System for detecting the shape of an endoscope using source coils and sense coils

[0002] 1. Field of the Invention

[0003] The present invention relates to a system for detecting the shape of an endoscope inserted into a lumen using source coils and sense coils.

[0004] 2. Description of the Related Art

[0005] In recent years, endoscopes have been widely used in the fields of medicine and industry alike. Endoscopes with a soft insertion unit may be inserted into a tortuous body cavity and thus make it possible to diagnose an organ located in a deep region in a body cavity without the necessity of incision. Moreover, the endoscopes with a soft insertion unit sometimes have, if necessary, a treatment appliance inserted into a channel and thus make it possible to conduct therapeutic treatments including resection of a polyp or the like.

[0006] The lumen in a body cavity such as a large intestine or small intestine, is tortuous. It is hard for an operator to see at what position the inserted insertion unit of an endoscope is located or in what form the insertion unit is inserted.

[0007] In this case, for example, when the lumen of the inferior alimentary track is examined by inserting the insertion unit through the anus, some expertise is needed for inserting the insertion unit into the tortuous lumen.

[0008] During insertion, for smoother insertion, a bendable part of the insertion unit must be bent in conformity with the curve of the lumen. The procedure would be more convenient if the position in a body cavity or if the current bent state of the insertion unit could be understood.

[0009] In the past, X rays have been irradiated externally to a subject, into which the insertion unit of an endoscope is inserted, in order to detect an inserted state. The inserted state means a position in a lumen, at which the insertion unit is located, or an inserted form of the insertion unit. However, X rays are harmful to a human body. Besides, a place of irradiation in which X rays are irradiated is limited. Thus, X rays are not a preferable means for detecting the inserted state of the insertion unit of an endoscope.

[0010] Various endoscopes and apparatuses have been proposed for detecting the inserted state of an insertion unit of an endoscope in a lumen in a body cavity or the inserted state of a catheter thereof by utilizing magnetic fields without a physiologically adverse effect upon a human body.

[0011] However, a technology for indicating the positional relationship of an endoscope to the exterior of a body cavity has not been disclosed. Since the positional relationship between the endoscope and the exterior of the body cavity is unknown, into what region in a patient's body the insertion unit of the endoscope has been inserted cannot be ascertained or in what direction the insertion unit should be inserted cannot be determined. Moreover, since the positional relationship between the insertion unit of the endoscope and an operator's hand is unknown, what region should be compressed manually cannot be determined.

[0012] Additionally, a conventional CRT generally adopted as a monitor deflects an electron beam using magnetic fields and therefore generates unwanted magnetic fields. Assuming that the CRT is used as an observation monitor for displaying an endoscopic image or a shape depiction monitor for depicting the shape of an endoscope for a conventional endoscope shape detection system employing magnetic fields, a magnetic detection device detects magnetic fields generated by a magnetic generation device while being affected by the unwanted magnetic fields generated by the CRT. This poses a problem in that the shape of the endoscope cannot be depicted on a stable basis.

[0013] According to a conventional display method implemented in a detection system for ascertaining the shape of an endoscope when inserted, if a patient changes his/her position, the positions and angles determining the shape of an endoscope are changed. Consequently, the user would have to re-set his/her line of sight. Moreover, the shape of the. entire endoscope is depicted on the display screen, including regions in which the user is not interested, for example, the exterior of a patient's body. The user has to therefore distinguish a region which the user wants to scrutinize, for example, an intracorporeal region of a patient, on the display screen.

[0014] According to the display method implemented in the conventional inserted state detection system, an estimated shape of an endoscope is depicted with a reference point in a space of detection aligned with the center of the display area on the monitor. A user must modify the setting of a magnification whenever the user finds it necessary. Showing the shape of an endoscope enlarged is known.

[0015] However, although an image showing the shape of an endoscope is not displayed in the center of the display area, if the image of the shape is enlarged, part of the image of the shape comes out of the display area. It is impossible to enlarge the image by making the most of the display area. Even when the image of the shape is enlarged and displayed in an easy-to-see manner, the position of the endoscope changes from one patient to another patient. Consequently, the position of the image of the shape of an endoscope differs and part of the image comes out of the display area. This causes a user to re-set a magnification of the displayed image.

[0016] When the insertion unit of an endoscope is inserted into the large intestine or the like, the insertion unit may loop spirally. The insertion of the looping insertion unit pains a patient. Whether an operator recognizes the looping depends conventionally on the operator's skill. This leads to a drawback when looping cannot be recognized readily.

[0017] Assuming that the insertion unit of an electronic endoscope is inserted into a body cavity, that a region to be observed is imaged, and that an endoscopic image is viewed through a monitor, it is a matter of common practice that endoscopic images are frozen to produce a still picture which is recorded and viewed for diagnosis. When an image to be frozen appears, a Freeze switch located on an operation unit is pressed. Thus, data of a desired image is recorded in a frame memory or the like in a video processor and then output to the monitor or the like. Eventually, a still picture is displayed.

[0018] However, when the Freeze switch is pressed, if the distal part of the insertion unit in which an imaging device is incorporated moves, a color mismatch will be observed in a still picture.

[0019] With conventional inserted state detection systems, a user converts images of the shape of an endoscope, which are displayed in the form of a motion picture, into a video signal. The images are then recorded on a video tape for future use in diagnosis or analysis succeeding an examination. The same picture as that viewed during the examination only can be reproduced from the recorded video tape. It is impossible to observe the shape of the endoscope in different directions.

[0020] When a conventional endoscope shape detection system is employed, extracorporeal markers are used to mark specified positions on the body surface of a patient for a better understanding of the positional relationship between an endoscope and the patient body. An operator or paramedic must manually affix the extracorporeal markers to the specified positions on the body surface, or fasten them using a tape or the like. The specified positions on the body surface serve as reference positions, for example, such as a position near the anus.

[0021] However, according to the foregoing fastening method, if the extracorporeal markers are removed to allow the patient to change his/her position, it is hard to place the markers at the same positions.

[0022] An object of the present invention is to provide an endoscope shape detection system making it possible to readily grasp the positional relationship between the shape of an insertion unit of an endoscope and the exterior of a body cavity.

[0023] Another object of the present invention is to provide an endoscope shape detection system capable of suppressing the adverse effect of unwanted magnetic fields generated by a monitor, and depicting the shape of an endoscope in a more stable manner.

[0024] Still another object of the present invention is to provide an endoscope shape detection system making it possible to view the shape of an endoscope with a line of sight set in an easy-to-see direction while being unaffected by a change in a patient's position.

[0025] Still another object of the present invention is to provide an endoscope shape detection system capable of depicting the shape of an endoscope without causing part of the shape to come out of a display area. Specifically, when the shape of an endoscope is depicted at a size permitting a user to find depiction easy-to-see, even if a subject is changed to another having a different size, no part of the shape will come out of the display area.

[0026] Yet another object of the present invention is to provide an endoscope shape detection system making it possible to recognize the looping of an insertion unit, as it is being inserted, in the course of detecting the shape of an endoscope.

[0027] Yet another object of the present invention is to provide an endoscope shape detection system capable of producing a desired still picture by freezing images according to a motion made by an insertion unit.

[0028] Yet another object of the present invention is to provide an endoscope shape detection system making it possible to observe the shape of an endoscope in an easy-to-see manner all the time even during diagnosis or analysis succeeding an examination.

[0029] Yet another object of the present invention is to provide an extracorporeal marker fastening device for an endoscope shape detection system. The extracorporeal marker fastening device makes it possible to set extracorporeal markers at the same positions even when they had to be removed to allow a patient to change his/her position.

[0030] An endoscope shape detection system in accordance with the present invention consists mainly of a first coil means, a second coil means, a third coil means, a transmitting and receiving means, and an arithmetic means. The first coil means is inserted into a subject. The second coil means is located at a predetermined position. The third coil means can be located at any position on the subject. The transmitting and receiving means permits transmission and reception of a first magnetic signal between the first coil means and second coil means, and transmission and reception of a second magnetic signal between the third coil means and second coil means. The arithmetic means calculates first position information corresponding to the position of the first coil means relative to the second coil means according to a first detection signal resulting from transmission and reception of the first magnetic signal. The arithmetic means calculates second position information corresponding to the position of the third coil means relative to the second coil means according to a second detection signal resulting from transmission and reception of the second magnetic signal.

[0031] In the endoscope shape detection system of the present invention, the arithmetic means calculates the first position information corresponding to the position of the first coil means relative to the second coil means according to the first detection signal resulting from transmission and reception of the first magnetic signal. The arithmetic means also calculates the second position information corresponding to the position of the third coil means relative to the second coil means according to the second detection signal resulting from transmission and reception of the second magnetic signal. Consequently, the endoscope shape detection system makes it possible to readily grasp the positional relationship between the shape of the insertion unit of an endoscope and the exterior of a body cavity.

[0032] Other features and advantages of the present invention will become apparent from the description made below.

[0033] The invention is described in detail below with reference to the following figures, throughout which similar reference characters denote corresponding features consistently, wherein:

[0034] FIG. 1A to FIG. 39 relate to the first embodiment of the present invention;

[0035] FIG. 1A is a schematic view of an endoscopic system constructed according to principles of the invention;

[0036] FIG. 1B is a diagrammatic view of the embodiment of FIG. 1A;

[0037] FIG. 1C is a partial schematic view, drawn to an enlarged scale, taken along line 1cin FIG. 1A;

[0038] FIG. 1D is a partial schematic view, drawn to an enlarged scale, taken along line 1din FIG. 1A;

[0039] FIG. 2 is a block diagram of the endoscope shape detection system shown in FIG. 1A;

[0040] FIG. 3 is a block diagram of the endoscope shape detection system shown in FIG. 2;

[0041] FIG. 4 is a block diagram of a two-port memory of the endoscope shape detection system shown in FIG. 3;

[0042] FIG. 5 is a graphical view of signals and the relative timing thereof with reference to actions made by the two-port memory shown in FIG. 4;

[0043] FIG. 6 is a flowchart of operations of the endoscopic system shown in FIG. 1;

[0044] FIG. 7 is a flowchart of a sequence of a fast Fourier transform (FFT) of. FIG. 6;

[0045] FIG. 8 is a graphical view of signals and the relative timing thereof with reference to the timing of parallel processing operations of the endoscopic system of FIG. 6;

[0046] FIG. 9 is a diagrammatic view. of source coil coordinates used in a coordinates-of-estimated source coil position calculation of FIG. 6;

[0047] FIG. 10 is a diagrammatic view of source coil coordinates used in the coordinates-of-estimated source coil position calculation of FIG. 6;

[0048] FIG. 11 is a diagrammatic view of source coil coordinates used in the coordinates-of-estimated source coil. position calculation of FIG. 6;

[0049] FIG. 12 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0050] FIG. 13 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0051] FIG. 14 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0052] FIG. 15A and 15B are a diagrammatic views of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0053] FIG. 16 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0054] FIG. 17 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0055] FIG. 18 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0056] FIG. 19 is a diagrammatic view of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0057] FIG. 20 is a flowchart describing a sequence of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0058] FIG. 21 is a flowchart describing a sequence of the coordinates-of-estimated source coil position calculation method of FIG. 6;

[0059] FIG. 22 is a flowchart describing a sequence of the position updating control for controlling updating of an estimated position of a source coil determined according to the method of FIG. 20 and FIG. 21;

[0060] FIG. 23 is a flowchart describing a sequence of the endoscope shape detection image display of FIG. 6:

[0061] FIG. 24 is a display view of a display produced in a normal mode according to the method of FIG. 23;

[0062] FIG. 25 is a flowchart describing a sequence in an enlargement mode of FIG. 23;

[0063] FIG. 26 is a display view of a display produced in the enlargement mode of FIG, 25;

[0064] FIG. 27 is a diagrammatic view of image models of three-dimensional models 1 and 2 produced by the endoscope shape detection image display method of FIG. 6;

[0065] FIG. 28 is a flowchart describing an image mode display method for displaying the three-dimensional models 1 and 2 shown in FIG. 27;

[0066] FIG. 29 is a diagrammatic view of image models of three-dimensional models 1 and 2 produced by the endoscope detection image display method of FIG. 6;

[0067] FIG. 30 is a flowchart describing a sequence of tone correction method of FIG. 29;

[0068] FIG. 31 is a diagrammatic view of an effect of the tone correction method of FIG. 30;

[0069] FIG. 32 is a flowchart describing a sequence of the tone correction in FIG. 29;

[0070] FIG. 33 is a diagrammatic view of an effect of tone correction described in FIG. 30;

[0071] FIG. 34 is a flowchart describing an image model display method for displaying a two-dimensional model produced by the endoscope shape detection image display method of FIG. 6;

[0072] FIG. 35 is a display view of an image resulting from the endoscope shape detection image display method displayed according to the sequence described in FIG. 34;

[0073] FIG. 36 is a flowchart describing an image model display method for displaying a twelve-point model produced by the endoscope shape detection image display method of FIG. 6;

[0074] FIG. 37 is a display view of an image resulting from the endoscope shape detection image display method displayed according to the sequence described in FIG. 36;

[0075] FIG. 38 is a flowchart describing an image model display method for displaying a linear model produced by the endoscope shape detection image display method of FIG. 6;

[0076] FIG. 39 is a display view of an image resulting from the endoscope shape detection image display method displayed according to the sequence described in FIG. 38;

[0077] FIG. 40 to FIG. 42 relate to the second embodiment of the present invention;

[0078] FIG. 40 is a graphical view of a coordinates-of-estimated source coil position calculation method according to the second embodiment of the invention;

[0079] FIG. 41 is a flowchart describing a sequence of the coordinates-of-estimated source coil position calculation method of FIG. 40;

[0080] FIG. 42 is a flowchart describing the sequence of the coordinates-of-estimated source coil position calculation method of FIG. 40;

[0081] FIG. 43A to FIG. 45 relate to the third embodiment of the present invention;

[0082] FIG. 43A is a graphical view of a coordinates-of-estimated source coil position calculation method according to the third embodiment of the invention;

[0083] FIG. 43B is a partial graphical view, drawn to an enlarged scale, taken along line 43bin FIG. 43A;

[0084] FIG. 44 is a flowchart describing a sequence of the coordinates-of-estimated source coil position calculation method performed using two sense coils that define an angle θ, as indicated in FIG. 43, approximate to an angle defined by orthogonal coils;

[0085] FIG. 45 is a flowchart describing a sequence of the coordinates-of-estimated source coil position calculation method performed using two sense coils that define an angle θ, as indicated in FIG. 43, approximate to an angle defined by orthogonal coils;

[0086] FIG. 46 and FIG. 47 relate to the fourth embodiment of the present invention;

[0087] FIG. 46 is a graphical view, in a three-dimensional space, at which a source coil for generating magnetic fields is located;

[0088] FIG. 47 is a graphical view of positions of sense coils relative to the position of the source coil shown in FIG. 46 in accordance with the fourth embodiment;

[0089] FIG. 48A to FIG. 55 relate to the fifth embodiment of the present invention;

[0090] FIG. 48A is a schematic view of. an endoscope shape detection system constructed according to the fifth embodiment of the invention;

[0091] FIG. 48B is a partial schematic view, drawn to an enlarged scale, taken along line 48bof FIG. 48A;

[0092] FIG. 48C is a partial schematic view, drawn to an enlarged scale, taken along line 48cof FIG. 48A;

[0093] FIG. 49 is a block diagram of a control unit shown in FIG. 48A;

[0094] FIG. 50 is a flowchart describing operations of the endoscope shape detection system shown in FIG. 48A;

[0095] FIG. 51A and FIG. 51B are schematic views of the markers shown in FIG. 48A, that is, an anus marker, a left marker, and a right marker;

[0096] FIG. 52 is a graphical view of a vector calculated using the anus marker, left marker, and right marker shown in FIG. 51B, and specifying a patient plane on which a patient lies;

[0097] FIG. 53 is a graphical view of a vector calculated using the anus marker, left marker and right marker shown in FIG. 51B, and specifying the patient plane;

[0098] FIG. 54 is a graphical view of a coordinate transformation to be performed with respect to the vector specifying the patient plane of FIG. 52 and FIG. 53;

[0099] FIG. 55 is a graphical view of a coordinate transformation to be performed with respect to the vector specifying the patient plane of FIG. 52 and FIG. 53;

[0100] FIG. 56 to FIG. 58 relate to the sixth embodiment of the present invention;

[0101] FIG. 56 is a flowchart describing operations of the endoscope shape detection system;

[0102] FIG. 57 is a display view of a monitor display produced by the sequences of the flowchart of FIG. 56;

[0103] FIG. 58 is a display view of a monitor display produced by the sequences of the flowchart of FIG. 56;

[0104] FIG. 59A to FIG. 61 relate to the seventh embodiment of the present invention;

[0105] FIG. 59A is a schematic view of an endoscope shape detection system according to the seventh embodiment of the invention;

[0106] FIG. 59B is a partial schematic view, drawn to an enlarged scale, taken along line 59bin FIG. 59A;

[0107] FIG. 59C is a partial schematic view, drawn to an enlarged scale, taken along line 59cin FIG. 59A;

[0108] FIG. 59D is a partial schematic view, drawn to an enlarged scale, taken along line 59din FIG. 59A;

[0109] FIG. 59E is a partial schematic view, drawn to an enlarged scale, taken along line 59ein FIG. 59A;

[0110] FIG. 60A is a diagrammatic view of a control unit shown in FIG. 59A;

[0111] FIG. 60B is a partial schematic view, drawn to an enlarged scale, taken along line 60bin FIG. 60A;

[0112] FIG. 61A is a schematic view of a variant of the endoscope shape detection system shown in FIG. 59A;

[0113] FIG. 61B is a partial schematic view, drawn to an enlarged scale, taken along line 61bin FIG. 61A;

[0114] FIG. 61C is a partial schematic view, drawn to an enlarged scale, taken along line 61cin FIG. 61A;

[0115] FIG. 61D is a partial schematic view, drawn to an enlarged scale, taken along line 61din FIG. 61A;

[0116] FIG. 62 to FIG. 64B relate to the eighth embodiment of the present invention;

[0117] FIG. 62A is a schematic view of an endoscope shape detection system according to the eighth embodiment of the invention;

[0118] FIG. 62B is a partial schematic view, drawn to an enlarged scale, taken along line 62bin FIG. 62A;

[0119] FIG. 62C is a partial schematic view, drawn to an enlarged scale, taken along line 62cin FIG. 62A;

[0120] FIG. 62D is a partial schematic view, drawn to an enlarged scale, taken along line 62din FIG. 61A;

[0121] FIG. 63 is a block diagram of a control unit shown in FIG. 62A;

[0122] FIG. 64A is a diagrammatic view of an operator, a patient and the control unit shown in FIG. 63;

[0123] FIG. 64B is a diagrammatic view of the positions of marking on the present endoscope and an operator's hand;

[0124] FIG. 65A through FIG. 66 relate to the ninth embodiment of the present invention;

[0125] FIG. 65A is a schematic view of operations of the endoscope shape detection system;

[0126] FIG. 65B is a diagrammatic view of endoscope shape modeling performed by the endoscope shape detection systems;

[0127] FIG. 66 is a diagrammatic view of endoscope shape modeling performed by a variant of the endoscope shape detection system of FIG. 65A;

[0128] FIGS. 67A and 67B are schematic views of a patient undergoing operations of the endoscope shape detection system in accordance with the tenth embodiment of the present invention;

[0129] FIG. 67C is a diagrammatic view of endoscope shape modeling performed by the endoscope shape detection system of FIG. 67A;

[0130] FIG. 68 to FIG. 71 relate to the eleventh embodiment of the present invention;

[0131] FIG. 68 is a flowchart describing operations of an endoscope shape detection system according to the eleventh embodiment of the invention;

[0132] FIG. 69 is a display view of a display viewed during steps S303 and S304 of FIG. 68;

[0133] FIG. 70 is a display view of a display viewed during steps S303 and S304 of FIG. 68;

[0134] FIG. 71 is a display view of a display viewed during steps S303 and S304 of FIG. 68;

[0135] FIG. 72 to FIG. 74 relate to the twelfth embodiment of the present invention;

[0136] FIG. 72 is a flowchart describing operations of the endoscope shape detection system according to the twelfth embodiment of the invention;

[0137] FIG. 73 is a schematic view of a patient undergoing step S312 of FIG. 72;

[0138] FIG. 74 is a diagrammatic view of an endoscope shape during step S316 of FIG. 72;

[0139] FIG. 75 and FIG. 76 relate to the thirteenth embodiment of the present invention;

[0140] FIG. 75 is a flowchart describing operations of the endoscope shape detection system according to the thirteenth embodiment of the invention;

[0141] FIGS. 76A and 76B are display views of displays viewed during step S323 of FIG. 75;

[0142] FIG. 77 and FIG. 78 relate to the fourteenth embodiment of the present invention;

[0143] FIG. 77 is a display view of a display viewed with the endoscope shape detection system according to the fourteenth embodiment of the invention;

[0144] FIG. 78 is a display view of a display viewed with the endoscope shape detection system according to the fourteenth embodiment of the invention;

[0145] FIG. 79A to FIG. 87 relate to the fifteenth embodiment of the present invention;

[0146] FIG. 79A is a schematic view of an endoscopic system according to the fifteenth embodiment of the invention;

[0147] FIG. 79B is a partial schematic view, drawn to an enlarged scale, taken along line 79bin FIG. 79A;

[0148] FIG. 79C is a partial schematic view, drawn to an enlarged scale, taken along line 79cin FIG. 79A;

[0149] FIG. 79D is a partial schematic view, drawn to an enlarged scale, taken along line 79din FIG. 79A;

[0150] FIG. 80 is a partial side elevational view of an endoscope showing the positions of source coils shown in FIG. 79A;

[0151] FIG. 81 is a partial side elevational view of a variant of an endoscope showing the positions of the source coils shown in FIG. 79A;

[0152] FIGS. 82 to 85 are graphical views of application of principles of detection on which the endoscope shape detection system of FIG. 79 detects the looping of an insertion unit;

[0153] FIG. 86 is a flowchart describing a sequence for detecting insertion unit looping by the endoscope shape detection system shown in FIG. 79A;

[0154] FIG. 87 is a flowchart describing a sequence for detecting insertion unit looping by the endoscope shape detection system shown in FIG. 79A;

[0155] FIG. 88 and FIG. 89 relate to the sixteenth embodiment of the present invention;

[0156] FIG. 88 is a schematic view of the configuration of a CCU;

[0157] FIG. 89 is a flowchart describing a sequence of freeze control to be performed by the CCU shown in FIG. 88;

[0158] FIG. 90 is a flowchart describing a sequence of freeze control to be performed by a CCU in accordance with the seventeenth embodiment of the present invention;

[0159] FIG. 91 to FIG. 95 relate to the eighteenth embodiment of the present invention;

[0160] FIG. 91 is a schematic view of the configuration of a control unit;

[0161] FIG. 92 is a flowchart describing operations of an endoscope shape detection system having the control unit shown in FIG. 91;

[0162] FIG. 93 is a flowchart continuous of the flowchart of FIG. 92;

[0163] FIG. 94 is a flowchart continuous of the flowchart of FIG. 92;

[0164] FIG. 95 is a diagrammatic view of a format with which data is recorded by the control unit according to the flowcharts of FIG. 92 to FIG. 94;

[0165] FIG. 96 and FIG. 97 relate to the nineteenth embodiment of the present invention;

[0166] FIG. 96 is a flowchart describing operations of the endoscope shape detection system according to the nineteenth embodiment of the invention;

[0167] FIG. 97 is a flowchart continuous of the flowchart of FIG. 96;

[0168] FIG. 98 and FIG. 99 relate to the twentieth embodiment of the present invention;

[0169] FIG. 98 is an environmental view of a marker placement sheet;

[0170] FIG. 99 is a schematic view of a variant of the marker placement sheet shown in FIG. 98;

[0171] FIG. 100 is an environmental view of a marker placement sheet in accordance with the twenty-first embodiment of the present invention;

[0172] FIG. 101 to FIG. 109 relate to an extracorporeal marker employed in the embodiments of the present invention;

[0173] FIG. 101 is a partial cross-sectional view of the extracorporeal marker;

[0174] FIG. 102 is a side elevational view of the extracorporeal marker shown in FIG. 101;

[0175] FIG. 103 is a top right front view of a first variant of the extracorporeal marker shown in FIG. 101;

[0176] FIG. 104 is a top right front view of a second variant of the extracorporeal marker shown in FIG. 101;

[0177] FIG. 105 is a top right front view of a third variant of the extracorporeal marker shown in FIG. 101;

[0178] FIG. 106 is a top right front view of a fourth variant of the extracorporeal marker shown in FIG. 101;

[0179] FIG. 107 is a top right front view of a fifth variant of the extracorporeal marker shown in FIG. 101;

[0180] FIG. 108 is a top right front view of a sixth variant of the extracorporeal marker shown in FIG. 101;

[0181] FIG. 109 is a top right front view of a seventh variant of the extracorporeal marker shown in FIG. 101;

[0182] To begin with, prior to presentation of exemplary embodiments, an algorithm for estimating the three-dimensional positions of source coils using a plurality of pairs of sense coils will be described with reference to the first to seventh embodiments.

[0183] According to a related art disclosed in, for example, PCT No. WO94/04938, three coils having mutually orthogonal axes and which are fixed to predetermined positions are used to successively generate alternating gradients, which exhibit mutually orthogonal vectors, in a space. A uniaxial coil having a position in the space is indicated with certain coordinates and is used to measure a voltage induced across itself due to the alternating gradients generated by the coils having three different axes. The spatial coordinates of the uniaxial coil are detected based on the measured data.

[0184] However, according to the related art disclosed in the PCT No. WO94/04938, unless the frequency of a radiofrequency signal used to generate magnetic fields agrees with that of a frequency component sampled by a frequency sampling means, the frequency component does not exhibit a value, which should be sampled, due to a change in ambient temperature or a time-sequential change. Consequently, the position of an endoscope determined based on the value disagrees with an actual position. This brings about a possibility that an inserted state may not be able to be detected accurately.

[0185] Japanese Unexamined Patent Publication No. 9-28661 has proposed an endoscope shape detection system capable of detecting an inserted state of an endoscope. Herein, a frequency adjusting means is included for harmonizing the frequency of a radio-frequency signal with that of a reference signal. Consequently, the frequencies of the radio-frequency signal and reference signal can be harmonized with each other even in such an environment that the frequencies may disagree with each other due to a change in ambient temperature or a time-seauential change. Thus, setting remains unsusceptible to the change.

[0186] According to the related art disclosed in the PCT NO. WO94/04938, the position of a magnetic generation device is estimated based on output values of a plurality of detection devices. For this purpose, a plurality of triaxial coils, each composed of three orthogonal single-core coils, is needed. This results in a complex configuration.

[0187] According to the Japanese Unexamined Patent Publication No. 9-28661, the an endoscope shape detection system is adapted to an endoscopic system. In this case, a plurality of triaxial coils, each composed of three orthogonal single-core coils, is needed for estimating the position of a magnetic generation device according to output values of a plurality of detection devices. This also results in a complex configuration.

[0188] According to the Japanese Unexamined Patent Publication No. 9-28661, there is difficulty in strictly harmonizing frequencies of signal components with frequencies observed through Fourier transform or the like in the course of vector analysis. This causes a “leakage” in the frequency domain. A window function or the like must be employed in order to alleviate an adverse effect of the “leakage.”

[0189] Japanese Unexamined Patent Publication No. 10-332309 has proposed a method of-measuring the position of a coil. A detection device (or a magnetic generation device) composed of at least four single-core coils placed at different positions in the same direction along the same straight light is used to estimate a space in which the magnetic generation device (or detection device) exists. The method is characteristic of the decreased number of variables to be estimated.

[0190] According to the Japanese Unexamined Patent Publication No. 10-332309, the aforesaid problem can be solved. However, since sense coils each composed of four single-core coils are juxtaposed, errors in circles of magnetic fields, that is, errors in the center and radius of each circle, estimated by the sense coils are observed in the same direction. This poses a problem in that an error in the estimated three-dimensional position of a source coil is observed to be large in a specified direction, thus when the distance between the source coils and sense coils increases, the estimated three-dimensional positions of the source coils become uncertain.

[0191] In conjunction with the first to seventh embodiments, a description will be made of an estimation algorithm effective in diminishing an estimation error occurring when the three-dimensional position of a source coil is estimated using sense coils each composed of a plurality of singlecore coils.

[0192] As shown in FIG. 1A, an endoscopic system 1 of this embodiment has an endoscope system 2 for use in endoscopic examinations, and an endoscope shape detection system 3 to be used as an aid for the endoscopic examinations. The endoscope shape detection system 3 is used as an insertion aid means when an insertion unit 7 of an electronic endoscope 6 is inserted into a body cavity of a patient 5.

[0193] The electronic endoscope 6 has an operation unit 8 formed at the back end of the elongated flexible insertion unit 7. The operation unit 8 has a bending knob. A universal cord 9 extends from the operation unit 8 coupled to a video imaging system (or a video processor) 10.

[0194] The electronic endoscope 6 has a light guide passing through it. Illumination light emanating from a light source unit in the video processor 10 is transmitted over the light guide, and irradiated through an illumination window formed on the distal plane of the insertion unit 7. An area of interest of a patient is thus illuminated. Light reflected from an illuminated object, such as a lesion, is converged on an imaging device located on the image plane of an objective locked in an observation window adjacent to the illumination window. The imaging device photoelectrically converts the optical image formed thereon into a data signal.

[0195] The signal resulting from photoelectric conversion is processed by a video signal processing unit in the video processor 10, whereby a standard video signal is produced. The images carried by the video signal are displayed on an image observation monitor 11 connected to the video processor 10.

[0196] As shown on FIGS. 1C and 1D, the electronic endoscope 6 has a forceps channel 12. A probe 15 having, for example, sixteen magnetic generation devices or source coils 14a, 14b, . . . , 14p(hereinafter, generically, 14i) is inserted through an insertion port 12aof the forceps channel 12. Thus, the source coils 14iare placed in the insertion unit 7.

[0197] A source cable 16 extends from the back end of the probe 15. A connector attached to the back end of the source cable 16 is coupled to a main unit 21 of the endoscope shape detection system 3 so that it can be uncoupled freely. A radio-frequency signal, or driving signal, is applied from the main unit 21 to the source coils 14i, serving as a magnetic generation means, by way of the source cable 16, serving as a radio-frequency transmitting means. This causes the source coils 11ito radiate electromagnetic waves identifiable with magnetic fields.

[0198] Referring also to FIG. 1B, magnetic detection devices, or sense coils, each made by arranging at least four single-core coils 22k, are coaxial and detect a magnetic field, in the same direction along the same straight line and are incorporated in the examining table 4 on which the patient 5 lies. For example, four sense coils 22a, 22b, 22c, and 22d(hereinafter, generically, 22j) are arranged in two rows and two columns. Specifically, the sense coils 22aand 22bare parallel to each other, and the sense coils 22cand 22dare orthogonal to the sense coils 22aand 22b. In this case, the number of single-core coils 22kis sixteen.

[0199] The sense coils 22jare connected to the main unit 21 through a connector attached to the examining table 4 with a sense cable 23 serving as a detection signal transmitting means. The main unit 21 has an operator panel 24 or keyboard to be manipulated by a user for operating the system. A monitor 25 serving as a display means for displaying an image showing the detected shape of an endoscope is connected to the main unit 21.

[0200] The configuration of the endoscope shape detection system 3 will be described below. The endoscope shape detection system 3 consists of, as shown in FIG. 2, a driving block 26 for actuating the source coils 14i, a detecting block 27 for detecting signals received by the sense coils 22j, and a host processor 28 for processing signals detected by the detecting block 27.

[0201] As shown in FIG. 3, the sixteen source coils 14ifor generating magnetic fields are arranged in the probe 15 placed at predetermined intervals in the insertion unit 7 of the electronic endoscope 6. The source coils 14iare connected to a source coil drive circuit unit 31 for producing sixteen different radio-frequency driving signals. The source coil drive circuit unit 31 forms the driving block 26.

[0202] The source coil drive circuit unit 31 drives the source coils 14iwith currents with characteristics of sine waves serving as driving signals of different frequencies. Driving frequencies at which the source coils 14iare driven are specified with driving frequency setting data or driving frequency data. The driving frequency setting data is stored in a driving frequency setting data storage means or a driving frequency setting data memory means (not shown) incorporated in the source coil drive circuit unit 31. The driving frequency data is stored in the driving frequency data storage means (not shown) in the source coil drive circuit unit 31 via a parallel input/output (PIO) circuit 33 under the control of a central processing unit (CPU) 32. The CPU 32 is incorporated in the host processor 28 for performing calculations required for estimating the shape of an endoscope and other processing.

[0203] The sixteen single-core coils 22kconstituting the four sense coils 22jare connected to a sense coil signal amplification circuit unit 34 in the detecting block 27.

[0204] As shown in FIG. 4, in the sense coil signal amplification circuit unit 34, the single-core coils 22kare connected to amplification circuits 35kon a one-to-one basis. Weak signals detected by the single-core coils 22kare amplified by the amplification circuits 35k. Filter circuits 36kpass a plurality of frequency components generated by the source coils and remove unwanted components of the signals. Thereafter, the signals are converted into digital signals readable by the host processor 28 by analog-to-digital (A/D) converters (ADCs) 38k.

[0205] As shown in FIG. 3, the detecting block 27 consists of the sense coil signal amplification circuit unit 24 and A/D converters 38k. As shown in FIG. 4, the sense coil signal amplification circuit unit 34 consists of the amplification circuits 35k, filter circuits 36k, and output buffers 37k.

[0206] Referring back to FIG. 3, outputs of the sixteen systems of circuits included in the sense coil signal amplification circuit unit 34 are transmitted to the sixteen A/D converters 38k. The outputs are converted into digital data that are sampled at intervals of a predetermined cycle synchronously with a clock pulse supplied from a control signal generation circuit unit 40. The digital data are written in a two-port memory 42 over a local data bus 41 in response to a control signal sent from the control signal generation circuit unit 40.

[0207] The two-port memory 42 is, as shown in FIG. 4, composed functionally of a local controller 42a, a first RAM 42c, a second RAM 42c, and a bus switch 42d. According to the timing of signals illustrated in FIG. 5, the A/D converters 38kstart digitization

[0208] in response to an A/D conversion start signal sent from the local controller 42a. The bus switch 42dswitches the RAMs 42band 42cin response to a switching signal sent from the local controller 42a, and thus uses the RAMs 42band 42calternately as a read memory and write memory respectively. After the endoscope shape detection system is energized, data is always acquired in response to a write signal.

[0209] Referring back to FIG. 3, the CPU 32 reads digital data written in the two-port memory 42 over an internal bus 46 in response to a control signal sent from the control signal generation circuit unit 40. As shown in FIG. 4, the internal bus 46 is composed of the local bus 43, a PCI controller 44, and a PCI bus 45. The CPU 32 uses a main memory 47 to sample frequencies (fast Fourier transform (FFT)) by processing the digital data as discussed below. Thus, the CPU 32 separates and samples magnetic detection information represented by frequency components having frequencies that correspond to the driving frequencies at which the source coils 14iare driven. The coordinates indicating the spatial positions of the source coils 14iplaced in the insertion unit 7 of the electronic endoscope 6 are calculated based on the digital data of the separated magnetic detection information.

[0210] An inserted state of the insertion unit 7 of the electronic endoscope 6 is estimated based on the data of the calculated coordinates of the positions. Display data according to which an image of the shape of the endoscope is displayed is produced and output to a video RAM 48. Data written in the video RAM 48 is read by a video signal generation circuit 49, and converted into an analog video signal or a video signal for a computer. The analog video signal is output to the monitor 25. The monitor 25 displays an image of an inserted state of the insertion unit 7 of the electronic endoscope 6 on the display screen according to the analog video signal.

[0211] The CPU 32 calculates magnetic detection information associated with the source coils 14i, that is, electromotive forces (amplitudes of sine-wave signals) developed at the four single-core coils 22kconstituting each sense coil 22jand phase information. The phase information indicates the positive or negative polarities of the electromotive forces.

[0212] A technique for estimating a space, in which source coils are present, from outputs of sense coils, and a technique for estimating the three-dimensional positions of each source coil using two orthogonal sense coils and two parallel sense coils are identical to those described in Japanese Patent Application No. 9-140603.

[0213] With respect to the present embodiment, another technique will be described. Specifically, a plurality of sense coils precisely detecting the space in which each source coil exists is selected from among the four sense coils arranged spatially. The three-dimensional position of each source coil is then estimated using the selected sense coils.

[0214] Referring to FIG. 6, after the endoscopic system 1 of the present embodiment is energized, system parameters are initialized based on the data contained in a parameter file at step S1. Hardware is initialized at step S2.

[0215] After the endoscopic system is energized, the same number of data as the number of time instants to be involved in FFT is always updated in the two-port memory 42 (see FIG. 5). At step S3, the CPU 32 acquires the same number of data as the number of time instants to be involved in FFT. At step S4, the data are corrected using a window function. The FFT, described below, is carried out at step S5. After the FFT is completed, frequency components having frequencies that correspond to driving frequencies are sampled at step S6. The amplitudes and phase differences of the frequency components are calculated at step S7. The calculated amplitudes and phase differences are corrected at step S8.

[0216] At step S9, the CPU 32 determines whether all signals sent from the eight A/D converters 38khave been detected. If the detection has not been completed, control is returned to step S3. If the detection has been completed, the amplitudes of all the signals are corrected according to the characteristics of the sense coils at step S10. At step Sit, the coordinates indicating the estimated positions of the source coils 14iare calculated based on the amplitudes and phase differences of all the signals according to a procedure described below.

[0217] Thereafter, at step S12, the CPU 32 determines whether a System End switch of the endoscopic system 1 is turned on. If the switch is not on, endoscope shape detection image display, described below, is carried out at step S13. Control is then returned to step S3. The above processing is repeated. At step S12, if the CPU 32 determines that the System End switch of the endoscopic system 1 is turned on, the system parameters are stored in the parameter file. The system is then de-energized.

[0218] During the FFT of step S5, as described in FIG. 7, the CPU 32 determines whether all the signals are time-varying, that is, whether the signals contain the same number of frequency components as the number of time instants to be involved in FFT. If the signals are time-varying, control is passed to step S22. If the signals are not time-varying, a standby state is established at step S23 and retained until the signals become time-varying. Control is then returned to step S22.

[0219] At step S22, the CPU 32 determines specific bits represented by a signal to be subjected to FFT. If the bit is a 0, the signal is processed this time; if the bit is a 1, the signal has already been processed. If the bit is a 0, FFT is carried out at step S24. After the FFT is completed, the bit is set to a 1 at step S25. At step S22, if the CPU 32 determines that the bit is a 1, control is returned to step S21. The processing is repeatedly performed on the subsequent signals. Thus, all signals are subjected to FFT.

[0220] At step 26 succeeding step S25, the CPU 32 determines whether specific bits represented by all of the signals are 1s. If the specific bits of all of the signals are not 1s, control is returned to step S21. Signals representing specific bits that are not 1s are subjected to FFT. At step S26, if the CPU 32 determines that the specific bits represented by all of the signals are 1s, the signals are brought to a non time varying state at step S27. A standby state is then established. When the signals each come to have the same number of frequency components as the number of time instants to be involved in FFT, the signals are brought to the time-varying state. Control is then returned to step S21.

[0221] In the processing described in FIG. 6, for attaining a high processing speed, as shown in FIG. 8, each processing unit is carried out as an object of parallel processing. Parallel processing is especially useful in carrying out FFT, an arithmetic operation that requires much processing time repeatedly with processing units having the same contents performing substantially concurrently. By adopting parallel

[0222] processing, time during which the CPU 32 is available is used effectively in order to attain the high processing speed.

[0223] As mentioned above, the CPU 32 carries out frequency sampling based on Fourier transform. What must be overcome is a phenomenon referred to as “leakage” which is dependent on the relationship between the frequencies f_iof sine waves with which the source coils 14iare driven and a width of truncation by which digital data is truncated.

[0224] Assuming that a width of truncation by which digital data to be sampled is truncated, which corresponds to the length of a time-varying signal, is equal to integral multiples of all of the driving frequencies f_i, the amplitudes and phases of sine waves of frequencies, can be calculated accurately. However, if the width of truncation is not equal to an integral multiple of any one of the driving frequencies f_i, “leakage” occurs. “Leakage” is reflected as an error in a calculated amplitude and phase. For preventing “leakage,” generally, a window function such as a Hamming function is adopted, as discussed in “The Fast Fourier Transform” written by E. Orangbrigham (Sec. 6).

[0225] However, the window function is used to merely alleviate an error deriving from “leakage.” Moreover, a value minimizing the adverse effect of “leakage” must be adopted as a driving frequency. This may place restrictions on system operations.

[0226] Frequency sampling to be performed at step S6 in FIG. 6 will be described below, wherein the adverse effect of the “leakage” is actively corrected by solving a simple algebra of matrices. Consequently, amplitudes and phases can be calculated highly precisely.

[0227] For brevity's sake, the discussion following assumes that the results of Fourier transform have already been normalized and multiplied by coefficients according to a window function (rectangular window or the like) and thus compensated for errors.

[0228] The Fourier transform, a complex discrete Fourier transform, F_kof a time-varying signal having sine waves of certain frequencies f_kis expressed as follows: $\begin{array}{cc}{F}_K=\sum _{n-0}^{N-1}{f}_n{e}^{\frac{-j2\pi \mathrm{kn}}{N}}& \left(1\right)\end{array}$

[0229] where N denotes the length of a sampled discrete signal, and j denotes an imaginary unit. F_kconsists of a real part Re{F_k} and an imaginary part IM{F_k}.

[0230] The condition that the aforesaid width of truncation by which digital data is truncated is equal to integral multiples of the driving frequencies f_iis equivalent to that frequencies fs_iobserved through the discrete Fourier transform are equal to the driving frequencies f_i. In other words, the frequencies f_icannot be observed. If all of the observed frequencies fs_iare equal to each other, the width of truncation corresponds to integral multiples of all the driving frequencies f_i. “Leakage” will therefore not occur.

[0231] Now, a description will be made of a procedure of defining the Fourier transform Fi that provides frequency components having frequencies that correspond to the driving frequencies f_ithat are equal to the observed frequencies fs_iprovided by performing the Fourier transform Fs_i.

[0232] A time-varying signal representing sampled digital data is assumed to be composed of M sine waves having frequencies equal to driving frequencies f_i(i=1, 2, . . . , M). In this case, the relationship between Fourier transforms providing frequency components having frequencies that correspond to the observed frequencies fs_iand driving frequencies f_irespectively can be expressed as follows: $\begin{array}{cc}\left[\begin{array}{c}\mathrm{Re}\left\{{\mathrm{Fs}}_1\right\}\\ \mathrm{Im}\left\{{\mathrm{Fs}}_1\right\}\\ \mathrm{RE}\left\{{\mathrm{Fs}}_2\right\}\\ \mathrm{Im}\left\{{\mathrm{Fs}}_2\right\}\\ ⋮\\ \mathrm{RE}\left\{{\mathrm{Fs}}_M\right\}\\ \mathrm{Im}\left\{{\mathrm{Fs}}_M\right\}\end{array}\right]=A·\left[\begin{array}{c}\mathrm{Re}\left\{F_1\right\}\\ \mathrm{Im}\left\{F_1\right\}\\ \mathrm{Re}\left\{F_2\right\}\\ \mathrm{Im}\left\{F_2\right\}\\ ⋮\\ \mathrm{RE}\left\{F_M\right\}\\ \mathrm{Im}\left\{F_M\right\}\end{array}\right]& \left(2\right)\end{array}$

[0233] wherein A denotes a matrix of 2M×2M in size composed of columns of coefficients defining the magnitudes of leakage occurring depending to the relationship between-each pair of Re{F₁} and Im{F₁} to Re{F_M} and Im{F_M}

[0234] The formula (2) is simplified as follows:

[0235] Y=A·X  (3)

[0236] where matrices X and Y are matrices of 2M×1 in size consisting of real parts and imaginary parts of the Fourier transforms providing the driving frequencies fi and observed frequencies fs_i(i=1, 2, . . . , M) respectively. When the matrix X is expressed as follows:

[0237] X=X₁=[1, 0, 0, 0, . . . , 0, 0, 0]^t  (4)

[0238] time-varying signal is composed only of sine waves having frequencies that correspond to driving frequencies fl, which are out of phase by π/2, and which have an amplitude 1, thus a cosine wave). “t” denotes transposition. When the matrix X is expressed as follows:

[0239] X=X₂=[0, 1, 0, 0, . . . , 0, 0, 0]^t  (5)

[0240] the time-varying signal is composed only of sine waves [whose] having frequencies that correspond to the driving frequencies f_iand have a phase shift of 0. Similarly, when the matrix X is expressed as follows:

[0241] X=X₃=[0, 0, 1, 0, . . . , 0, 0, 0]^t,

[0242] X=X₄=[0, 0, 0, 1, . . . , 0, 0, 0]^t,

[0243] X=X_2M−1=[0, 0, 0, 0, . . . , 0, 1, 0]^t,

[0244] X=X_2M=[0, 0, 0, 0, . . . , 0, 0, 1]^t  (6)

[0245] sine waves having frequencies that correspond to the driving frequencies f₁(where i=2, 3, . . . , M), which have a phase shift of π/2 or 0, and which have an amplitude 1 are generated as the time-varying signal.

[0246] Assuming that when the matrices X₁, X₂etc. and X_2Mare assigned to the formula (3), the matrix Y is expressed as matrices Y₁, Y₂, etc., and Y_2M, the matrices Y₁, Y₂, etc., and Y_2Mcontain “leakage” and express Fourier transforms of digital data represented by time-varying signals having frequency components that are given primarily as the matrices X₁, X₂, etc. and X_2M. Herein, “leakage” is construed as such a phenomenon that a matrix X contains a term, which should provide 0 but actually provides any value other than 0, or that a value that should be a 1 is not actually a 1. The matrices Y=Y₁, Y₂, etc., and Y_2Mserve only as terms constituting columns of the matrix A, and are therefore expressed as follows:

[0247] A=[Y₁, Y₂, Y₃, . . . , Y_2M]  (7)

[0248] Summarily, the matrix X stemming from the Fourier transform F₁that provides the driving frequencies f_i, primarily, frequency sampling information used to calculate amplitudes and phases, is expressed as follows:

[0249] X=A⁻¹·Y  (8)

[0250] The matrix X is calculated as a product of the matrix Y by an inverse matrix A⁻¹of the matrix A. The matrix Y stems from the Fourier transform Fs_iproviding the observed frequencies fs_i. The matrix A can be, as mentioned above, composed of the matrices Y=Y₁, Y₂, Y₃, . . . , Y_2Mstemming from Fourier transforms of time-varying signals having frequency components that are given primarily as the matrices X=X₁, X₂, X₃, etc., and X_2M.

[0251] Consequently, the matrix Y stemming from the Fourier transform of a time-varying signal is multiplied by the inverse matrix A⁻¹of the matrix A that is obtained in advance. Frequencies can therefore be sampled more accurately. Eventually, the positions of the source coils 14ican be estimated highly precisely.

[0252] A matrix Q of 2M×N in size may be defined as a matrix that should stem from Fourier transform and should be multiplied by the inverse matrix A⁻¹. Digital data of N×1 long may be multiplied directly by the matrix Q to obtain matrix X.

[0253] Thus, the endoscope shape detection system can ensure high precision in estimating the positions of the source coils. This also expands the freedom in selecting driving frequencies.

[0254] Next, coordinates-of-estimated source coil position referred to at step S11 in FIG. 6 will be described. To begin with, a procedure of calculating coordinates of the estimated position of each source soil will be described. Thereafter, the contents of processing will be described more particularly.

[0255] As shown in FIG. 9, a round coil is assumed to have a very small radius and to be very thin. As described in Japanese Unexamined Patent Publication No. 9-84745, when a current flows into the round coil, similarly to when a current flows into a magnetic dipole, a magnetic potential at a point P in a three-dimensional space is expressed as the formula below. $\begin{array}{cc}{U}_P\mp =\frac{\mu {\mathrm{IN}}_1\pi a^2}{4\pi \mu }\mp \frac{z}{{\left(x^2+y^2+z^2\right)}^{\frac{3}{2}}}& \left(9\right)\end{array}$

[0256] where μ denotes a magnetic permeability, N₁denotes the number of windings of a round coil, a denotes the radius of the round coil, and I denotes a current flowing into the round coil. A magnetic field (H_px, H_py, H_pz) having vectors with the same directions as the X, Y. and Z axes are expressed as follows: $\begin{array}{cc}\begin{array}{c}{H}_{\mathrm{Px}}=-\frac{\partial U_P}{\partial x}=\frac{{\mathrm{IN}}_1{a}^2}{4}\frac{3\mathrm{xz}}{{\left(x^2+y^2+z^2\right)}^{\frac{5}{2}}}=k_g\frac{3\mathrm{xz}}{r^5}\\ H_{\mathrm{Py}}=-\frac{\partial U_P}{\partial y}=\frac{{\mathrm{IN}}_1{a}^2}{4}\frac{3\mathrm{yz}}{{\left(x^2+y^2+z^2\right)}^{\frac{5}{2}}}=k_g\frac{3\mathrm{yz}}{r^5}\\ H_{\mathrm{Pz}}=-\frac{\partial U_P}{\partial z}=\frac{{\mathrm{IN}}_1{a}^2}{4}\frac{2z^2-x^2-y^2}{{\left(x^2+y^2+z^2\right)}^{\frac{5}{2}}}=k_g\frac{2z^2-x^2-y^2}{r^5}\end{array}& \left(10\right)\end{array}$

[0257] In a three-dimensional space like the one shown in FIG. 10 (hereinafter, a world coordinate system Xw−Yw−Zw), the position of a single-core coil (hereinafter, a source coil) for generating magnetic fields shall be indicated with coordinates (x_gw, y_gw, z_gw). Any position in the three dimensional space shall be point P (x_pw, y_pw, z_pw).

[0258] Assuming that a coordinate system defined with the source coil as a reference is a local coordinate system X_L−Y_L−Z_L, the coordinates (x_p1, y_p1, z_p1) indicating the point P in the local coordinate system are expressed as follows: $\begin{array}{cc}\begin{array}{c}{P}_1=R^{-1}\left(P_W-G_W\right)\\ \left(\begin{array}{c}{x}_{\mathrm{Pl}}\\ y_{\mathrm{Pl}}\\ z_{\mathrm{Pl}}\end{array}\right)=\left(\begin{array}{ccc}{R}_{00}& R_{10}& R_{20}\\ R_{01}& R_{11}& R_{21}\\ R_{02}& R_{12}& R_{22}\end{array}\right)=\left(\begin{array}{c}{x}_{\mathrm{PW}}-x_{\mathrm{gW}}\\ y_{\mathrm{PW}}-y_{\mathrm{gW}}\\ z_{\mathrm{PW}}-z_{\mathrm{gW}}\end{array}\right)\end{array}& \left(11\right)\end{array}$

[0259] where P₁denotes a vector extending from an origin 0 to the point P in the local coordinate system, P_wdenotes a vector extending from the origin 0 to the point P in the world coordinate system, G_wdenotes a vector oriented towards the position of the source coil in the world coordinate system, and R denotes a rotation matrix.

[0260] The rotation matrix R in a polar coordinate system shown in FIG. 11 is expressed as follows: $\begin{array}{cc}R=\left(\begin{array}{ccc}{R}_{00}& R_{10}& R_{02}\\ R_{01}& R_{11}& R_{12}\\ R_{02}& R_{21}& R_{22}\end{array}\right)=\left(\begin{array}{ccc}\cos \alpha & \sin \alpha \cos \beta & \sin \alpha \sin \beta \\ -\sin \alpha & \cos \alpha \cos \beta & \cos \alpha \sin \beta \\ 0& -\sin \beta & \cos \beta \end{array}\right)& \left(12\right)\end{array}$

[0261] where a denotes a magnitude of rotation with the Z_waxis as a center, and β denotes a magnitude of rotation with the X_waxis as a center. In the local coordinate system with the source coil as a reference, a magnetic field H₁(H_px1, H_py1, H_pz1) developed at the point P is expressed based on the formula (10) as follows: $\begin{array}{cc}\begin{array}{c}{H}_{\mathrm{Pxl}}=\frac{k_g}{r^5}3x_{\mathrm{Pl}}{z}_{\mathrm{Pl}}\\ H_{\mathrm{Pyl}}=\frac{k_g}{r^5}3y_{\mathrm{Pl}}{z}_{\mathrm{Pl}}\\ H_{\mathrm{Pzl}}=\frac{k_g}{r^5}\left(2z_{\mathrm{Pl}}^2-x_{\mathrm{Pl}}^2-y_{\mathrm{Pl}}^2\right)\end{array}& \left(13\right)\end{array}$

[0262] A magnetic field H_w(H_pxw, H_pyw, H_pzw) having components with the same directions as the X_w, Y_wand Z_waxes and which is developed at the point P in the world coordinate system is expressed as follows: $\begin{array}{cc}\begin{array}{c}{H}_W={\mathrm{RH}}_l\\ \begin{array}{c}\begin{array}{c}{H}_{\mathrm{PxW}}=\frac{k_g}{r^5}[\{2{\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)}^2-{\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)}^2-\\ {\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)}^2\}\sin \alpha \sin \beta +\\ 3\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)\cos \alpha \sin \beta +\\ 3\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)\cos \beta ]\end{array}\\ \begin{array}{c}{H}_{\mathrm{PyW}}=\frac{k_g}{r^5}[3\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)\sin \alpha \sin \beta +\\ \{2{\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)}^2-{\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)}^2-\\ {\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)}^2\}\cos \alpha \cos \beta +\\ 3\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)\cos \beta ]\end{array}\\ \begin{array}{c}{H}_{\mathrm{PzW}}=\frac{k_g}{r^5}[3\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)\sin \alpha \sin \beta +\\ 3\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)\cos \alpha \sin \beta +\\ \left\{2{\left(z_{\mathrm{PW}}-z_{\mathrm{gW}}\right)}^2-{\left(y_{\mathrm{PW}}-y_{\mathrm{gW}}\right)}^2-{\left(x_{\mathrm{PW}}-x_{\mathrm{gW}}\right)}^2\right\}\cos \beta ]\end{array}\end{array}\end{array}& \left(14\right)\end{array}$

[0263] As shown in FIG. 12, assuming that a source coil is placed at a proper position (X_g, Y_g, Z_g) in the three-dimensional space, and that a single-core coil (hereinafter, a sense coil) is placed at a position (X_d, Y_d, Z_d) on the Y axis, the single-core coil has the same direction as the Y axis and detects a magnetic field generated by the source coil as an electromotive force. A magnetic field H_ydeveloped at the position of the sense coil is expressed based on the formula (14) as follows: $\begin{array}{cc}\begin{array}{c}{H}_y=\frac{k_g}{r^5}[3\left(x_d-x_g\right)\left(y_d-y_g\right)\sin \alpha \sin \beta +\\ \left\{2{\left(y_d-y_g\right)}^2-{\left(z_d-z_g\right)}^2-{\left(x_d-x_g\right)}^2\right\}\cos \alpha \sin \beta +\\ 3\left(z_d-z_g\right)-\left(y_d-y_g\right)\cos \beta ]\end{array}& \left(15\right)\end{array}$

[0264] Furthermore, an electromotive force V_ydeveloped at the sense coil is expressed as a formula below that provides the partial differential of the magnetic field H_ywith respect to a time t. $\begin{array}{cc}\begin{array}{c}{V}_y=-\mu N_2\pi b^2\frac{d}{dt}{H}_y\\ =-\mu N_2\pi b^2\frac{\omega I_{\max }\cos \left(\omega t+\varphi \right)N_1{a}^2}{4r^5}[3\left(x_d-x_g\right)\left(y_d-y_g\right)\\ \sin \alpha \sin \beta +\left\{2{\left(y_d-y_g\right)}^2-{\left(z_d-z_g\right)}^2-{\left(x_d-x_g\right)}^2\right\}\\ \cos \alpha \sin \beta +3\left(z_d-z_g\right)\left(y_d-y_g\right)\cos \beta ]\\ =\frac{k_s}{r^5}[3\left(x_d-x_g\right)\left(y_d-y_g\right)\sin \alpha \sin \beta +\\ \left\{2{\left(y_d-y_g\right)}^2-{\left(z_d-z_g\right)}^2-{\left(x_d-x_g\right)}^2\right\}\cos \alpha \sin \beta +\\ 3\left(z_d-z_g\right)\left(y_d-y_g\right)\cos \beta ]\end{array}& \left(16\right)\end{array}$

[0265] where N₂denotes the number of windings of the sense coil. Calculating ωI_maxcos(ωt+φ) provides a value by differentiating a current I_maxsin(ωt+φ)flowing into the source coil with respect to the time t.

[0266] As shown in FIG. 13, assuming that a circle is drawn with the Y axis as a center and that the source coil is moved along the circle, a constant electromotive force is detected at the sense coil all the time. However, the orientation of the sense coil relative to the Y axis remains unchanged.

[0267] Therefore, when a plurality of sense coils is lined on the Y axis, a space in which the source coil exists, that is, the circle with the Y axis as a center can be estimated.

[0268] As shown in FIG. 14, assuming that four sense coils are placed on the Y axis and that a coordinate system defined on a plane y determined with the Y axis and the position of the source coil is X′−Y′, an electromotive force V_yideveloped at each sense coil is expressed as follows: $\begin{array}{cc}\begin{array}{c}{V}_{\mathrm{yi}}=\frac{k_{\mathrm{si}}}{r_i^5}[3\left(x_{\mathrm{di}}-x_g^{\prime }\right)\left(y_{\mathrm{di}}-y_g^{\prime }\right)g_x+\{2{\left(y_{\mathrm{di}}-y_g^{\prime }\right)}^2-\\ {\left(x_{\mathrm{di}}-x_g^{\prime }\right)}^2\}{g}_y]\end{array}& \left(17\right)\end{array}$

[0269] where g_xand g_ydenote terms specified with the plane y and the orientation of the source coil, X_diand y_diindicate the position of each sense coil in the coordinate system X′−Y′, and x_g′ and y_g′ indicate the position of the source coil.

[0270] The formula (17) includes four unknowns g_x, g_y, x_g′, and y_g′. When at least four sense coils are aligned along the Y axis, four equations are defined. The position of the source coil in the coordinate system X′−Y′ can be determined by solving the equations.

[0271] Specifically, as shown in FIGS. 15A and 15B, a source coil for generating magnetic fields is placed at a proper position in a three-dimensional space, and four sense coils are placed along the Y axis. A plane γ determined with the source coil and four sense coils is regarded as an X′−Y′ plane. The position of the source coil shall be indicated as (x_g, y_g), and the positions of the sense coils shall be indicated as (X_d0, y_d0), (x_d1, y_d1), (x_d2, y_d2), and (x_d3, y_d3), respectively.

[0272] Electromotive forces V_y0, V_y1, V_y2, and V_y3developed at the sense coils C_s0, C_s1, C_s2, and C_s3are expressed based on the formula (17) as follows: $\begin{array}{cc}\begin{array}{c}{V}_{\mathrm{y0}}=\frac{k_{\mathrm{s0}}}{r_0^5}[3\left(x_{\mathrm{d0}}-x_g\right)\left(y_{\mathrm{d0}}-y_g\right)g_x+\{2{\left(y_{\mathrm{d0}}-y_g\right)}^2-\\ {\left(x_{\mathrm{d0}}-x_g\right)}^2\}{g}_y]\end{array}& \left(18\right)\\ \begin{array}{c}{V}_{\mathrm{y1}}=\frac{k_{\mathrm{s1}}}{r_1^5}[3\left(x_{\mathrm{d1}}-x_g\right)\left(y_{\mathrm{d1}}-y_g\right)g_x+\{2{\left(y_{\mathrm{d1}}-y_g\right)}^2-\\ {\left(x_{\mathrm{d1}}-x_g\right)}^2\}{g}_y]\end{array}& \left(19\right)\\ \begin{array}{c}{V}_{\mathrm{y2}}=\frac{k_{\mathrm{s2}}}{r_2^5}[3\left(x_{\mathrm{d2}}-x_g\right)\left(y_{\mathrm{d2}}-y_g\right)g_x+\{2{\left(y_{\mathrm{d2}}-y_g\right)}^2-\\ {\left(x_{\mathrm{d2}}-x_g\right)}^2\}{g}_y]\end{array}& \left(20\right)\\ \begin{array}{c}{V}_{\mathrm{y3}}=\frac{k_{\mathrm{s3}}}{r_3^5}[3\left(x_{\mathrm{d3}}-x_g\right)\left(y_{\mathrm{d3}}-y_g\right)g_x+\{2{\left(y_{\mathrm{d3}}-y_g\right)}^2-\\ {\left(x_{\mathrm{d3}}-x_g\right)}^2\}{g}_y]\end{array}& \left(21\right)\end{array}$

[0273] k_si(i=0, 1, 2, 3) denotes a constant determined with a current flowing into the source coil and the number of windings of each sense coil.

[0274] The formulas (19) and (20) are rewritten into matrix format as follows: $\begin{array}{cc}\left(\begin{array}{c}{V}_{\mathrm{y1}}\\ V_{\mathrm{y2}}\end{array}\right)=\left(\begin{array}{cc}\frac{k_{\mathrm{s1}}}{r_1^5}3\left(x_{\mathrm{d1}}-x_g\right)\left(y_{\mathrm{d1}}-y_g\right)& \frac{k_{\mathrm{s1}}}{r_1^5}\left\{2{\left(y_{\mathrm{d1}}-y_g\right)}^2-{\left(x_{\mathrm{d1}}-x_g\right)}^2\right\}\\ \frac{k_{\mathrm{s2}}}{r_2^5}3\left(x_{\mathrm{d2}}-x_g\right)\left(y_{\mathrm{d2}}-y_g\right)& \frac{k_{\mathrm{s2}}}{r_2^5}\left\{2{\left(y_{\mathrm{d2}}-y_g\right)}^2-{\left(x_{\mathrm{d2}}-x_g\right)}^2\right\}\end{array}\right)\left(\begin{array}{c}{g}_x\\ g_y\end{array}\right)& \left(22\right)\end{array}$

[0275] The terms specified with the positions of the sense coils and source coil are rewritten as a matrix A as follows: $\begin{array}{cc}\left(\begin{array}{c}{V}_{\mathrm{y1}}\\ V_{\mathrm{y2}}\end{array}\right)=\left(\begin{array}{cc}{a}_{00}& a_{01}\\ a_{10}& a_{11}\end{array}\right)\left(\begin{array}{c}{g}_x\\ g_y\end{array}\right)& \left(23\right)\\ A=\left(\begin{array}{cc}{a}_{00}& a_{01}\\ a_{10}& a_{11}\end{array}\right)& \end{array}$

[0276] An inverse matrix A⁻¹of the matrix A is defined according to Cramer's formula as follows: $\begin{array}{cc}{A}^{-1}=\frac{1}{a_{00}{a}_{11}-a_{00}{a}_{10}}\left(\begin{array}{cc}{a}_{11}& -a_{01}\\ -a_{10}& a_{00}\end{array}\right)& \left(24\right)\end{array}$

[0277] Consequently, g_xand g_yare expressed as follows: $\begin{array}{cc}\left(\begin{array}{c}{g}_x\\ g_y\end{array}\right)=\frac{1}{a_{00}{a}_{11}-a_{01}{a}_{10}}\left(\begin{array}{cc}{a}_{11}& -a_{01}\\ -a_{10}& a_{00}\end{array}\right)\left(\begin{array}{c}{V}_{\mathrm{y1}}\\ V_{\mathrm{y2}}\end{array}\right)& \left(25\right)\end{array}$

[0278] The determinant of the inverse matrix A⁻¹is solved, and g_xand g_yare assigned to the formulas (18) and (21). $\begin{array}{cc}{V}_{\mathrm{y0}}=\frac{\begin{array}{c}{K}_1{R}_1\left(2Y_{02}+X_{02}\right)\left({\mathrm{XY}}_{02}-{\mathrm{XY}}_{20}\right)-\\ K_2{R}_2\left(2Y_{01}+X_{01}\right)\left({\mathrm{XY}}_{01}-{\mathrm{XY}}_{10}\right)\end{array}}{R_3\left(2Y_{12}+X_{12}\right)\left({\mathrm{XY}}_{12}-{\mathrm{XY}}_{21}\right)}& \left(26\right)\\ V_{\mathrm{y3}}=\frac{\begin{array}{c}{K}_3{R}_1\left(2Y_{32}+X_{32}\right)\left({\mathrm{XY}}_{32}-{\mathrm{XY}}_{23}\right)-\\ K_4{R}_2\left(2Y_{31}+X_{31}\right)\left({\mathrm{XY}}_{31}-{\mathrm{XY}}_{13}\right)\end{array}}{R_3\left(2Y_{12}+X_{12}\right)\left({\mathrm{XY}}_{12}-{\mathrm{XY}}_{21}\right)}& \left(27\right)\end{array}$

[0279] Thus, $\begin{array}{cc}\begin{array}{cccc}{K}_1=\frac{k_{\mathrm{s0}}}{k_{\mathrm{s1}}}{V}_{\mathrm{y1}}& K_2=\frac{k_{\mathrm{s0}}}{k_{\mathrm{s2}}}{V}_{\mathrm{y2}}& K_3=\frac{k_{\mathrm{s3}}}{k_{\mathrm{s1}}}{V}_{\mathrm{y1}}& K_4=\frac{k_{\mathrm{s3}}}{k_{\mathrm{s2}}}{V}_{\mathrm{y2}}\\ R_0=r_0^5& R_1=r_1^5& R_2=r_2^5& R_3=r_3^5\end{array}\begin{array}{cc}{X}_{01}=\left(x_{\mathrm{d0}}-x_g\right)\left(x_{\mathrm{d1}}-x_g\right)& X_{02}=\left(x_{\mathrm{d0}}-x_g\right)\left(x_{\mathrm{d2}}-x_g\right)\\ X_{31}=\left(x_{\mathrm{d3}}-x_g\right)\left(x_{\mathrm{d1}}-x_g\right)& X_{32}=\left(x_{\mathrm{d3}}-x_g\right)\left(x_{\mathrm{d2}}-x_g\right)\\ Y_{01}=\left(y_{\mathrm{d0}}-y_g\right)\left(y_{\mathrm{d1}}-y_g\right)& Y_{02}=\left(y_{\mathrm{d0}}-y_g\right)\left(y_{\mathrm{d2}}-y_g\right)\\ Y_{31}=\left(y_{\mathrm{d3}}-y_g\right)\left(y_{\mathrm{d1}}-y_g\right)& Y_{32}=\left(y_{\mathrm{d3}}-y_g\right)\left(y_{\mathrm{d2}}-y_g\right)\\ X_{12}=\left(x_{\mathrm{d1}}-x_g\right)\left(x_{\mathrm{d2}}-x_g\right)& Y_{12}=\left(y_{\mathrm{d1}}-y_g\right)\left(y_{\mathrm{d2}}-y_g\right)\\ {\mathrm{XY}}_{01}=\left(x_{\mathrm{d0}}-x_g\right)\left(y_{\mathrm{d1}}-y_g\right)& {\mathrm{XY}}_{10}=\left(x_{\mathrm{d1}}-x_g\right)\left(y_{\mathrm{d0}}-y_g\right)\\ {\mathrm{XY}}_{02}=\left(x_{\mathrm{d0}}-x_g\right)\left(y_{\mathrm{d2}}-y_g\right)& {\mathrm{XY}}_{20}=\left(x_{\mathrm{d2}}-x_g\right)\left(y_{\mathrm{d0}}-y_g\right)\\ {\mathrm{XY}}_{31}=\left(x_{\mathrm{d3}}-x_g\right)\left(y_{\mathrm{d1}}-y_g\right)& {\mathrm{XY}}_{13}=\left(x_{\mathrm{d1}}-x_g\right)\left(y_{\mathrm{d3}}-y_g\right)\\ {\mathrm{XY}}_{32}=\left(x_{\mathrm{d3}}-x_g\right)\left(y_{\mathrm{d2}}-y_g\right)& {\mathrm{XY}}_{23}=\left(x_{\mathrm{d2}}-x_g\right)\left(y_{\mathrm{d3}}-y_g\right)\\ {\mathrm{XY}}_{12}=\left(x_{\mathrm{d1}}-x_g\right)\left(y_{\mathrm{d2}}-y_g\right)& {\mathrm{XY}}_{21}=\left(x_{\mathrm{d2}}-x_g\right)\left(y_{\mathrm{d1}}-y_g\right)\end{array}& \left(28\right)\end{array}$

[0280] The formulas (26) and (27) are nonlinear equations having unknowns x_gand y_gx_gand y_gare obtained by solving the two equations according to Newton's method.

[0281] Assuming that electromotive forces developed at sense coils are V_y0′ and V_y3′, respectively, and that V_y0′ and V_y3′ given by the formulas (26) and (27) are estimated values, differences are expressed as follows:

[0282] f₁(X_g, Y_g)=V_y0−V_y0′  (29)

[0283] f₂(X_g, Y_g)=V_y3−V_y3′  (30)

[0284] Assuming that the electromotive forces V_y0and V_y3developed at the sense coils are measured accurately and that x_gand y_gused to calculate the estimated values V_y0and V_y3perfectly agree with those indicating the position of the source coil, the right sides of the formulas (29) and (30) sum to 0.

[0285] For estimating the position of the source coil, therefore, x_gand y_gwith which f₁=0 and f₂=0 are satisfied should be obtained.

[0286] The partial differentials of f₁and f₂are solved with respect to x_gand y_grespectively, whereby a Jacobian determinant J is defined as follows: $\begin{array}{cc}J=\left(\begin{array}{cc}\frac{\partial f_1}{\partial x_g}& \frac{\partial f_1}{\partial y_g}\\ \frac{\partial f_2}{\partial x_g}& \frac{\partial f_2}{\partial y_g}\end{array}\right)& \left(31\right)\end{array}$

[0287] An inverse matrix J⁻¹of the Jacobian determinant J is defined according to [the] Cramer's formula, and regarded as a matrix C. $\begin{array}{cc}{J}^{-1}=\left(\begin{array}{cc}{C}_{00}& C_{01}\\ C_{10}& C_{11}\end{array}\right)& \left(32\right)\end{array}$

[0288] Newton's method is an iteration of a nonlinear equation f(X)=0 and defined as follows:

[0289] X^(k+1)=X^(k)−ΔX^(k)

[0290] A correction value ΔX^(k)is determined while being linearly approximated to the solution of f(X) calculated with X^(k)specified as X.

[0291] ΔX^(k)=J⁻¹(X^(k))f(X^(k))

[0292] Assuming that proper initial values of x_gand y_gare x_g0and y_g0respectively, the approximate values of x_gand y_g, x_g1and y_g1, are obtained according to the following formulas:

[0293] X_g1=X_g0−{C₀₀f₁(x_g0, y_g0)+C₀₁f₂(x_g0, y_go)}  (33)

[0294] y_g1=y_g0−{C₁₀f₁(x_g0, y_g0)+C₁₁f₂(x_g0, y_g0)}  (34)

[0295] x_g1and y_g1are assigned to the formulas (29) and (30). If f₁and f₂are not equal to 0, x_g1and y_g1are specified as x_g0and y_g0in the formulas (33) and (34). This results in xg2 and yg2, respectively. Thereafter, f₁and f₂are obtained again. This operation is repeated, whereby f₁and f₂each are approximated as 0. Consequently, x_gand y_gare obtained.

[0296] Newton's method is adopted to solve the nonlinear equations. Alternatively, the least squares method may be adopted.

[0297] The position of a source coil on a plane defined with sense coils and the source coil can be estimated from outputs of the sense coils each having at least four single-core coils aligned along the same straight line. In other words, a position of a circle in a three-dimensional space in which the source coil exists can be estimated.

[0298] Out of the at least four sense coils aligned along the same straight line, at least two sense coils are arranged in a space to estimate the three-dimensional position of a source coil as a point of intersection between two circles in the space.

[0299] Each sense coil 22kconsists of four single-core coils. The largest output among the outputs of the four single-core coils is acquired as a maximum output, and two sense coils having maximum outputs that are the largest are selected.

[0300] The three-dimensional position of each source coil is estimated according to the condition for placement of the two selected sense coils, that is, whether the two sense coils are placed orthogonally or parallel.

[0301] In the present embodiment, a point of intersection between two circles in the space is detected. If the two circles do not intersect, two circumferential points separated by the shortest distance are detected. The two circles may not intersect due to noise or the like.

[0302] To begin with, a procedure of estimating the three-dimensional position of each source coil will be described on the assumption that two selected sense coils are placed orthogonally. As shown in FIG. 16, a source coil is placed at a proper position, and sense coils are aligned along the X axis and Y axis alike. Circles C₁and C₂on which the source coil exists are determined from the outputs of the sense coils.

[0303] Assuming that C₁is a circle lying on a plane x=a₁, centered on (a₁, 0, 0), and having a radius r₁, and that C₂is a circle lying on a plane y=b₂, centered on (0, b₂, 0) and having a radius r₂, then:

[0304] C₁: (x−a₁)²+y²+z²=r₁²  (35)

[0305] C₂: x²+(y−b₂)²+z²=r₂²  (36)

[0306] As shown in FIG. 17, the coordinates of a point Q descending perpendicularly from any point P(x₁, y₁, z₁,) to the plane y=b₂are (x₁, b₂, z₁).

[0307] A straight line m lying on the plane y=b₂end passing through the point Q and the center (O, b₂, O) of the circle C₂is expressed using a real variable t as follows:

[0308] x=x₁+tx₁

[0309] y=b₂

[0310] z=z₁+tz₁  (37)

[0311] When the formula (35) is assigned to the equation (36) of the circle C₂,

[0312] (x₁+tx₁)²=(z₁+tz₁)²=r₂²  (38)

[0313] The real variable t is obtained as follows: $\begin{array}{cc}t=\pm \frac{r_2}{{\left(x_1^2+z_1^2\right)}^{\frac{1}{2}}}-1& \left(39\right)\end{array}$

[0314] The straight line m and circle C₂intersect at two points. Herein, t>0 shall be satisfied. When the formula (39) is assigned to the formula (37), $\begin{array}{cc}x=\frac{x_1{r}_2}{{\left(x_1^2+z_1^2\right)}^{\frac{1}{2}}}y=b_{2}z=\frac{z_1{r}_2}{{\left(x_1^2+z_1^2\right)}^{\frac{1}{2}}}& \left(40\right)\end{array}$

[0315] The formula (40) expresses, as shown in FIG. 17, a point P′ on the circle C₂nearest the point P.

[0316] If the point P (x₁, y₁, z₁) lies on the circle C₁, x₁, y₁, and z₁: are expressed as follows:

[0317] x₁=a₁

[0318] y₁=r₁cos θ

[0319] z₁=r₁sinθ  (41)

[0320] The formula (41) is assigned to the formula (40), thus defining the formula (42). $\begin{array}{cc}x=\frac{a_1{r}_2}{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}y=b_{2}z=\frac{r_1{r}_2\sin \theta }{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}& \left(42\right)\end{array}$

[0321] The square D of the distance between a point on the circle C₁and a point on the circle C₂is expressed as follows: $\begin{array}{cc}\begin{array}{c}D={\left\{\frac{a_1{r}_2}{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}-a_1\right\}}^2+{\left(b_2-r_1\cos \theta \right)}^2+\\ {\left\{\frac{r_1{r}_2\sin \theta }{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}-r_1\sin \theta \right\}}^2\\ ={\left\{r_2-{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}\right\}}^2+{\left(b_2-r_1\cos \theta \right)}^2\end{array}& \left(43\right)\end{array}$

[0322] When the formula (43) is differentiated with respect to θ, the following formula results: $\begin{array}{cc}\frac{dD}{dD}=-2\left\{\frac{r_1{r}_2\cos \theta }{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}-b_2\right\}{r}_1\sin \theta & \left(44\right)\end{array}$

[0323] The condition under which the solution of the formula (44) is 0 is as follows:

[0324] sine θ=0

[0325] However, the two circles do not intersect at any point, and r₁<a₁and r₁<b₂, r₁<b₂, or r₂<a₁is established. An alternative condition is expressed as follows: $\begin{array}{cc}\frac{r_1{r}_2\cos \theta }{{\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}}-b_2=0& \left(45\right)\end{array}$

[0326] Solving the formula (45) for θ is as follows: $\begin{array}{cc}{r}_1{r}_2\cos \theta ={b_2\left(a_1^2+r_1^2{\sin }^2\theta \right)}^{\frac{1}{2}}{\cos }^2\theta =\frac{\left(b_2^2{a}_1^2+b_2^2{r}_1^2\right)}{\left(r_1^2{r}_2^2+b_2^2{r}_1^2\right)}\theta ={\cos }^{-1}\left\{\pm \sqrt{\frac{\left(b_2^2{a}_1^2+b_2^2{r}_1^2\right)}{\left(r_1^2{r}_2^2+b_2^2{r}_1^2\right)}}\right\}& \left(46\right)\end{array}$

[0327] The coordinates of the points on the circumferences of the circles C₁and C₂can be derived from. the formulas (46), (40) and (41).

[0328] Assuming that the point on the circle C₁. has the coordinates (x_c1, y_c2, z_c1) and the point on the circle c₂has the coordinates (x_c2, y_c2, z_c2), the position (x_g, y_g, z_g) of the source coil can be obtained as averages of the coordinates of the points on the circles as follows: $\begin{array}{cc}{x}_g=\frac{x_{\mathrm{c1}}+x_{\mathrm{c2}}}{2}{y}_g=\frac{y_{\mathrm{c1}}+y_{\mathrm{c2}}}{2}{z}_g=\frac{z_{\mathrm{c1}}+z_{\mathrm{c2}}}{2}& \left(47\right)\end{array}$

[0329] Consequently, the spatial position of the source coil can be estimated using two sense coils each having four single-core coils aligned along the same straight line.

[0330] As shown in FIG. 19, sense coils are aligned in parallel n order to estimate the position of a source coil. A circle C₁determined using the sense coils is expressed as follows:

[0331] x=a₁

[0332] y=b₁+r₁cos θ

[0333] z=r₁sin θ  (48)

[0334] A circle C₂[shall be] is expressed as follows:

[0335] x=a₂

[0336] y=b₂+r₂cos φ

[0337] z=r₂sin φ  (49)

[0338] Assuming that a point P₁on the circle C₁has the coordinates (x₁, y₁, z₁.) and a point on the circle C₂has the coordinates (x₂, y₂, z₂), the condition under which the points on the circles C₁and C₂intersect or approach most closely is provided as follows:

[0339] y₁=y₂and z₁=z₂  (50)

[0340] When the formulas (48) and (49) are solved with the conditional expression (50), the following formulas result:

[0341] b₁+r₁cos θ=b₂+r₂cosθ  (51)

[0342] r₁sin θ=r₂sin φ  (52)

[0343] When both the sides of the formula (52) are squared,

[0344] r₁²(1−cos²θ)=r₂²(1−cos²φ)  (52a)

[0345] Solving the formula (51) with this formula, such may be rewritten as: $\begin{array}{cc}\cos \theta =\frac{r_2^2-r_1^2-{\left(b_1-b_2\right)}^2}{2r_1\left(b_1-b_2\right)}\theta ={\cos }^{-1}\left\{\frac{r_2^2-r_1^2-{\left(b_1-b_2\right)}^2}{2r_1\left(b_1-b_2\right)}\right\}& \left(53\right)\end{array}$

[0346] A point of intersection (a₁=a₂) between the circles C₁and C₂or two most closely located points can be determined from the formulas (48) and (53), the formulas (49) and (52), and the formula (53).

[0347] When the two most closely located points on the circles C₁and C₂are determined, similarly to when the circles intersect, the x coordinates of the points are averaged in order to estimate one x coordinate. One y coordinate and one z coordinate are determined according to the formula (53).

[0348] Next, a description will be made of coordinates-of-estimated source coil position calculation carried out by the CPU 32 according to the aforesaid method.

[0349] As shown in FIG. 1, four sense coils 22j, each having four single-core coils aligned along the same straight line, are incorporated in the couch 4. The probe 15, in which the source coils 14imade by concatenating sixteen single-core coils are placed, is passed through the forceps channel 12 in the electronic endoscope 6.

[0350] In the endoscope shape detection system 3, the maximum amplitudes and phases of voltages developed at the sense coils 22jassociated with the source coils 14iare measured in order to determine the positive or negative polarities of the voltages, which are attained at the maximum amplitudes, according to the phases.

[0351] Referring to FIG. 20, the CPU 32 initializes the serial numbers of those of the source coils 14iand sense coils 22j, which are treated first, at step S32 and step S32. At step S31, i is reset to 0. At step S32, j is reset to 0.

[0352] First, the 0-th source coil and 0-th sense coil are selected. Voltages V₀₀, V₀₁, V₀₂, and V₀₃developed at the four single-core coils of the 0-th sense coil are acquired at step S33. At step S34, the highest one of the four acquired voltages acquired at step S33 is detected as a maximum voltage V=_max.

[0353] At step S35, the CPU 32 determines whether the maximum voltages of all sense coils have been detected. If the detection has not been completed, control is passed to step S36 and j is incremented by one. Control is then returned to step S33.

[0354] When step S35 is completed, control is passed to step S37. The absolute values of the maximum voltages detected at the sense coils are compared. Two sense coils at which the highest maximum voltages are detected are selected.

[0355] At step S38, the two-dimensional positions (X′_g00, Y′_g00) and (x′_g01, y′_g01) of the 0-th source coil on planes defined with the 0-th source coil and the two sense coils selected at step S37 are obtained.

[0356] At step S39, the CPU 32 determines whether the two sense coils selected at step S37 intersect. If so, control is passed to step S40 in FIG. 21. Otherwise, control is passed to step S41 in FIG. 21.

[0357] As described in FIG. 21, at step S40, a circle on which the source coil exists and which is determined with two intersecting sense coils is specified. At step S41, a circle on which the source coil exists and which is determined using two sense coils is determined.

[0358] At step S42, points most closely located on the circumferences of the two circles determined under the condition for placement of the two sense coils are determined.

[0359] At step S43, the three-dimensional position (x_g0, y_g0, z_g0) of the 0-th source coil is obtained from the two points determined at step S42. At step S44,the CPU 32 determines whether the three-dimensional positions (x_gi, y_gi, z_gi) of all of the source coils have been obtained. If the three-dimensional positions of all of the source coils (x_gi, y_gi, z_gi) have not been obtained, i is incremented at step S45 in FIG. 20 and control is returned to step S32. The processing is repeated until the three-dimensional positions of all of the source coils (x_gi, y_gi, z_gi) have been obtained. The processing is then terminated.

[0360] Consequently, the spatial position of each source coil can be estimated using sense coils each having four single-core coils aligned along the same straight line.

[0361] In the present embodiment, necessary sense coils are selected based on the absolute values of the maximum voltages detected at the sense coils. Alternatively, the position of each source coil on a plane determined with the four sense coils and the source coil may be estimated, and two sense coils located most closely to the source coil, where the radii of the associated circles are smallest, may be detected. The three-dimensional position of the source coil may then be estimated.

[0362] The estimated spatial positions of the source coils are updated continuously. Position update control described in FIG. 22 is extended relative to the position of each source coil. Specifically, as described in FIG. 22, assuming that the spatial (three-dimensional) position of the 0-th source coil has coordinates (x₀, y₀, z₀), the three-dimensional estimated coordinates (x₀, y₀, z₀) are entered at step S51. At step S52 the CPU 32 determines whether the three-dimensional estimated coordinates (x₀, y₀, z₀) were calculated through the first coordinates-of-estimated source coil position calculation. If not, control is passed to step S53. If so, control is passed to step S54.

[0363] If the three-dimensional estimated coordinates (x₀, y₀, z₀) are the initial three-dimensional estimated coordinates, the three-dimensional estimated coordinates (x₀, y₀, z₀) are stored as the previous three-dimensional estimated coordinates (x_B, y_B, z_B) at step S54. At step S55, the three-dimensional estimated coordinates (x₀, y₀, z₀) are regarded as resulting from a coordinates-of-estimated source coil position calculation. The processing is then terminated.

[0364] The following is based on the assumption that the three-dimensional estimated coordinates are not the initial ones, but the ones calculated through the second processing. At step S51, the three-dimensional estimated coordinates (x₀, y₀, z₀) are entered. At step S52, the CPU 32 determines whether the three-dimensional estimated coordinates (x₀, y₀, z₀) are the initial three-dimensional estimated coordinates calculated through the first coordinates-of-estimated source coil position calculation. Since they are not the initial three-dimensional estimated coordinates, control is passed to step S53. The CPU 32 then determines whether the absolute values of differences between the current three-dimensional estimated coordinates (x₀, y₀, z₀) and the previous three-dimensional estimated coordinates (x_B, y_B, z_B) exceed predetermined variation limits x_S, y_S, z_S) for the x, y, and z coordinates. If not, similarly to the initial three-dimensional estimated coordinates, the three-dimensional estimated coordinates (x₀, y₀, z₀) are stored as the previous ones (x_B, y_B, z_B). At step S55, the three-dimensional estimated coordinates (x₀, y₀, z₀) are regarding as resulting from a coordinates-of-estimated source coil position calculation. The processing is then terminated.

[0365] At step S53, if the CPU 32 determines that the absolute values of the differences between the current three-dimensional estimated coordinates (x₀, y₀, z₀) and the previous ones (x_B, y_B, z_B) exceed the predetermined variation limits x_S, y_S, and z_S) for the x, y, and z coordinates, control is passed to step S56. The previous three-dimensional estimated coordinates (x_B, y_B, z_B) are regarded as resulting from coordinates-of-estimated source coil position calculation. The processing is then terminated.

[0366] As mentioned above, the position of each source coil is updated based on the predetermined variation limits x_S, y_S, and z_S) for the x, y, and z coordinates.

[0367] The foregoing concerned the 0-th source coil. The same applies to all of the source coils.

[0368] Next, a description will be made of endoscope shape detection image display referred to at step S13 in FIG. 6.

[0369] The endoscope shape detection image display is, as described in FIG. 23, at step S61, constructed from data into a model representing the shape of an endoscope according to the coordinates of the position in a space of a source coil resulting from coordinates-of-estimated source coil position calculation. At step S62, the CPU 32 evaluates the depiction mode according to which the model representing the shape of the endoscope is depicted based on the model data pertaining to the shape of the endoscope. If a normal mode is designated, normal-mode processing is carried out at step S63. If an enlargement mode is designated, enlargement-mode processing is carried out at step S64. The processing is then terminated.

[0370] In the normal mode, a model representing the shape of the endoscope like the one shown in FIG. 24 is displayed on the monitor 25 of the endoscope shape detection system 3.

[0371] In the enlargement mode, as shown in FIG. 25, the operator panel 24 is used to acquire the coordinates of, for example, the left upper and right lower points LU and RL, respectively, defining a range to be enlarged at step S71. The range is defined with respect to the model representing the shape of the endoscope and displayed on the monitor 25 (see FIG. 24). At step S72, the CPU 32 determines whether the coordinates of the left upper and right lower points are the same. If the coordinates are the same, the range of enlargement cannot be determined. Control is therefore returned to step S71. If the coordinates of the selected left upper and right lower points are not the same, control is passed to step S73.

[0372] At step S73, the current model representing the shape of the endoscope is moved so that the center thereof will be aligned with the center of the defined range. At step S74, the defined range is enlarged in conformity with a display window on the monitor 25. The processing is then terminated.

[0373] Consequently, the model representing the shape of the endoscope displayed as shown in FIG. 24 on the monitor 25 is enlarged and displayed on the monitor 25 as shown in FIG. 26.

[0374] An image showing the shape of the endoscope can be displayed by selecting any of models listed below.

[0375] (1) Three-dimensional model 1 and three-dimensional model 2

[0376] (2) Two-dimensional model

[0377] (3) Twelve-point model

[0378] (4) Linear model

[0379] For depicting the shape of the endoscope using the three-dimensional model 1 or 2, coordinates indicating points on the source coils are, as shown in FIG. 27, interpolated in order to produce a three-dimensional image showing the shape of an endoscope. Then, a combination of cubic function curve approximation and natural line interpolation, a third-order B spline interpolation method, or a second-order B spline interpolation method is adopted. Normal vectors observed at any points on two source coils are then calculated in order to complete the model.

[0380] As described in FIG. 28, at step S81, surfaces abcd and cdef shown in FIG. 29 are depicted in that order according to the model data pertaining to the shape of the endoscope. At step S82, the surfaces are shaded based on the normal vectors observed at the points according to a smooth shading method. Consequently, a three-dimensional image showing the shape of the endoscope is displayed.

[0381] Thereafter, at step S83, the CPU 32 determines whether tone correction should be carried out to improve a sense of three-dimensionality. For the tone correction, the plane of the monitor 25 is regarded as an XY plane and a depth direction is regarded as the direction of the Z axis. Tone is corrected by assigning proper gray-scale levels to z coordinates. If the CPU 32 determines that tone correction should be carried out, tone correction is carried out at step S84. The processing is then terminated.

[0382] The tone correction of step S84 falls into first tone correction and second tone correction. The first tone correction is such that tone is corrected within a range of measurement performed by the endoscope shape detection system 3. The second tone correction is such that tone is corrected within a domain in which the model representing the shape of the endoscope is present.

[0383] The first tone correction is, as described in FIG. 30, such that maximum and minimum values of z coordinates are acquired from a range of measurement at step S91. Tone is determined from the model data pertaining to the shape of the endoscope at step S92. Colors that can be displayed are selected based on the tone determined at step S93. Consequently, as shown in FIG. 31, the tone is corrected within the range of measurement defined with z coordinates.

[0384] The second tone correction is, as described in FIG. 32, such that maximum and minimum values of z coordinates are acquired from a domain in which the model representing the shape of the endoscope is present at step S95. Tone is determined from the model data pertaining to the shape-of the endoscope at step S96. At step S97, colors that can be displayed are selected based on the tone determined at step S97. Consequently, as shown in FIG. 33, the tone is corrected within the domain in which the shape-of-endoscope model is present. The second tone correction ensures finer tone correction for the model representing the shape. of the endoscope than the first tone correction.

[0385] When a two-dimensional model is designated in order to display an image of the shape of the endoscope, a circle is drawn about each of the source coils at step S101 in FIG. 34. Each circle is always oriented in the direction of a line of sight. At step S102, the CPU 32 determines whether tone correction should be carried out. If the CPU 32 determines tone correction should be carried out, tone correction is carried out at step S103. The processing is then terminated. Consequently, an image showing the shape of the endoscope like in FIG. 35 is displayed on the monitor 25.

[0386] When the twelve-point model is designated to display an image showing the shape of the endoscope, all of the points in a coordinate system indicating the positions of all of the source coils are linked by lines at step S105 in FIG. 36. A cross (x) is drawn at all the points at step S106. The processing is then terminated. Consequently, an image showing the shape of the endoscope like in FIG. 37 is displayed on the monitor 25.

[0387] When the linear model is designated to display an image showing the shape of the endoscope, points in a coordinate system indicating the positions of all the source coils are linked by lines at step S108 in FIG. 38. A tiny blackened square is generated at all the points at step S109. The processing is then terminated. Consequently, an image representing the shape of the endoscope like in FIG. 39 is displayed on the monitor 25.

[0388] As described so far, according to the present embodiment, sense coils having outputs that permit estimation of the most precise spaces in which a source coil is present are selected from among a plurality of sense coils in a three-dimensional space. The three-dimensional position of the source coil is thus estimated. Consequently, the three-dimensional positions of all of the source coils can be estimated accurately.

[0389] The configuration of the second embodiment is identical to that of the first embodiment. A difference lies in a method of estimating the three-dimensional position of each source coil. The same reference numerals will therefore be assigned to the components identical to those of the first embodiment.

[0390] A circle on which a source coil is present and which is estimated using one sense coil is, as shown in FIGS. 15A and 15B, determined by solving an equation of a circle expressed using each point (x_g, y_g). The point (x_g, y_g) satisfies the two nonlinear equations (26) and (27) defined using the coordinates indicating the positions of any of the single-core coils C_s0to C_s2and any of the single-core coils C_s1, to C_s3. The point (x_g, y_g) is determined according to Newton's method.

[0391] When the source coils and sense coils approach each other, as shown in FIG. 40, a plurality of points of intersection satisfies the equations (26) and (27). One of the points of intersection is selected based on an estimated initial value (x_g, y_g).

[0392] Assuming that the second-point of intersection shown in FIG. 40 is selected, a circle on which a source coil is present is determined based on the coordinates of the second point of intersection.

[0393] The determined circle C₁is shown in FIG. 18. The circle C₂is determined in terms of another sense coil. The most closely located points P and P′ on the two circumferences are obtained to determine the three-dimensional position of the source coil.

[0394] If the second point of intersection is correct, the distance between the points P and P′ approaches zero. If the second point of intersection is incorrect, the distance between the points P and P′ increases. A pair of sense coils permitting estimation of two circles, points on which being separated by the shortest distance, is selected to estimate the three-dimensional position of a source coil.

[0395] FIG. 41 and FIG. 42 are flowcharts describing estimation of the three-dimensional position of a source coil.

[0396] As described in FIG. 41, at steps S120 and S121, the serial numbers of the source coils 14iand sense coils 22jare initialized.

[0397] First, the 0-th source coil and the 0-th sense coil are selected. Voltages V₀₀, V₀₁, V₀₂, and V₀₃developed at four single-core coils constituting the 0-th sense coil are acquired at step S122. At step S123, the CPU 32 determines whether or not all the four voltages acquired at step S122 are 0 V.

[0398] At step S123, the CPU 32 determines that all the voltages are 0 V, a flag j associated with the 0-th sense coil is reset to 0 at step S128. At step S129, j is incremented and control is returned to step S122. Control is thus passed to handling of the first sense coil.

[0399] At step S123, if the CPU 32 determines that all of the voltages are not 0 V, the flag j associated with the 0-th sense coil is set to 1 at step S124.

[0400] The position (x_g00, y_g00) of the 0-th source coil on a plane determined with the 0-th sense coil and 0-th source coil is obtained at step S125. A circle on which the source coil is present is estimated at step S126.

[0401] When it is found at step S127 that handling the 0-th sense coil is completed, control is passed to step S129 where j is incremented.

[0402] A step S127, when the CPU 32 detects that handling all of the sense coils in relation to the 0-th source coil is completed, control is passed to step S131 in FIG. 42.

[0403] As described in FIG. 42, sense coils having associated flags set to 1 are selected at step S131. Relative to all of the pairs of selected sense coils, most closely located points on two circumferences are detected and the distance between the two points is calculated.

[0404] A pair of sense coils found to have the shortest distance between them at step S131 is selected at step S132. The three-dimensional position of the source coil is estimated using the two selected sense coils.

[0405] At step S134, the CPU 32 determines whether the positions of all the source coils have been estimated. If not, i is incremented at step S130 in FIG. 41. Control is then returned to step S121. The processing is repeated until the positions of all the source coils have been estimated. When the three-dimensional positions of the sixteen source coils have been determined, the processing is terminated.

[0406] From the distance between points on two circles estimated using a pair of sense coils, whether the circles have been estimated correctly can be determined. Even if the sense coils and source coils approach each other, the three-dimensional position of each source coil can be estimated correctly.

[0407] The configuration of the third embodiment is identical to that of the first embodiment. A difference lies in a method of estimating the three-dimensional position of a source coil. The same reference numerals will be assigned to components identical to those of the first embodiment.

[0408] According to the present embodiment, from the conditions under which two curves intersect, the CPU 32 determines whether a circle estimated using a sense coil consisting of four single-core coils has been determined highly precisely. Two sense coils are selected to estimate the three-dimensional position of each source coil.

[0409] A circle estimated using one sense coil and having a source coil present thereon is determined by determining a point (x_g, y_g) which satisfies two nonlinear equations (26) and (27), according to Newton's method. The nonlinear equations (26) and (27) are defined using the coordinates indicating the positions of any of the single-core coils C_s0to C_s2and any of the single-core coils C_s1to C_s3shown in FIGS. 15A and 15B.

[0410] The partial differentials of V_y0and V_y3determined according to the formulas (26) and (27) with respect to x_g, and y_gare expressed as follows: $\begin{array}{cc}\frac{\partial V_{\mathrm{y0}}}{\partial x_g},\frac{\partial V_{\mathrm{y0}}}{\partial y_g}& \left(54\right)\\ \frac{\partial V_{\mathrm{y3}}}{\partial x_g},\frac{\partial V_{\mathrm{y3}}}{\partial y_g}& \left(55\right)\end{array}$