of a three-dimensional position and orientation tracking control program executed by the computer of the system of FIG. 1, in accordance with the principles of the present invention;

[0020] FIG. 3 is a diagram of a relationship between a pixel position in an image plane and a point in real space with respect to a lens of a camera of the computer interface system of FIG. 1;

[0021] FIG. 4 is a representation of the coordinates of a user's extended finger in the image coordinate system; and

[0022] FIG. 5 is a diagram of an image plane and lens axis of each of the lenses of the cameras of the computer interface system of FIG. 1.

[0023] The system and method of the present invention are based on the above incorporated Segen system and preferably include substantially all of the functionality of the Segen system. Accordingly, certain functions of the Segen system that are used in the inventive system are not described in detail herein. While incorporating the functionality of the Segen system, the system of the present invention provides important additional inventive features that cannot be implemented in the Segen system and that are essential for enabling user control of applications requiring three-dimensional interaction. In particular, the system of the present invention provides three-dimensional tracking of a user's finger with five degrees of freedom.

[0024] To facilitate a better understanding of the inventive system it would be helpful to provide a summary of the Segen system on which the inventive system is based. In summary, the Segen system comprises a video input device, such as a camera, for capturing images of a user's hand, that is connected to a computer with image processing capabilities for operatively interpreting the received images. The image processing capabilities of the computer act upon predetermined recognized hand gestures as computer commands. Hand images from the camera are converted to a digital format and input to the computer for processing. The results of the processing and attempted recognition of each image are sent to an application or the like for performing various functions or operations. The process of identifying a hand gesture in an image frame is illustratively performed by conducting a region extraction step which defines a set of regions containing or possibly containing an image of a hand in a video frame of a video input signal. The set of regions in the image is then processed by a screening step which detects and selects one of the defined regions of the image as a hand image (i.e. identifies one of the regions in the image as a hand). If a hand image is thereby identified, then the set of localized features of the hand image is computed from the boundary of the selected hand region. The boundary of the hand region is thereafter analyzed to correlate the boundary data with one of a stored plurality of hand gestures. Once a particular gesture is determined or identified in the frame, an application or operation associated with the identified gesture is performed by or under the control of the computer. When the identified gesture is a pointing gesture, the Segen system also determines pixel coordinates p(x, y) of the tip and the orientation angle of the user's extended finger.

[0025] Referring now to the drawings, and initially to FIG. 1 thereof, a video gesture-based three-dimensional interface system 10 is there shown. The system 10 is formed of or includes a computer 12, such as a personal computer or a workstation, for controlling the system's operation, a display unit 14, such as a display monitor, for displaying a three-dimensional environment to a user, and a pair of video acquisition devices, such as cameras, 16 and 18, connected to computer 12, for monitoring a predefined identification zone 28 to enable the computer 10 to identify hand gestures and movements made by a user's hand 26, finger 24, and fingertip 25 within the identification zone 28 and for converting the hand gestures and hand movements to digital signals that are preferably interpreted as various commands. A lens axis 20 is representative of the orientation of the camera 16; similarly, a lens axis 22 is representative of the orientation of the camera 18. The cameras 16 and 18 are preferably synchronized to work in unison so that, at any given time, the images from all of the cameras correspond to the same scene.

[0026] While the system 10 is described herein with specific reference to a pair of cameras 16 and 18 connected to the computer 10, it should be understood that three or more cameras may be connected to the computer 10 and oriented to observe the user's hand 26 within the identification zone 28 without departing from the spirit of the invention. For example, a third camers 19 may be attached to the computer 12 and also oriented toward the identification zone 28. Preferably, the identification zone 28 is defined in an area proximate to the user so that the user may comfortably make desired hand gestures and movements within the zone 28. Proximity of the identification zone 28 to the display unit 14, while often convenient or desirable, is not required. Furthermore, while the system 10 is described herein with specific reference to the user's hand 26, finger 24, and fingertip 25, it should be understood that another elongated object with a tip may be used as a substitute for the user's hand 26 and extended finger 24 without departing from the spirit of the invention.

[0027] The operation of the system 10 is generally controlled by a control program stored in a memory (not shown) of the computer 12 and executed by the computer. This control program typically consists of multiple integrated program modules, with each module bearing responsibility for controlling one or more functions of the system 10. For example, one program module may provide one or more functions of the Segen system, such as identifying a hand gesture in an image frame, while another module may control determination of pixel coordinates of a finger 24 of the user's hand 26. In effect, each program module is a control program dedicated to a specific function or set of functions of the system 10.

[0028] FIG. 2 depicts a logic flow diagram representing the control program, executed by the computer 12 of FIG. 1 in accordance with the present invention, for controlling the identification and interpretation of the user's hand gestures and for tracking three-dimensional position and orientation coordinates of an extended finger 24 of the user's hand 26 with five degrees of freedom. Preferably, the control program is executed in conjunction with a Segen system program module that correlates a predetermined hand gesture to the identified hand image for effectuating a particular computer operation or function. Thus, in effect the present invention incorporates the most important feature of the Segen system, i.e. the correlation of a hand image to a hand gesture and effectuation of a computer command associated with the gesture, in addition to the advantageous features of the present invention described below in connection with FIG. 2. The correlation program module is not shown herein, but is described in detail in the above-incorporated co-pending Segen application.

[0029] The computer 12 first initializes the system 10 at a step 100 and then proceeds to a step 102 at which the cameras 16 and 18 are calibrated to determine perspective projection matrices M₁and M₂, respectively. The matrices M₁and M₂, are used by the computer 12 to compensate for any differences between internal parameters and external parameters of the respective cameras 16 and 18. Camera calibration is a complex task that may be performed in accordance with any of a variety of well known calibration approaches. However, it may prove helpful to describe a preferred approach as is done below. The image of a scene observed by a camera is generated in accordance with two types of parameters called internal parameters and external parameters. Internal parameters include the focal length of the camera lens, the distance between the pixels, size of each pixel and lens distortion coefficients. External parameters include the camera's position and orientation relative to the scene observed by the camera or with respect to a user-defined “real world” coordinate system. These parameters determine the image pixel coordinates (x, y) in a camera image plane, a pixel representation of the scene observed by the camera, that correspond to a three-dimensional point (X, Y, Z) in the real world coordinate system.

[0030] This arrangement is illustrated in FIG. 3 in perspective projection form for the camera 16. Light bouncing off of a three-dimensional point P(X, Y, Z) in the real world passes through the center of a camera lens 30 along a line 34 and meets an image plane 32 at pixel location p(x, y) to form the image of point P. A three-dimensional point can potentially lie anywhere along the line 34 and still be imaged at point P. When a camera's internal and external parameters are known it is possible to trace the line 34 for each image point p(x, y) backwards through the lens 30 center into the 3D world. In other words, for each image point p, we can compute the line 34 in the world coordinate system on which the corresponding three-dimensional point P should lie. It should be noted that when using a single camera 16, only the line 34 may be computed—the computer 12 cannot determine the exact position of three-dimensional point P along line 34. In order to determine the exact three-dimensional coordinate position of point P, at least two cameras 16 and 18 are necessary, as described below in connection with FIG. 5.

[0031] Calibration of a camera typically involves determining a perspective projection matrix M that encapsulates the camera's internal and external parameters using data derived from a perspective projection arrangement as shown in FIG. 3 and described above. There are several known approaches for determining the matrix M of a camera. A superior approach is shown below as Equation 1.

[0032] Mathematically, the matrix M can be derived from the relationship between the coordinates of three-dimensional point P (X, Y, Z) and image point p (x, y) which can be expressed as follows:

[0033] [s*x, s*y, s]{circumflex over ( )}T=M*[X, Y, Z,1]{circumflex over ( )}T  [Equation 1]

[0034] where M is a 4×4 perspective projection matrix, ‘s’ is a scale parameter and [. . . ]{circumflex over ( )}T denotes matrix transpose.

[0035] Preferably, the computer 12 determines the matrices M₁and M₂for the cameras 16 and 18, respectively, in accordance with Equation 1. Alternately, the camera calibration may be performed using other well-known techniques that do not require determination of the matrices M₁and M₂.

[0036] At a step 104, the computer 12 acquires IMAGE_1 of the user's hand 26 from a current image frame of the camera 16 and IMAGE_2 of the user's hand 26 from the current image frame of camera 18. The computer 12 also analyzes IMAGE_1 and IMAGE_2 to determine whether each of the images contains a hand gesture made by the user's hand 26. The image acquisition and analysis to determine presence of hand gestures is performed in a similar manner to the Segen system as described above, except that two images IMAGE_1 and IMAGE_2 are obtained and analyzed instead of a single image. Once presence of a hand gesture is determined in each image IMAGE_1 and IMAGE_2. the computer 12 proceeds to a step 106.

[0037] At step 106, the computer 12 classifies the hand gesture in each image IMAGE_1 and IMAGE_2 as belonging to a particular predetermined class of hand gestures. The system 10 recognizes four classes of hand gestures:

[0038] (1) POINT: This class corresponds to a gesture in which one finger 24 (typically, the index finger) of the user's hand 26 is extended while the other fingers are closed. Preferably, if the user extends two proximal fingers, then the computer 12 identifies a left of the two fingers as a primary extended finger when the hand 26 is a right hand, and identifies a right of the two fingers as a primary extended finger when the hand 26 is a left hand.

[0039] (2) REACH: This class corresponds to the gesture in which all of the fingers of the hand 26 are outstretched or extended.

[0040] (3) FIST: This class corresponds to a gesture in which the hand 26 is closed to form a fist.

[0041] (4) GROUND: This class covers all other gestures and the case where the hand 26 is not visible.

[0042] Once each hand gesture in IMAGE_1 and IMAGE_2 is classified, the computer 12 proceeds to a test 108, where it determines whether both hand gestures in IMAGE_1 and IMAGE_2 belong to the POINT class. If only one hand gesture in IMAGE_1 and IMAGE_2 belongs to the POINT class, then the computer 12 returns to the step 104 to acquire a next set of images IMAGE_1 and IMAGE_2 corresponding to a next image frame in each respective camera 16 and 18. Similarly if neither hand gesture in IMAGE_1 and IMAGE_2 belongs to the POINT if class, then the computer also returns to the step 104. On the other hand, if both hand gestures in IMAGE_1 and IMAGE_2 belong to the POINT class, then computer 12 proceeds to a step 110.

[0043] At the step 110, the computer 12 identifies and locates the fingertip 25 of the extended finger 24, preferably in accordance with the fingertip identification technique used by the Segen system. Identification of the fingertip 25 is important because the user uses the fingertip 25 to point out and manipulate three-dimensional objects in a computer application requiring three-dimensional interaction. The computer 12 then determines a first set of pixel coordinates p₁(x₁,y₁) of the fingertip 25 and an orientation angle of the finger 24 from IMAGE_1, and also determines a second set of pixel coordinates p₂(x₂,y₂) of the fingertip 25 and an orientation angle of the finger 24 from IMAGE_2. A diagram representing exemplary coordinates and orientation angle is shown in FIG. 4. Preferably, the determination of pixel coordinates and the orientation angle for each image is performed in accordance with the technique used by the Segen system.

[0044] Given p₁(x₁,y₁) from IMAGE_1, and p₂(x₂,y₂) of IMAGE_2, the first objective of the computer 12 is to determine P(X, Y, Z) i.e. the three-dimensional coordinates of a point P in the real world scene in the identification zone 28 that corresponds to the actual position of the fingertip 25 in the scene. At a step 112, the computer 12 determines P(X,Y,Z) from the pixel point coordinates p₁(x₁,y₁) and p₂(x₂,y₂) as follows. Referring to FIG. 5, the image plane 32, lens 30, and lens axis 20 of the camera 16 are shown along with an image plane 38, lens 36, and lens axis 22 of the camera 18. The pixel point coordinates p₁(x₁,y₁) and p₂(x₂,y₂) are shown on image planes 32 and 38, respectively. The computer 12 computes the parameters of a first virtual line 40 using any well-known geometric line equation with parameters corresponding to p₁(x₁,y₁), and similarly generates the parameters of a second virtual line 42 using parameters corresponding to p₂(x₂,y₂). The computer 12 then determines P(X, Y, Z) by locating a “virtual intersection” 44 of the virtual lines 40 and 42 that corresponds to the fingertip 25 three-dimensional position. By “virtual intersection” of two lines, we mean a point that is equidistant from both lines while being as close to each line as possible. Thus, the computer 12 continually tracks the three-dimensional coordinates P(X, Y, Z) of the fingertip 25 by generating the parameters of virtual lines 40 and 42 and determining, for each image frame of IMAGE_1 and IMAGE_2, the coordinates of the virtual intersection 44.

[0045] Given p₁(x₁,y₁), from IMAGE_1, and p₂(x₂,y₂), from IMAGE_2, the second objective of the computer 12 is to determine AZIMUTH and ELEVATION angles of the finger 24, that together with the point P coordinates X, Y, and Z enable the system 10 to track the user's hand 26 and finger 24 with five degrees of freedom. At a step 114, the computer 12 determines a first parameter vector representative of a first three-dimensional virtual plane defined along a first projected axis of the finger 24 derived by the computer 12 from IMAGE_1, and also generates a second parameter vector representative of a second three-dimensional virtual plane defined along a second projected axis of the finger 24 derived by the computer 12 from IMAGE_2. The computer 12 then generates the parameters of the first and second virtual planes in accordance with the respective first and second parameter vectors and determines an intersection line corresponding to a line formed by intersection of the first and second virtual planes. The computer then derives AZIMUTH and ELEVATION angles of the finger 24 from parameters of the intersection line using a set of any well-known trigonometric angle determination expressions. Thus, the computer 12 continually tracks the three-dimensional orientation (AZIMUTH and ELEVATION angles) of the finger 24 by generating the parameters of the first and second virtual planes and determining, for each image frame of IMAGE_1 and IMAGE_2, the parameters of the intersection line formed by the intersection of the first and second virtual planes. The computer 12 then returns to step 104 where image frames of IMAGE_1 and IMAGE_2 are acquired.

[0046] While there are a number of well known mathematical approaches for determining the above-described first and second parameter vectors from p₁(x₁,y₁), and p₂(x₂,y₂), respectively, an exemplary approach is illustrated below.

[0047] A line in a two dimensional plane can be represented by the equation

[0048] a*x+b*y+c=0  [Equation 2]

[0049] where (x,y) is an arbitrary point on the line and where a, b and c denote the parameters of the line. Equation 2 can be expressed in matrix notation as

[0050] v{circumflex over ( )}T*u=0  [Equation 3]

[0051] where v=[x, y, 1]{circumflex over ( )}T and u=[a, b, c]{circumflex over ( )}T. The vector u denotes the parameters of the line and [. . . ]{circumflex over ( )}T denotes matrix transpose.

[0052] Similarly, a three-dimensional plane may be represented by the equation

[0053] a′*x+b′*y+c′*z+d′=0  [Equation 4]

[0054] where (x, y, z) is an arbitrary point on the three-dimensional plane and where a′, b′, c′, and d′ denote the parameters of the plane. In matrix notation, the above equation becomes

[0055] v′{circumflex over ( )}T*u′=0  [Equation 5]

[0056] where v′=[x, y, z, 1]{circumflex over ( )}T and u′=[a′, b′, c′d′]{circumflex over ( )}T. The vector u′ denotes the parameters of the line.

[0057] Given an image p₁(x₁,y₁), of the fingertip 25 and its orientation () in IMAGE_1, an image line representing the projected axis of the finger 24, as it appears in IMAGE_1, has the parameters given by:

[0058] a=cos(), b=−sin(), and c=y₁*sin()−x₁*cos().  [Equations 6]

[0059] Hence, in accordance with Equation 3, the projected axis may be expressed as follows:

[0060] v{circumflex over ( )}T*u=0  [Equation 7]

[0061] where v=[x, y, 1]{circumflex over ( )}T, u=[cos(), −sin(), y₁*sin()−x₁*cos()] and where (x, y) is an arbitrary point on the projected axis.

[0062] Multiplying both sides of Equation 7 by a scale factor “s”, the equation may be rewritten as:

[0063] w{circumflex over ( )}T*u=0  [Equation 8]

[0064] where w=[s*x, s*y, s]{circumflex over ( )}T.

[0065] An arbitrary three-dimensional point (X, Y, Z) in the real world coordinate system is then selected. The relationship between the “real-world” point (X, Y, Z) and the image point p(x,y) is shown in Equation 1. Equation 1 may be rewritten in a matrix notation as:

[0066] w=M1*W  [Equation 9]

[0067] where w=[s*x, s*y, s]{circumflex over ( )}T, W=[X, Y, Z, 1] and M₁is the projection matrix determined at the step 102 for camera 16.

[0068] Since point (x, y) lies on the projection axis of finger 24, it satisfies Equation 8. Combining Equations 8 and 9, the following equation is formed:

[0069] W{circumflex over ( )}T*M1{circumflex over ( )}T*u=0  [Equation 10]

[0070] Comparing the above equation with Equation 4, the first parameter vector for the first three-dimensional virtual plane containing the projection axis of the finger 24 from IMAGE_1 may be determined as:

[0071] u′=M1{circumflex over ( )}T*u  [Equation 11]

[0072] The above-described procedure is then similarly performed using p₂(x₂,y₂), to determine the second parameter vector for the second three-dimensional virtual plane.

[0073] Thus, while there have shown and described and pointed out fundamental novel features of the invention as applied to preferred embodiments thereof, it will be understood that various omissions and substitutions and changes in the form and details of the methods disclosed and devices illustrated, and in their operation, may be made by those skilled in the art without departing from the spirit of the invention. For example, although the preferred embodiment employs images of a hand to effectuate computer control, images of other objects may, likewise, be employed such, for example, as images of a pencil or other pointing-type object to mark or designate menu entries or features on a computer screen, etc. Accordingly, it is expressly intended that all combinations of those elements and/or method steps which perform substantially the same function in substantially the same way to achieve the same results are within the scope of the invention. It is the intention, therefore, to be limited only as indicated by the scope of the claims appended hereto.