Patent Publication Number: US-9836891-B2

Title: Shape data generation method and apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuing application, filed under 35 U.S.C. section 111(a), of International Application PCT/JP2012/068636, filed on Jul. 23, 2012, the entire contents of which are incorporated herein by reference. 
    
    
     FIELD 
     This invention relates to a shape data generation technique. 
     BACKGROUND 
     In the medical field, simulators such as operation simulators, organ simulators and the like are used to determine treatment plans, perform diagnosis, predict postoperative conditions, develop medical supplies and equipment and the like. In the simulation using these kinds of simulators, 3-dimensional shape data of an organ is used, however, often the generation of the 3-dimensional shape data of the organ is not easy. This is because the organs are located inside the body, so visual observation and direct measurement of the organs are not possible, and the shapes of the organs are very complex, fundamentally. 
     There is a conventional technique for generating a target shape by transforming a reference shape of an organ. Here, a method described below is known as a method for generating a target 3-dimensional shape data. Specifically, a doctor or the like observes tomographic images such as Computer Tomography (CT) images, Magnetic Resonance Imaging (MRI) images or the like, sets the boundaries of each portion of the organ, and draws boundary lines. Then, 3-dimensional shape data of an organ is obtained by laminating the tomographic images with the boundary lines. 
     However, it takes a long time to take one image in MRI or the like, sometimes sufficient tomographic images are not obtained (namely, a slice interval becomes long.) In this case, it is not possible to generate 3-dimensional shape data with high accuracy. For example, a shape that does not exist in a real organ is formed. 
     Moreover, there&#39;s a technique for transforming a reference model so as to optimize a predetermined evaluation function by using a model fitting method to generate a model of a target object. However, when resolution of a 3-dimensional image used for extracting the target object is low, 3-dimensional shape data with high accuracy is not obtained. 
     Moreover, there&#39;s a technique using a transformable model whose surface is formed by a mesh network. In this method, by repeatedly executing a step of newly calculating a position of a network point, 3-dimensional structured segmentation is generated from 2-dimensional images. 
     Moreover, there&#39;s a transformation method using a landmark as a parameter. In this transformation method, source landmarks are deployed on an area to be transformed, target landmarks are deployed on positions after transformation, and then transformation is executed. A method like this causes a problem that an unnatural shape would form unless the source landmarks and the target landmarks are set properly. 
     Furthermore, even though conventional techniques described above are used, accuracy of 3-dimensional shape data becomes low when sufficient tomographic images are not obtained. 
     Patent Document 1: Japanese Laid-open Patent Publication No. 2007-098028 
     Patent Document 2: Japanese Laid-open Patent Publication No. 2007-164592 
     Patent Document 3: Japanese Laid-open Patent Publication No. 2002-329216 
     Patent Document 4: Japanese Laid-open Patent Publication No. 2004-265337 
     Non-Patent Document 1: “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations”, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, Fred L. Bookstein, VOL. 11, NO. 6, June 1989 
     Non-Patent Document 2: “Laplacian surface editing”, SGP &#39;04 Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, O. Sorkine, Tel Aviv University, D. Cohen-Or, Tel Aviv University, Y. Lipman, Tel Aviv University, M. Alexa, Darmstadt University of Technology, C. Roessl, Max-Planck Institut fuer Informatik, Saarbruecken, H.-P. Seidel, Max-Planck Institut fuer Informatik, Saarbruecken 
     SUMMARY 
     A shape data generation method relating to this invention includes: generating data of a target shape of transformation from plural tomographic images of an object; first specifying, from among plural vertices of a first shape that is a reference shape of the object and is to be transformed, plural first vertices, each first vertex of which satisfies a condition that a normal line of the first vertex passes through a certain point that is located on the target shape and is located on a boundary of the object in any one of the plural tomographic images; second specifying, for each of the plural first vertices, a second vertex that internally divides a segment between the first vertex and the certain point; transforming the first shape so as to put each of the plural first vertices on a corresponding second vertex; setting a shape after the transforming to the first shape; and executing the first specifying, the second specifying, the transforming and the setting a predetermined number of times. 
     The object and advantages of the embodiment will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the embodiment, as claimed. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a functional block diagram of a shape data generation apparatus relating to this embodiment; 
         FIG. 2  is a diagram depicting an example of a segment image; 
         FIG. 3  is a diagram depicting a processing flow of a processing for generating 3-dimensional data of a target shape; 
         FIG. 4  is a diagram depicting an example of a bounding box that contains the target shape; 
         FIG. 5  is a diagram depicting a processing flow of a mapping processing; 
         FIG. 6  is a diagram to explain results of the mapping processing; 
         FIG. 7  is a diagram depicting a processing flow of a boundary setting processing; 
         FIG. 8  is a diagram to explain results of the boundary setting processing; 
         FIG. 9  is a diagram depicting a processing flow of an interpolation processing; 
         FIG. 10  is a diagram to explain the interpolation processing; 
         FIG. 11  is a diagram depicting a processing flow of the interpolation processing; 
         FIG. 12  is a diagram to explain results of the interpolation processing; 
         FIG. 13  is a diagram depicting a processing flow of a processing for generating 3-dimensional data; 
         FIG. 14  is a diagram depicting a processing flow of a primary transformation processing; 
         FIG. 15  is a diagram to explain TPS Warp; 
         FIG. 16  is a diagram depicting an example that the primary transformation processing is applied to a left ventricle; 
         FIG. 17  is a diagram depicting an example that the primary transformation processing is applied to a right ventricle; 
         FIG. 18  is a diagram to explain a problem in case where target landmarks are placed on a target shape; 
         FIG. 19  is a diagram to explain an outline of a secondary transformation processing; 
         FIG. 20  is a diagram depicting a processing flow of the secondary transformation processing; 
         FIG. 21  is a diagram depicting a processing flow of a landmark setting processing; 
         FIG. 22  is a diagram depicting a processing flow of a boundary point search processing; 
         FIG. 23  is a diagram depicting a relationship between a position of a vertex “v” and a brightness value; 
         FIG. 24  is a diagram depicting the relationship between the position of the vertex “v” and the brightness value; 
         FIG. 25  is a diagram depicting an example of a case where the search point passes through a shape before the transformation; 
         FIG. 26  is a diagram depicting an example of a case where the search point passes through the shape before the transformation; 
         FIG. 27  is a diagram to explain a search method of the boundary point; 
         FIG. 28  is a diagram depicting a shape after the secondary transformation and the target shape; 
         FIG. 29  is a diagram depicting 3-dimensional data of the target shape obtained when processings of this embodiment are not executed; 
         FIG. 30  is a diagram depicting final 3-dimensional data obtained when processings of this embodiment are not executed; 
         FIG. 31  is a diagram depicting final 3-dimensional data obtained when processings of this embodiment are executed; 
         FIG. 32  is a diagram depicting the shape after the secondary transformation processing, being superimposed on the target shape; 
         FIG. 33  is a diagram depicting final 3-dimensional data obtained when processings of this embodiment are not executed and final 3-dimensional data obtained when processings of this embodiment are executed; and 
         FIG. 34  is a functional block diagram of a computer. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
       FIG. 1  illustrates a functional block diagram of a shape data generation apparatus  1  relating to this embodiment. In the example in  FIG. 1 , the shape data generation apparatus  1  includes a reference shape data storage unit  101 , a primary transformation unit  102 , an image data storage unit  103 , a primary transformation data storage unit  104 , a first landmark data storage unit  105 , a landmark processing unit  106 , a secondary transformation unit  107 , a second landmark data storage unit  110 , a secondary transformation data storage unit  111 , a display unit  112 , a generator  113 , a target shape data storage unit  114 , and a post-transformation data storage unit  115 . Moreover, the landmark processing unit  106  includes a setting unit  108  and a boundary point search unit  109 , and the generator  113  includes a definition unit  1130 , a mapping unit  1131 , a boundary setting unit  1132 , and an interpolation unit  1133 . 
     The generator  113  generates 3-dimensional shape data of a target shape from segment image data stored in the image data storage unit  103 , and stores the 3-dimensional shape data of the target shape in the target shape data storage unit  114 . Specifically, the definition unit  1130  executes a processing for defining a bounding box that contains the target shape. The mapping unit  1131  relates voxels in a bounding box that contains segment images to voxels in the bounding box that contains the target shape. The boundary setting unit  1132  executes a processing for allocating a predetermined brightness value (“2” in this embodiment) to a boundary between a heart and non-heart. The interpolation unit  1133  allocates a brightness value to each voxel that does not have a brightness value and is in the bounding box, which contains the target shape. 
     The primary transformation unit  102  uses data that is stored in the reference shape data storage unit  101  and data that is stored in the target shape data storage unit  114  to perform a primary transformation processing, which will be described later, and stores the processing results in the primary transformation data storage unit  104  and first landmark data storage unit  105 . In addition, the primary transformation unit  102  instructs the display unit  112  to display a screen for causing a user to designate a landmark. The setting unit  108  and the boundary point search unit  109  in the landmark processing unit  106  use data that is stored in the target shape data storage unit  114 , data that is stored in the primary transformation data storage unit  104 , data that is stored in the post-transformation data storage unit  115 , data that is stored in the first landmark data storage unit  105 , and data that is stored in the second landmark data storage unit  110  to perform a landmark setting processing and a boundary search processing which will be described later, and stores the processing results in the second landmark data storage unit  110 . The secondary transformation unit  107  uses data that is stored in the target shape data storage unit  114 , data that is stored in the primary transformation data storage unit  104 , data that is stored in the post-transformation data storage unit  115 , and data that is stored in the second landmark data storage unit  110  to perform a processing, and stores the processing results in the post-transformation data storage unit  115  or the secondary transformation data storage unit  111 . In addition, the secondary transformation unit  107  instructs the display unit  112  to display data that is stored in the secondary transformation data storage unit  111 . The display unit  112  displays data on a display device in reaction to instructions from the primary transformation unit  102  and the secondary transformation unit  107 . 
     The data for the standard shape of a heart (herein after, referred to as reference shape) is stored in the reference shape data storage unit  101 . More specifically, Standard Triangulated Language (STL) data that contains information about the vertices of the shape and information about connections of vertices is stored. However, the data format is not limited to the format of the STL data. 
       FIG. 2  illustrates an example of segment image data that is stored in the image data storage unit  103 . In  FIG. 2 , data of four segment images are illustrated. The segment image data is obtained by painting over a portion surrounded by the boundary with a predetermined brightness value (hereinafter, referred to a label value) for CT images of a certain patient&#39;s heart. Each point in the segment image data has 3-dimensional coordinates. In this embodiment, the segment image data is used to obtain 3-dimensional shape data of the target shape that is a target of transformation. 
     Next, the operation of the shape data generation apparatus  1  that is illustrated in  FIG. 1  will be explained by using  FIGS. 3 to 33 . The shape data generation apparatus  1  performs a processing for transforming the reference shape so as to come close to the target shape. Firstly, a processing for generating the 3-dimensional shape data of the target shape from the segment image data will be explained by using  FIGS. 3 to 12 . 
     The definition unit  1130  in the generator  113  reads out the segment image data stored in the image data storage unit  103 . Then, the definition unit  1130  defines the bounding box that contains the target shape ( FIG. 3 : step S 1 ). 
     When a slice interval of the segment images is long, the length in a z-axis direction of a voxel in the bounding box that contains the target shape becomes longer. Incidentally, a voxel is an element of a grid image in the 3-dimensional image data, which corresponds to a pixel that is an element of a rectangular image in the 2-dimensional image data. Therefore, a processing described below is executed to make a voxel in the bounding box that contains the target shape a cube. 
     (1) The lower limit coordinates of the bounding box that contains the target shape=the lower limit coordinates of the bounding box that contains the segment images; 
     (2) the upper limit coordinates of the bounding box that contains the target shape=the upper limit coordinates of the bounding box that contains the segment images; and 
     (3) the number of grid points in the x-axis direction of the bounding box that contains the target shape=1+(the upper limit x-coordinate of the bounding box−the lower limit x-coordinate of the bounding box)/voxel size in the x-axis direction or voxel size in the y-axis direction of the bounding box that contains the segment images (incidentally, the numbers of grid points in the y-axis direction and the z-axis direction can be calculated similarly). 
     However, if a setting is performed as described above, the position of the upper limit is sometimes misaligned. Therefore, the upper limit is reset as described below: 
     (4) the upper limit x-coordinate of the bounding box that contains the target shape=the lower limit x-coordinate of the bounding box that contains the segment images+voxel size in the x-axis direction or voxel size in the y-axis direction of the bounding box that contains the segment images*(the number of grid points in the x-axis direction−1) (incidentally, the y-coordinate and the z-coordinate can be calculated similarly). 
     Here, the bounding box that contains the segment images is defined by column vector B min =(x min , y min , z min ) T  that represents lower limit and column vector B max =(x max , y max , z max ) T  that represents upper limit. x min  is the minimum of the x-coordinates, and x max  is the maximum of the x-coordinates. y min  is the minimum of the y-coordinates, and y max  is the maximum of the y-coordinates. z min  is the minimum of the z-coordinates, and z max  is the maximum of the z-coordinates. 
     A bounding box illustrated in  FIG. 4  is defined by the processing of the step S 1 . A range of the bounding box is defined by the coordinate of the lower limit and the coordinate of the upper limit. For each voxel in the bounding box, the length of the voxel in the x-axis direction is v′ x , the length of the voxel in the y-axis direction is v′ y , and the length of the voxel in the z-axis direction is v′ z . Here, v′ x =v′ y =v′ z  holds. The number of grid points in each axis is the number of voxels in the axis plus 1. Data of the bounding box defined is stored in the target shape data storage unit  114 . “0” is set to each voxel in the bounding box as an initial value of the brightness value. 
     Returning to the explanation of the  FIG. 3 , the mapping unit  1131  executes a mapping processing (step S 3 ). The mapping processing is explained using  FIGS. 5 and 6 . 
     The mapping unit  1131  sets z=0 as the z-coordinate of the position u(x, y, z) in the segment images ( FIG. 5 : step S 11 ). Here, assume uε[0, n x ]*[0, n y ]*[0, n z ]⊂I 3 . n x  is the number of voxels in the x-axis direction in the bounding box that contains the segment images, n y  is the number of voxels in the y-axis direction, and n z  is the number of voxels in the z-axis direction. I(x, y, z) is a brightness value of u(x, y, z) in the bounding box that contains the segment images. 
     Moreover, in the following, when n x  is the number of voxels in the x-axis direction in the bounding box that contains the segment images, n y  is the number of voxels in the y-axis direction, and n z  is the number of voxels in the z-axis direction, the number of grid points in the z-axis direction is n z +1. And assume that the size of the voxel in the x-axis direction is v x , the size of the voxel in the y-axis direction is v y , and the size of the voxel in the z-axis direction is v z . 
     Returning to the explanation of the  FIG. 5 , the mapping unit  1131  sets y=0 as the y-coordinate of u (step S 13 ). 
     The mapping unit  1131  sets x=0 as the x-coordinate of u (step S 15 ). 
     The mapping unit  1131  determines whether (u)==the label value holds (step S 17 ). Namely, the mapping unit  1131  determines whether u is a voxel that is inside the boundary (namely, inside the heart). When I(u)==the label value does not hold (step S 17 : No route), processing shifts to a processing of step S 23 . When I(u)==the label value holds (step S 17 : Yes route), the mapping unit  1131  sets M′ −1 *(M*u) as a position u′ in the bounding box that contains the target shape (step S 19 ). In addition, the mapping unit  1131  sets “1” as a brightness value I′(u′) at a position u′ in the bounding box that includes the target shape (step S 21 ). Here, u′ε I 3  holds. Moreover, M and M′ are defined as described below. 
     
       
         
           
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     Incidentally, the brightness value of the position u′, which is stored in the target shape data storage unit  114 , is updated from 0 to 1 by the processing of the step S 21 . 
     The mapping unit  1131  determines whether x≧n x  holds (step S 23 ). When x≧n x  does not hold (step S 23 : No route), the mapping unit  1131  increments the x-coordinate of u(x, y, z) by 1 (step S 25 ), then the processing returns to the processing of the step S 17 . 
     When x≧n x  holds (step S 23 : Yes route), the mapping unit  1131  determines whether y≧n y  holds (step S 27 ). When y≧n y  does not hold (step S 27 : No route), the mapping unit  1131  increments the y-coordinate of the u(x, y, z) by 1 (step S 29 ), then the processing returns to the processing of the step S 15 . 
     When y≧n y  holds (step S 27 : Yes route), the mapping unit  1131  determines whether z≧n z  holds (step S 31 ). When z≧n 2  does not hold (step S 31 : No route), the mapping unit  1131  increments the z-coordinate of the u(x, y, z) by 1 (step S 33 ), then the processing returns to the processing of the step S 13 . On the other hand, when z≧n z  holds (step S 31 : Yes route), the processing returns to the calling source processing. 
     By performing the aforementioned processing, a brightness value of a boundary between a heart or the inside of the heart and non-heart becomes 1, and a brightness value of the portion that is not the heart becomes 0 in the target shape. Upper part of  FIG. 6  illustrates 3-dimensional shape data of the target shape in this phase, lower part of  FIG. 6  illustrates a portion of generated 3-dimensional shape data observed from the z-axis direction. In this phase, in the z-axis direction, a brightness value is given to only a portion in which the segment image data exist. 
     Returning to the explanation of  FIG. 3 , the boundary setting unit  1132  executes the boundary setting processing (step S 5 ). The boundary setting processing is explained using  FIGS. 7 and 8 . 
     Firstly, the boundary setting unit  1132  sets “z=0” as the z-coordinate of the brightness value I′(x, y, z) at the position u′ on the target shape ( FIG. 7 : step S 41 ). 
     The boundary setting unit  1132  sets “y=0” as the y-coordinate of I′ (step S 43 ). 
     The boundary setting unit  1132  sets “x=0” as the x-coordinate of I′ (step S 45 ). 
     The boundary setting unit  1132  determines whether a following expression holds (step S 47 ):
 
| I ′( x, y, z )− I ′( x− 1,  y, z )|+| I ′( x, y, z )− I ′( x+ 1,  y, z )|+| I ′( x, y, z )− I ′( x, y− 1,  z )|+| I ′( x, y, z )− I ′( x, y+ 1,  z )|&gt;0
 
     In the step S 47 , the boundary setting unit  1132  determines whether or not there&#39;s a difference between the brightness value of u′(x, y, z) and brightness values of points around u′(x, y, z).
 
| I ′( x, y, z )− I ′( x− 1,  y,z )|+| I ′( x, y, z )− I ′( x+ 1,  y, z )|+| I ′( x, y, z )− I ′( x, y− 1,  z )|+| I ′( x, y, z )− I′ ( x, y+ 1,  z )|&gt;0
 
     When the aforementioned expression does not hold (step S 47 : No route), the processing shifts to a processing of step S 53  because u′ is not the boundary between the heart and the non-heart.
 
| I ′( x, y, z )− I ′( x− 1,  v, z )|+| I ′( x, y, z )− I ′( x+ 1,  y, z )|+| I ′( x, y, z )− I ′( x, y− 1,  z )|+| I ′( x, y, z )− I ′( x, y+ 1,  z )|&gt;0
 
     When the aforementioned expression holds (step S 47 : Yes route), the boundary setting unit  1132  determines whether I′(x, y, z)==1 holds (step S 49 ). 
     When I′(x, y, z)==1 does not hold (step S 49 : No route), the processing shifts to a processing of the step S 53  because u′ is not the boundary between the heart and the non-heart. On the other hand, when I′(x, y, z)==1 holds (step S 49 : Yes route), u′ is the boundary between the heart and the non-heart. Therefore, the boundary setting unit  1132  sets “2” as I′(x, y, z) (step S 51 ). The brightness value of the position u′, which is stored in the target shape data storage unit  114 , is updated from 1 to 2 by the processing of the step S 51 . 
     The boundary setting unit  1132  determines whether x≧n x  holds (step S 53 ). When x≧n x  does not hold (step S 53 : No route), the boundary setting unit  1132  increments the x-coordinate of u′(x, y, z) by 1 (step S 55 ), then the processing returns to the processing of the step S 47 . 
     When x≧n x  holds (step S 53 : Yes route), the boundary setting unit  1132  determines whether y≧n y  holds (step S 57 ). When y≧n y  does not hold (step S 57 : No route), the boundary setting unit  1132  increments the y-coordinate of u′(x, y, z) by 1 (step S 59 ), then the processing returns to the processing of the step S 45 . 
     When y≧n y  holds (step S 57 : Yes route), the boundary setting unit  1132  determines whether z≧n z  holds (step S 61 ). When z≧n z  does not hold (step S 61 : No route), the boundary setting unit  1132  increments the z-coordinate of u′(x, y, z) by 1 (step S 63 ), then the processing returns to the processing of the step S 43 . On the other hand, when z≧n z  holds (step S 61 : Yes route), the processing returns to the calling source processing. 
     By performing the aforementioned processing, the brightness value inside the heart in the target shape becomes 1, the brightness value on the boundary between the heart and the non-heart becomes 2, and the brightness value of the non-heart becomes 0. Upper part of  FIG. 8  illustrates 3-dimensional shape data of the target shape at this phase, and lower part of  FIG. 8  illustrates a part of the generated 3-dimensional shape data observed from the z-axis direction. The upper part of  FIG. 8  illustrates only the boundary in order to make it easy to recognize the boundary. At this phase, in the z-axis direction, the brightness value is given only to a portion in which the segment image data exist. 
     Returning to the explanation of  FIG. 3 , the interpolation unit  1133  executes an interpolation processing (step S 7 ). The interpolation processing is explained by  FIGS. 9 to 12 . 
     Firstly, the interpolation unit  1133  sets “1” as a value of a variable j for counting the number of the segment images ( FIG. 9 : step S 71 ). 
     The interpolation unit  1133  sets List_i′[j]=List_i[j](v z /v′ z ) (step S 73 ). Here, List_i[j] is the z-coordinate of the j-th segment image data. Therefore, List_i′[j] is the z-coordinate of the bounding box that contains the target shape, and the z-coordinate corresponds to the z-coordinate of the j-th segment image data. As illustrated in upper and middle parts of  FIG. 10 , segment image data whose z-coordinate is List_i[j+1] is associated with the target shape data whose z-coordinate is List_i′[j+1]. In addition, segment image data whose z-coordinate is List_i[j] is associated with the target shape data whose z-coordinate is List_i′[j]. 
     The interpolation unit  1133  determines whether j≧n_List holds (step S 75 ). n_List is the number of the segment images actually created by the user. When j≧n_List does not hold (step S 75 : No route), the interpolation unit  1133  increments j by 1 in order to process the next segment image data (step S 77 ), then the processing returns to the processing of the step S 73 . 
     On the other hand, when j≧n_List holds (step S 75 : Yes route), the interpolation unit  1133  sets “1” as the value of the variable  1  for counting the segment images (step S 79 ). 
     The interpolation unit  1133  sets List_i′[j+1]−List_i′[j] as n (step S 81 ). n is the number of z-coordinates that do not have brightness values and are located in between the z-coordinate List_i′[j+1] and the z-coordinate List_i′[j]. The processing shifts to step S 83  in  FIG. 11  through terminal A. 
     Shifting to the explanation of the processing in  FIG. 11 , the interpolation unit  1133  sets “1” as a value of a variable k for counting the number of z-coordinates (step S 83 ). 
     The interpolation unit  1133  determines whether I′(x, y, List_i′[j])&gt;0 holds (step S 85 ). When I′(x, y, List_i′[j])&gt;0 holds (step S 85 : Yes route), the interpolation unit  1133  sets “1” as I′(x, y, List_i′[j]+k) (step S 67 ). The brightness value of the position u′, which is stored in the target shape data storage unit  114 , is updated from 0 to 1 by the processing of the step S 87 . On the other hand, when I′(x, y, List_i′[j])&gt;0 does not hold (step S 85 : No route), the interpolation unit  1133  sets “0” as I′(x, y, List_i′[j]+k) (step S 89 ). In the processing of the step S 89 , “0” is maintained for the brightness value of the position u′, which is stored in the target shape data storage unit  114 . 
     The interpolation unit  1133  determines whether k&gt;(n/2) holds (step S 91 ). When k&gt;(n/2) does not hold (step S 91 : No route), the interpolation unit  1133  increments k by 1 (step S 93 ), then the processing returns to the processing of the step S 85 . 
     In the processing from the steps S 83  to  393 , the brightness value that is the same as I′(x, y, List_i′[j]) is set because z-coordinate List_i′[j]+k is closer to z-coordinate List_i′[j] than z-coordinate List_i′[j+1]. For example, as illustrated in the lower part of  FIG. 10 , when List_i′[j]+k=z A  holds, because z A  is closer to z-coordinate List_i′[j] than z-coordinate List_i′[j+I]. However, even though I′(x, y, List_i′[j])=2 holds, not “2” but “1” is set as I′(x, y, List_i′[j]+k) in order to give brightness value “2” only to the boundary set by the user. 
     Returning to the explanation of the  FIG. 11 , when k&gt;(n/2) holds (step S 91 : Yes route), the interpolation unit  1133  sets (n/2)+1 as the value of the variable k for counting the number of z-coordinates (step S 95 ). 
     The interpolation unit  1133  determines whether I′(x, y, List_i′[j+1])&gt;0 holds (step S 97 ). When I′(x, v, List_i′[j+1])&gt;0 holds (step S 97 : Yes route), the interpolation unit  1133  sets “1” as I′(x, y, List_i′[j]+k) (step S 99 ). The brightness value of the position u′, which is stored in the target shape data storage unit  114 , is updated from 0 to 1 by the processing of the step S 99 . On the other hand, when I′(x, y, List_i′[j])&gt;0 does not hold (step S 97 : No route), the interpolation unit  1133  sets “0” as I′(x, y, List_i′[j]+k) (step S 101 ). In the processing of the step S 101 , “0” is maintained for the brightness value of the position which is stored in the target shape data storage unit  114 . 
     The interpolation unit  1133  determines whether k&gt;n holds (step S 103 ). When k&gt;n does not hold (step S 103 : No route), the interpolation unit  1133  increments k by 1 (step S 105 ), then the processing returns to the processing of the step S 97 . 
     In the processing from the step S 95  to step S 105 , the brightness value that is the same as I′(x, v, List_i′[j+1]) is set because z-coordinate List_i′[j]+k is closer to z-coordinate List_i′[j+1] than z-coordinate List_i′[j]. For example, as illustrated in the lower part of  FIG. 10 , when List_i′[j]+k=z B  holds, I′(x, y, List_i′[j]+k)=I′(x, y, List_i′[j+1]) is set because z B  is closer to z-coordinate List_i′[j+1] than z-coordinate List_i′[j]. However, even though I′(x, y, List_i′[j+1])=2 holds, not “2” but “1” is set as I′(x, y, List_i′[j]+k) in order to give brightness value “2” only to the boundary set by the user. 
     Returning to the explanation of the  FIG. 11 , when k&gt;n holds (step S 103 : Yes route), the interpolation unit  1133  determines whether j&gt;n_List−1 holds (step S 107 ). When j&gt;n_List−1 does not hold (step S 107 : No route), the interpolation unit  1133  increments j by  1  (step S 109 ) because there&#39;s a grid point that does not have a brightness value. The processing returns to the processing of the step S 81  in  FIG. 9  through terminal B. On the other hand, when j&gt;n_List−1 holds (step S 107 : Yes route), the processing returns to the calling source processing. 
     Returning to the explanation of the  FIG. 3 , the processing ends when the interpolation processing (step S 7 ) ends. 
     As described above, by preliminarily arranging to be able to discern the boundary between the heart and the non-heart in the target shape (namely, preliminarily setting “2” as the brightness value), transformation is performed so as to match the boundary of the target shape to the boundary of the reference shape. This enables to generate 3-dimensional shape data with high, accuracy. Incidentally, by performing the processing so far, 3-dimensional shape data, for example illustrated in  FIG. 12 , is generated. In this 3-dimensional shape data, a shape of a voxel becomes a cubic shape. 
     Next, a transformation processing of a reference shape which is carried out so as to come close to the target shape will be explained using  FIGS. 13 to 33 . 
     First, the primary transformation unit  102  performs the primary transformation processing ( FIG. 13 : step S 111 ). The primary transformation processing will be explained by using  FIGS. 14 to 17 . In the primary transformation processing, rough alignment of the reference shape and target shape is performed. 
     First, the primary transformation unit  102  reads the reference shape data from the reference shape data storage unit  101 , and reads the target shape data from the target shape data storage unit  114 . Then, the primary transformation unit  102  instructs the display unit  112  to display a landmark setting screen that includes the reference shape data and the target shape data. The display unit  112  displays the landmark setting screen on the display device in response to the instruction from the primary transformation unit  102  ( FIG. 14 : step S 112 ). 
     The user watches the landmark setting screen that is displayed on the display device and carries out rough alignment of the reference shape and the target shape. More specifically, (1) the user sets source landmarks at predetermined positions in the reference shape. (2) The user then sets target landmarks at positions in the target shape, which correspond to the positions where the source landmarks are arranged. The predetermined positions are characteristic positions of the heart, for example, the four annular valves, apex, bottom section of the right ventricle fluid surface, myocardial boundary (for example, the boundary between the right ventricle and left ventricle), the end surfaces of the four pulmonary veins, superior vena cava, and the inferior vena cava. 
     The primary transformation unit  102  then accepts settings for the source landmarks and target landmarks, and stores the data for the source landmarks and the target landmarks (for example, 3-dimensional coordinates) in the first landmark data storage unit  105  (step S 113 ). 
     Then the primary transformation unit  102  performs a processing, using a method such as the Thin Plate Spline (TPS) Warp method, which will be described later, to transform the reference shape according to the landmark data stored in the first landmark data storage unit  105  (step S 114 ). The primary transformation unit  102  then stores the processing results, which are data of the shape after the primary transformation, in the primary transformation data storage unit  104 . The processing then returns to the calling-source processing. 
     Here, the TPS Warp method will be explained. As illustrated in  FIG. 15 , in TPS Warp, when the source landmarks and the target landmarks that correspond to the source landmarks are given, the transformation is performed so that the source landmarks lay on the corresponding target landmarks. For details on the TPS Warp, refer to Fred L. Bookstein, “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations”, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 11 NO. 6, PP. 567-585, June 1989, for example. This document is incorporated into this specification by reference. 
     The format of the data stored in the primary transformation data storage unit  104  is the same as the format of the data that is stored in the reference shape data storage unit  101 . Moreover, the source landmarks that were used in the primary transformation processing (namely, points overlap with the target landmarks in the primary transformation processing) are handled as fixed points in the secondary transformation processing. In other words, the source landmarks that were used in the primary transformation processing do not move in the secondary transformation processing, and the positions are kept the same. 
       FIG. 16  illustrates an example of applying the primary transformation processing described above to the left ventricle. An object in the upper left part represents the reference shape with the source landmarks (circles), and an object in the upper right part represents the target shape with the target landmarks (circles). An object in the lower left part represents a shape after the primary transformation, which is the result of the TPS Warp using the reference shape and the target shape. An object in the lower right part represents the target shape observed from a different viewpoint. 
       FIG. 17  illustrates an example of applying the primary transformation processing described above to the right ventricle. An object in the upper left part represents the reference shape with the source landmarks (circles), and an object in the upper right part represents the target shape with the target landmarks (circles). Meshed lines  60  represent a shape after the primary transformation, which is the result of the TPS Warp using the reference shape and the target shape. In the lower part, the shape after the primary transformation, which is illustrated by the meshed lines  60 , is superimposed on the target shape in order to easily compare. 
     As described above, by performing the rough alignment in advance according to the setting of the landmark settings, which are accepted from the user, it becomes possible to more effectively perform the detailed transformation that will be performed later. 
     Returning to the explanation of  FIG. 13 , the secondary transformation unit  107  performs a secondary transformation processing (step S 115 ). The secondary transformation processing will be explained using  FIGS. 18 to 27 . 
     First, a summary of the secondary transformation processing will be given. In case where the transformation processing is performed according to the TPS Warp method, when considering that typically the heart has a rounded shape, setting the target landmarks on the normal lines of the source landmarks is thought to be effective. For example, as illustrated in  FIG. 18 , it is considered that the transformation processing based on the TPS Warp method is performed by placing the source landmarks on the shape before the transformation (i.e. shape after the primary transformation), and placing the target landmarks at the intersecting points between normal lines of the source landmarks and the target shape. However, when such a situation occurs that the normal lines cross as illustrated in  FIG. 18 , an unnatural portion that differs from the target shape may occur in the shape after the transformation. 
     Therefore, in the secondary transformation processing in this embodiment, as illustrated in  FIG. 19 , the target landmarks are placed at points that internally divide the line segments that connect the source landmarks, which are placed on the shape before the transformation (i.e. shape after the primary transformation), with the aforementioned intersecting points, and then the transformation processing is carried out according to the TPS Warp method. Furthermore, by repeating such a transformation processing, the shape gradually approaches the target shape. In this way, an unnatural portion dose not easily occur in the shape after the transformation, and it becomes easier that the direction of the normal lines faces toward the portion that should be originally target. 
     The secondary transformation processing will be explained in detail using  FIGS. 20 to 27 . First, the secondary transformation unit  107  sets the initial value of a variable t for counting the number of times of the transformations as t=0 ( FIG. 20 : step S 121 ). Next, the secondary transformation unit  107  counts the number of times of the transformations by incrementing the variable t such that t=t+1, and sets the initial value of a variable m as m=0 (step S 123 ). Here, m is a variable for counting the number of vertices that were processed. 
     The secondary transformation unit  107  then increases the variable m so that m=m+1 (step S 125 ), and instructs the landmark processing unit  106  to perform the landmark setting processing. Then, the landmark processing unit  106  performs the landmark setting processing (step S 127 ). The landmark setting processing will be explained using  FIGS. 21 to 27 . 
     First, the setting unit  108  of the landmark processing unit  106  identifies one vertex “v” at random front data stored in the post-transformation data storage unit  115  or data stored in the primary transformation data storage unit  109  ( FIG. 21 : step S 141 ). In the step S 141 , the setting unit  108  identifies one vertex “v” from data stored in the post-transformation data storage unit  115  when data is stored in the post-transformation data storage unit  115  (in other words, the transformation processing is performed at least one time in the secondary transformation processing). The setting unit  108  identifies one vertex “v” from data stored in the primary trans formation data storage unit  104  when data is not stored in the post-transformation data storage unit  115 . 
     Then, the setting unit  108  uses the data of the source landmarks moved in the primary transformation processing and the data of the source landmarks that are stored in the second landmark data storage unit  110  to calculate the Euclidean distances between the vertex “v” and the respective source landmarks. Initially, the second landmark data storage unit  110  sometimes stores no data. In this case, data of the source landmarks stored in the first landmark data storage unit  105  are used to calculate the distance. The setting unit  108  then determines whether the minimum distance of the Euclidean distances between the vertex “v” and the respective source landmarks is equal to or less than a threshold value D (step S 143 ). The step S 143  is performed in order to uniformly place the vertex “v” on the shape before the secondary transformation processing. At the step S 143 , whether or not a following equation is satisfied is determined. 
     
       
         
           
             
               
                 min 
                 i 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 d 
                 ⁡ 
                 
                   ( 
                   
                     v 
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                       i 
                     
                   
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             ≤ 
             D 
           
         
       
     
     Here, d(v, v i ) indicates the Euclidean distance between the point “v” and the point “v i ”. The point “v i ” is a fixed point (or in other words, a source land mark moved in the primary transformation processing), or is a source landmark (a vertex whose data is stored in the second landmark data storage unit  110 ). 
     When it is determined that the minimum of the Euclidean distances between the vertex “v” and the respective source landmarks is equal to or less than the threshold value D (step S 143 : Yes route), the processing returns to the calling-source processing. On the other hand, when it is determined that the minimum of the Euclidean distances between the vertex “v” and the respective source landmarks is greater than the threshold value D (step S 143 : NO route), the setting unit  108  instructs the boundary point search unit  109  to carry out a boundary point search processing. Then, the boundary point search unit  109  carries out the boundary point search processing (step S 145 ). The boundary point search processing will be explained using  FIGS. 22 to 27 . 
     First, the boundary point search unit  109  calculates the unit normal vector n(v) ( FIG. 22 : step S 161 ). Here, n(v) is the normal unit vector with respect to the surface H at the vertex v(εH). The unit normal vector is a normal vector with the length “1”. H (⊂V) is a shape surface that is specified by data stored in the primary transformation data storage unit  104  or data stored in the post-transformation data storage unit  115 , and V(⊂R 3 ) is a voxel space that is specified by the 3-dimensional shape data of the target shape. Moreover, R 3  represents a real number space. 
     The boundary point search unit  109  also determines whether or not the vertex “v” exists on a boundary (step S 163 ). At the step S 163 , it is determined whether or not the following equation is satisfied.
 
 f ( v )=2
 
     Here, mapping from the voxel space V to the real number space R (f: V−&gt;R) is defined as follows. According to this mapping f, the elements of 3-dimensional shape data of the target shape, which are included in the voxel space V, are correlated with the real number space R.
 
 f ( p )= I  
 
     Here, I is the brightness value of a voxel that includes a point p(εV). 
     When it is determined that the vertex “v” exists on the boundary (step S 163 : Yes route), the processing shifts to the step S 150  in  FIG. 21  through terminal C. Then, the setting unit  108  sets the vertex “v” as a target landmark (step S 150 ). The setting unit  108  sets the vertex “v” as a source landmark (step S 151 ). Furthermore, the setting unit  108  adds a set of data of the set target landmark and data of the set source landmark to the second landmark data storage unit  110 . 
     By doing the processing aforementioned above, the vertex “v” does not move when the vertex “v” exists on the boundary. Therefore, the boundary between the heart and the non-heart is simulated with high accuracy. 
     On the other hand, when the vertex “v” does not exist on the boundary (step S 163 : No route), the boundary point search unit  109  determines whether the vertex “v” exists inside the target shape (step S 165 ). At the step S 165 , it is determined whether or not the following equation is satisfied.
 
 f ( v )=1
 
     The processing at the step S 165  will be explained using  FIGS. 23 and 24 . As illustrated in  FIG. 23 , when the brightness value f(v) in the voxel space, which corresponds to the vertex “v”, is equal to 1, the vertex “v” is located on the inside of the target shape. Therefore, by setting a coefficient k in the processing at step S 177 , which will be described later, so as to be incremented by “1” at a time, the boundary point is searched for in the direction going toward the outside from the inside of the target shape. On the other hand, as illustrated in  FIG. 24 , when the brightness value f(v) in the voxel space, which corresponds to the vertex “v” becomes 0, the vertex “v” is located on the outside of the target shape. Therefore, by setting the coefficient k in the processing at step S 191 , which will be described later, so as to be decremented by “1” at a time, the boundary point is searched for in the direction going toward the inside from outside the target shape. 
     Returning to the explanation of  FIG. 22 , when it is determined that the vertex “v” exists on the inside of the target shape (step S 165 : Yes route), the boundary point search unit  109  sets the coefficient k as k=0 (step S 167 ). In addition, the boundary point search unit  109  sets a point (hereafter referred to as a search point) for which a determination will be made as to whether or not the point is a boundary point as described below (step S 169 ).
 
 v+kn ( v )
 
     The boundary point search unit  109  then determines whether or not the search point exists inside the voxel space specified by the 3-dimensional shape data of the target shape (step S 171 ). At the step S 171 , it is determined whether or not the following equation is satisfied.
 
 v+kn ( v )ε V  
 
     When it is determined that the search point does not exist inside the voxel space specified by the 3-dimensional shape data of the target shape (step S 171 : NO route), the processing returns to the calling-source processing. This is because the search point has reached outside the voxel space, so it is possible to determine that the normal line for the vertex “v” does not cross the target shape. 
     On the other hand, when it is determined that the search point exists inside the voxel space that is specified by the 3-dimensional shape data of the target shape (step S 171 : Yes route), the boundary point search unit  109  determines whether or not the search point passed through the shape before the transformation (step S 173 ). At the step S 173 , it is determined whether or not the following equation is satisfied.
 
 g ( v ), g ( v+kn ( v )))&lt;0
 
     Here, mapping g: V−&gt;R 3  is defined as follows. This mapping g correlates the elements of the segment image data that is included in the voxel space V with the real number space R 3 . 
     
       
         
           
             
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               ⁡ 
               
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                 p 
                 ) 
               
             
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                           ⁢ 
                           
                               
                           
                           ⁢ 
                           p 
                         
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     Be aware that the limit g| H  of the mapping g becomes n(v). 
     The processing of the step S 173  will be explained using  FIGS. 25 and 26 . When the search point passes through the shape before the transformation before reaching the boundary point, there is a possibility that searching for the boundary point is not carried out properly. For example, the case where the search point passes through the shape before the transformation before reaching the boundary point may be the case illustrated in  FIG. 25  or the case illustrated in  FIG. 26 . In other words, there is a case where the boundary point does not exist in the search direction according to the extent of the transformation of the target shape with respect to the shape before the transformation. In any case, there is a possibility that the boundary point is not detected, or that the boundary point is detected at a location that is not suitable. Therefore, at the step S 173 , the inner product of the normal vector for the vertex “v” and the normal vector for the search point is calculated, and when the inner product is less than 0 (in other words, when the angle between normal vectors is greater than 90 degrees), the search point is determined to have passed through the shape before the transformation. 
     Returning to the explanation of  FIG. 22 , when it is determined that the search point passed through the shape before the transformation (step S 173 : Yes route) it is not possible to detect a suitable boundary point, so the processing returns to the calling-source processing. On the other hand, when it is determined that the search point does not pass through the shape before the transformation (step S 173 : NO route), the boundary point search unit  109  determines whether the brightness value in the voxel space, which corresponds to the search point, is “2” (step S 175 ). 
     When the brightness value of the search point is not “2” (step S 175 : No route), the boundary point search unit  109  increments the coefficient k by “1” (step S 177 ), then the processing returns to the processing of the step S 169 . 
     In this way, as illustrated in  FIG. 27 , it is possible to determine whether or not the search point is on the boundary while moving the search point one voxel at a time in the normal direction from the vertex “v”. 
     On the other hand, when the brightness value is “2” (step S 175 : Yes route), the boundary point search unit  109  sets the search point as the boundary point (step S 179 ). At the step S 179 , data of the search point (for example, the value of k) is stored in a storage device such as a main memory. Then the processing returns to the calling-source processing. 
     In regards to this, the processing that is performed at the step S 165  when it is determined that the vertex “v” is located on the outside of the target shape (step S 165 : NO route) will be explained. The processing in this case differs only in the direction of the search, so the contents of the basic processing are as described above. In other words, the processing of the step S 181  is the same as the processing of the step S 167 , the processing of the step S 183  is the same as the processing of the step S 169 , the processing of the step S 185  is the same as the processing of the step S 171 , the processing of the step S 187  is the same as the processing of the step S 173 , and the processing of the step S 189  is the same as the processing of the step S 175 . Therefore, detailed explanations of the processing from steps S 181  to step S 189  are omitted. 
     Then, at the step S 191 , the boundary point search unit  109  decrements the coefficient k as k=k−1 (step S 191 ), and the processing returns to the processing of the step S 183 . As a result, the search point is moved one voxel at a time in the normal direction from the outside of the target shape toward the inside. In addition, the processing of the step S 193  is the same as the processing of the step S 179 . 
     By performing the processing such as described above, it becomes possible to detect the crossing point between the target shape and the normal line with respect to the vertex “v” as the boundary point, whose brightness value is “2”. 
     Returning to the explanation of  FIG. 21 , the setting unit  108  determines whether a boundary point was not detected by the boundary point search unit  109  (step S 147 ). When it is determined that a boundary point was not detected by the boundary point search unit  109  (step S 147 : NO route), the processing returns to the calling-source processing in order to perform a processing for the next vertex. 
     On the other hand, when it is determined that a boundary point was detected (step S 147 : Yes route), the setting unit  108  sets an internal dividing point on the line segment that connects the vertex “v” and the boundary point “v+kn(v)” as a target landmark (step S 149 ). More specifically, a point as described below is set as the target landmark. 
     
       
         
           
             v 
             + 
             
               
                 t 
                 T 
               
               ⁢ 
               
                 kn 
                 ⁡ 
                 
                   ( 
                   v 
                   ) 
                 
               
             
           
         
       
     
     Then, the setting unit  108  sets the vertex “v” as a source landmark (step S 151 ). The setting unit  108  stores the data for the set of source landmark and the target landmark in the second landmark data storage unit  110 . Then, the processing returns to the calling-source processing. 
     By performing the processing such as described above, it is possible to set an internal dividing point on the line segment that connects a vertex in the shape before the transformation and a boundary point in the target shape as a target landmark. In addition, it is possible to simulate a boundary in the segment image data and generate the other portion so as to become a typical shape that is similar to the reference shape, because only a portion that is on the boundary in the segment image data is set as the target landmark. As a result, it becomes 3-dimensional shape data with high accuracy. 
     Returning to the explanation of  FIG. 20 , the secondary transformation unit  107  determines whether or not m&lt;N is satisfied for the variable m (step S 129 ). Here, N is the total number of vertices in the shape after the primary transformation (or the shape after transforming the shape after the primary transformation). When it is determined that m&lt;N is satisfied (step S 129 : Yes route), the processing returns to the processing of the step S 125  in order to perform the processing for the next vertex. 
     On the other hand, when it is determined m&lt;N is not satisfied for the variable m (step S 129 : NO route) the secondary transformation unit  107  performs the transformation based on the TPS Warp according to the data for the source landmarks and target landmarks that are stored in the second landmark data storage unit  110 , and stores the data for the transformed shape in the post-transformation data storage unit  115  (step S 131 ). If a result of transformation processing in the previous time is stored in the post-transformation data storage unit  115 , the result is overwritten with the data for the transformed shape. 
     As described above, in the transformation processing at the step S 131 , a point that was a source landmark in the primary transformation processing is handled as a fixed point and is not moved. Then, the secondary transformation unit  107  deletes the data of the source landmark and the target landmark, which are stored in the second landmark data storage unit  110 . 
     The secondary transformation unit  107  then determines whether t&lt;T is satisfied for variable t (step S 133 ). When it is determined that t&lt;T is satisfied (step S 133 : Yes route), the processing returns to the processing of the step S 123  in order to perform further transformation processing. Here, T is the total number of times of the transformation, and may be set beforehand by an administrator or the like (for example, T=500). 
     On the other hand, when it is determined t&lt;T is not satisfied for variable t (step S 133 : NO route), the transformation has been performed T times, so the secondary transformation unit  107  stores the data for the shape after the secondary transformation processing in the secondary transformation data storage unit  111 , and the processing returns to the calling-source processing. 
     By performing the processing such as described above, the shape after the primary transformation approaches the target shape, and it becomes possible to obtain 3-dimensional shape data having high precision. Moreover, with such a kind of transformation method, the processing time becomes comparatively short. 
     Returning to the explanation of  FIG. 13 , after the secondary transformation processing has been performed, the display unit  112  displays the data stored in the secondary transformation data storage unit  111  on the display device or the like (step S 117 ). The processing then ends. 
       FIG. 28  illustrates an example of data that is displayed on the display device or the like. In the example in  FIG. 28 , the target shape and the shape after the secondary transformation, which is indicated by mesh lines, are displayed. The figure on the left side is a figure illustrating the entire transformed portion, and the figure on the right side is an enlarged view of part of the transformed portion. 
     Incidentally, when the number of the segment images is little and slice intervals are long, 3-dimensional shape data of the target shape illustrated in  FIG. 29  are generated unless processings in this embodiment are performed. Voxels are elongate in the z-axis direction in this 3-dimensional shape data. When the transformation processing is performed using this 3-dimensional shape data of the target shape, 3-dimensional shape data illustrated in  FIG. 30  is generated. As illustrated in  FIG. 30 , concavity and convexity which do not exist in a real heart occur. 
     On the other hand, even though the number of the segment images is little and slice intervals are long, 3-dimensional shape data of the target shape has no concavity or convexity and becomes smooth as illustrated in  FIG. 31  when processings in this embodiment are performed. 
       FIG. 32  illustrates a shape after the secondary transformation processing illustrated in  FIG. 31 , which is superimposed on the target shape illustrated in  FIG. 12 . Voxels in the target shape, which have a brightness value “2” (in other words, voxels on a boundary in the segment image data), correspond to a boundary of a shape after the secondary transformation processing. 
       FIG. 33  illustrates 3-dimensional shape data obtained when the method of this embodiment is not used (in other words, when the processings of the steps from S 1  to S 7  are not performed) and 3-dimensional shape data obtained when the method of this embodiment is used side by side. In  FIG. 33 , slice intervals becomes longer (in other words, the number of the segment images becomes less) from left to right. When the method of this embodiment is not used, the longer slice intervals become, the more concavity and convexity that do not exist in a real heart occur. On the other hand, when the method of this embodiment is used, even though slice intervals become longer, a smooth shape with no concavity and convexity is maintained. 
     Although the embodiment of this technique was explained above, this technique is not limited to this embodiment. For example, the functional block diagram of the shape data generation apparatus  1  explained above does not necessarily have to correspond to an actual program module configuration. 
     Moreover, in the processing flow explained above, the order of steps may be changed as long as the processing results do not change. Furthermore, as long as the processing results do not change, the steps may be executed in parallel. 
     In the example described above, the segment image data is displayed on the landmark setting screen to set the target landmarks. However, for example, tomographic images such as CT images may be displayed to set the target landmarks. 
     The processing such as described above is not only applicable to the heart, but can also be applied to other objects. 
     In addition, the aforementioned the shape data generation apparatus  1  is a computer device as illustrated in  FIG. 34 . That is, a memory  2501  (storage device), a CPU  2503  (processor), a hard disk drive (HDD)  2505 , a display controller  2507  connected to a display device  2509 , a drive device  2513  for a removable disk  2511 , an input unit  2515 , and a communication controller  2517  for connection with a network are connected through a bus  2519  as illustrated in FIG.  34 . An operating system (OS) and an application program for carrying out the foregoing processing in the embodiment, are stored in the HDD  2505 , and when executed by the CPU  2503 , they are read out from the HDD  2505  to the memory  2501 . As the need arises, the CPU  2503  controls the display controller  2507 , the communication controller  2517 , and the drive device  2513 , and causes them to perform predetermined operations. Moreover, intermediate processing data is stored in the memory  2501 , and if necessary, it is stored in the HDD  2505 . In this embodiment of this technique, the application program to realize the aforementioned functions is stored in the computer-readable, non-transitory removable disk  2511  and distributed, and then it is installed into the HDD  2505  from the drive device  2513 . It may be installed into the HDD  2505  via the network such as the Internet and the communication controller  2517 . In the computer as stated above, the hardware such as the CPU  2503  and the memory  2501 , the OS and the application programs systematically cooperate with each other, so that various functions as described above in details are realized. 
     The aforementioned embodiment is summarized as follows: 
     A shape data generation method includes; (A) generating data of a target shape of transformation from plural tomographic images of an object; (B) first specifying, from among plural vertices of a first shape that is a reference shape of the object and is to be transformed, plural first vertices, each first vertex of which satisfies a condition that a normal line of the first vertex passes through a certain point that is located on the target shape and is located on a boundary of the object in anyone of the plural tomographic images; second specifying, for each of the plural first vertices, a second vertex that internally divides a segment between the first vertex and the certain point; (C) transforming the first shape so as to put each of the plural first vertices on a corresponding second vertex; (D) setting a shape after the transforming to the first shape; and (E) executing the first specifying, the second specifying, the transforming and the setting a predetermined number of times. 
     By doing the aforementioned processing, it becomes possible to transform the reference shape so as to match to the boundary in the tomographic images, and to generate the other portion so as to approach the reference shape of the object. In addition, by trans forming gradually, an unnatural shape is unlikely to occur in a shape after the transformation. Therefore, it becomes possible to generate 3-dimensional shape data with high accuracy when sufficient tomographic images are not obtained. 
     Moreover, the generating may include: (a1) setting a brightness value of the certain point to a predetermined brightness value. By doing the aforementioned processing, it becomes possible to discern a boundary between the object and the non-object in the tomographic images. 
     Moreover, the generating may include; (a2) allocating brightness values of first voxels in a first voxel space to corresponding second voxels in a second voxel space, wherein the first voxel space contains the plural tomographic images, the second voxel space contains the target shape, and each of the first voxels is located in any one of the plurality of tomographic images; and (a3) allocating, to each voxel which is in the second voxel space and does not have a brightness value, a brightness value of a third voxel which is the closest to the voxel. By doing the aforementioned processing, the target shape becomes a proper shape because proper brightness values are allocated to voxels that do not have brightness values. 
     Moreover, the second vertex may be specified by a vector calculated by multiplying a unit normal vector for the first vertex by a certain value, which is calculated by dividing the number of times that the transforming has already been executed by the predetermined number. By doing the aforementioned processing, it becomes possible to transform the reference shape gradually so as not to occur an unnatural shape such as overlapping. 
     Moreover, the setting may include; (b1) moving a vertex to be processed to a point of destination that is located on a normal line of the vertex to be processed, wherein a distance between the vertex to be processed and the point of destination is a predetermined second distance; (b2) first determining whether the point of destination is included in a first voxel space that contains the plurality of tomographic images; (b3) upon determining that the point of destination is included in the first voxel space, second determining, based on an inner product of a normal vector of the vertex to be processed and a normal vector of the point of destination, whether the point of destination passes through the first shape; (b4) upon determining that the point of destination does not pass through the first shape, third determining whether a brightness value at the point of destination is the predetermined brightness value; (b5) upon determining that the brightness value at the point of destination is the predetermined brightness value, fourth determining that the vertex to be processed is the first vertex; and (b6) upon determining that the brightness value at the point of destination is not the predetermined brightness value, executing the moving and the first to third determining again for the point of destination. By doing the aforementioned processing, it becomes possible to specify the first vertex from the target shape. 
     Moreover, the first specifying may include: (b7) each of the plural first vertices so as to become at a second distance or more from any of the other first vertices. By doing the aforementioned processing, because portions to be transformed are uniform, a shape after the transformation becomes smooth and it becomes possible to generate a shape data with higher accuracy. 
     Moreover, the shape data generation method may further include: (E) outputting data of the first shape after the executing. By doing the aforementioned processing, it becomes possible for a user to determine whether the first shape after the executing is proper or not. 
     Moreover, the transforming may include: (c1) transforming using TPS Warp (Thin Plate Splines Warp). By using TPS Warp, it becomes possible to transform so as to match the first vertex to the second vertex. 
     Incidentally, it is possible to create a program causing a computer to execute the aforementioned processing, and such a program is stored in a computer readable storage medium or storage device such as a flexible disk, CD-ROM, DVD-ROM, magneto-optic disk, a semiconductor memory, and hard disk. In addition, the intermediate processing result is temporarily stored in a storage device such as a main memory or the like. 
     All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present inventions have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.