Patent Publication Number: US-10762648-B2

Title: Image processing apparatus, image processing method, image processing system, and program

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a national phase application of international application PCT/JP2016/005253, filed on Dec. 28, 2016, and claims the benefit of, and priority to, Japanese Patent Application No. 2016-016446, filed Jan. 29, 2016, and Japanese Patent Application No. 2016-224383, filed Nov. 17, 2016, which applications are hereby incorporated by reference herein in their entireties. 
     TECHNICAL FIELD 
     The disclosure of the present specification relates to an image processing apparatus, an image processing method, an image processing system, and a program. 
     BACKGROUND ART 
     A technology for registering and displaying a plurality of images and analyzing the images is used, for example, for image diagnosis in a medical field. To improve an accuracy of the registration, information of corresponding points that are points corresponding between the images on which the registration is performed may be obtained and used for the registration in some cases. In view of the above, a technology for supporting the obtainment of the information of the corresponding points has been proposed. For example, PTL 1 and PTL 2 describes that, with regard to images obtained by imaging different shapes of the same object (subject), coordinate transformation based on a particular structure of the object is applied to each of images to have the corresponding points easily identified by visual observation, and the information of the corresponding points between the images to which the coordinate transformation has been applied is obtained. 
     When the coordinate transformation is applied to each of the plurality of images such that it is facilitated to identify the corresponding points, the shape of the object drawn in the deformed image may be significantly different from the shape of the object drawn in the original image in some cases. Therefore, while it is facilitated to identify the corresponding points by performing the application of the coordinate transformation, it may become difficult to perform the registration between the deformed images on which the coordinate transformation has been applied in some cases. 
     CITATION LIST 
     Patent Literature 
     PTL 1: US2016/314587 
     PTL 2: US2015/235369 
     SUMMARY OF INVENTION 
     According to some embodiments of the present invention, an image processing apparatus includes a coordinate transformation obtaining unit configured to obtain information related to coordinate transformation for respectively transforming a first image and a second image obtained by imaging an object to a first transformed image and a second transformed image, an information obtaining unit configured to obtain correspondence information that serves as information of a corresponding point included in each of the first transformed image and the second transformed image, a transformation unit configured to transform a position of the corresponding point included in the first transformed image to a position in the first image and a position of the corresponding point included in the second transformed image to a position in the second image on the basis of the correspondence information obtained by the information obtaining unit and the coordinate transformation obtained by the coordinate transformation obtaining unit, and a deformation transformation obtaining unit configured to obtain information related to deformation transformation for performing registration of the first image and the second image on the basis of information of the corresponding points respectively transformed to the positions in the first image and the second image by the transformation unit. 
     Advantageous Effects of Invention 
     According to the exemplary embodiments of the present invention, since the information of the corresponding points obtained between the images on which the coordinate transformation for the deformation to the images with which it is facilitated to identify the corresponding points is transformed to the information of the corresponding points in the original images, it is possible to obtain the deformation transformation for the registration in accordance with the features of the original images. As a result, since the registration can be performed on the basis of the original images in a case where it is difficult to perform the registration between the images deformed so as to be facilitated to identify the corresponding points, it is possible to provide the system for performing the highly accurate registration. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates an example of a function configuration of an image processing apparatus according to first to third exemplary embodiments. 
         FIG. 2  illustrates an example of a hardware configuration of the image processing apparatus according to the first to third exemplary embodiments. 
         FIG. 3  is a flow chart illustrating an example of processing performed by the image processing apparatus according to the first to third exemplary embodiments. 
         FIG. 4A  illustrates an example of the processing performed by the image processing apparatus according to the first to third exemplary embodiments. 
         FIG. 4B  illustrates an example of the processing performed by the image processing apparatus according to the first to third exemplary embodiments. 
         FIG. 5  is a flow chart illustrating an example of processing performed by the image processing apparatus according to the first to third exemplary embodiments. 
         FIG. 6  illustrates an example of processing performed by the image processing apparatus according to an exemplary embodiment of the present invention. 
         FIG. 7  illustrates an example of a function configuration of the image processing apparatus according to a fourth exemplary embodiment. 
         FIG. 8  is a flow chart illustrating an example of processing performed by the image processing apparatus according to the fourth exemplary embodiment. 
         FIG. 9  is a flow chart illustrating an example of processing performed by the image processing apparatus according to the fourth exemplary embodiment. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, exemplary embodiments of the present invention will be described with reference to the drawings. 
     First Embodiment 
     A plurality of images including images obtained by a plurality of image acquisition apparatuses (modalities), on different dates and times, at different body postures, or in different image acquisition modes may be used for image diagnosis in a medical field, for example. In the image diagnosis using the plurality of images, registration between the plurality of images may be performed in some cases. To improve the accuracy of the registration, for example, an operation is performed to associate structures and the like drawn in the respective images with one another as corresponding points by visual observation. 
     While a first image and a second image obtained by imaging an object in mutually different deformed states are used as inputs, an image processing apparatus  100  according to a first exemplary embodiment obtains deformation transformation for performing registration between the first image and the second image. Subsequently, the image processing apparatus  100  generates a deformed image obtained by deforming one of the first image and the second image to be matched with the other image. To obtain the corresponding points used for the registration and to obtain corresponding positions between the input images, the first image and the second image are projected onto a space where it is easy to compare the images with each other to be observed, for example, a normalized space. Subsequently, the image processing apparatus  100  obtains the corresponding positions by an operation input by a user or the like on the images projected onto the normalized space. Furthermore, the image processing apparatus  100  projects the positions corresponding to the obtained positions onto a space of the original image, that is, a real space, which is a space defined by the coordinate system of the modality, and performs registration on the basis of regularization information for taking into account an appropriateness of physical deformation of the object (applying regularization to the deformation) in this real space. Herein, the real space refers to a space defined by image coordinate systems of the first image and the second image corresponding to a space defined by the image coordinate systems of the images obtained through the image acquisition by the modalities. The normalized space will be described below in detail. 
       FIG. 1  is a block diagram illustrating an example of a function configuration of the image processing apparatus  100 . The image processing apparatus  100  is connected to a data server  170 , an operation unit  180 , and a display unit  190 . 
     The data server  170  is connected to the image processing apparatus  100  via a network and holds the first image and the second image which will be described below. The first image and the second image held by the data server  170  are medical images obtained by imaging the object in advance in respectively different conditions. Herein, the conditions include types of the modalities used for the image acquisition, image acquisition modes, dates and times, body postures of the object, and the like. The medical images include, for example, three-dimensional tomographic images acquired by modalities such as a magnetic resonance imaging (MRI) apparatus, an X-ray computed tomography (CT) apparatus, an ultrasonic imaging apparatus, a photoacoustic tomography (PAT) apparatus, a positron emission tomography (PET)/single photon emission computed tomography (SPECT) apparatus, an optic coherence tomography (OCT) apparatus, and the like. According to the first exemplary embodiment, the first image and the second image are two medical images obtained by imaging the same object corresponding to the two medical images obtained by imaging the same object in respectively different deformation states. The first image and the second image may also be, for example, images acquired by different modalities or in different image acquisition modes around the same time or images obtained by imaging the same patient by the same modality in the same body posture at different dates and times for follow-up observations. The first image and the second image are input to the image processing apparatus  100  via a data obtaining unit  102 . 
     The operation unit  180  is, for example, a mouse or a key board. The operation unit  180  accepts an operation by a user. Positions of corresponding points between the first image and the second image which are specified by the user through an operation input via the operation unit  180  are input to the image processing apparatus  100  via a corresponding point obtaining unit  106 . 
     The display unit  190  performs display on the basis of an output from the image processing apparatus  100 . For example, a display image generated by the image processing apparatus  100  is displayed, and a graphical user interface (GUI) for obtaining an instruction from the user is provided. 
     The image processing apparatus  100  includes the data obtaining unit  102 , a normalization transformation obtaining unit  103 , a normalized image generation unit  104 , a display control unit  105 , the corresponding point obtaining unit  106 , a corresponding point transformation unit  107 , a deformation transformation obtaining unit (deformation estimation unit)  108 , and the deformed image generation unit  109 . The data obtaining unit  102  obtains the first image and the second image which are the input images from the data server  170 . The normalization transformation obtaining unit  103  obtains coordinate transformation for deforming each of the first image and the second image to an image with which it is facilitated to identify the corresponding point. According to the first exemplary embodiment, the normalization transformation obtaining unit  103  obtains normalization transformations with regard to the first image and the second image. The normalization transformation obtaining unit  103  is an example of a coordinate transformation obtaining unit. The normalized image generation unit  104  generates a first normalized image and a second normalized image corresponding to normalized images obtained by performing the normalization transformation of each of the first image and the second image. The normalized image generation unit  104  is an example of a transformed image generation unit. The display control unit  105  displays the first image, the second image, the first normalized image, the second normalized image, and a deformed image which will be described below on the display unit  190  in accordance with the operation input by the user or control based on a predetermined program. The display control unit  105  also functions as an output unit configured to output data of the image that can be displayed to an external apparatus such as the display unit  190 , for example. The corresponding point obtaining unit  106  obtains information related to the positions of the corresponding points on the first normalized image and the second normalized image in accordance with the operation input on the operation unit  180  by the user. The corresponding point obtaining unit  106  is an example of an information obtaining unit. The corresponding point transformation unit  107  transforms corresponding point positions from the normalized space to the real space corresponding to the space of the respective original images on the basis of the normalization transformations. The corresponding point transformation unit  107  is an example of a transformation unit. The deformation transformation obtaining unit (deformation estimation unit)  108  performs deformation estimation between the first image and the second image on the basis of the corresponding point positions in the transformed real space and obtains the deformation transformation between the images. The deformation transformation obtaining unit  108  is an example of a deformation transformation obtaining unit. The deformed image generation unit  109  deforms at least one of the first image and the second image on the basis of the deformation transformation between the images to generate a registered first deformed image or a registered second deformed image. Hereinafter, a position of a single point or each of a plurality of points on one image specified as the corresponding point that is a point indicating the position corresponding to the position on the other image will be referred to as a corresponding point position. 
       FIG. 2  illustrates an example of a hardware configuration of the image processing apparatus  100 . The image processing apparatus  100  is a computer, for example. The image processing apparatus  100  includes a central processing unit (CPU)  201  functioning as a main control unit, a random access memory (RAM)  202  functioning as a memory unit, a read only memory (ROM)  203 , a solid state drive (SSD)  204 , a communication circuit  205  functioning as a communication unit, a universal serial bus (USB)  206  functioning as a connection unit, High Definition Multimedia Interface (HDMI) (registered trademark)  208 , and a graphics processing unit (GPU)  207  functioning as a graphics control unit, and these components are connected via an internal bus so as to be communicable to one another. The CPU  201  is a control circuit configured to control the image processing apparatus  100  and the respective units connected to the image processing apparatus  100  in an integrated manner. The RAM  202  is a memory that stores a program for executing processing in the image processing apparatus  100  and the respective units connected to the image processing apparatus  100  and various parameters used for image processing. The RAM  202  stores a control program executed by the CPU  201  and temporarily stores various pieces of data used when the CPU  201  executes various controls. The ROM  203  stores a program or data in which a control procedure by the CPU  201  is stored. The SSD  204  stores a program or data. The communication circuit  205  performs a communication with the data server  170  via a network. The USB  206  is connected to the operation unit  180 . The GPU  207  is an image processing unit and executes image processing in accordance with the control from the CPU  201 . An image obtained as a result of the image processing is output to the display unit  190  via the HDMI (registered trademark)  208  to be displayed. The operation unit  180  and the display unit  190  may be integrated with each other as a touch panel monitor. 
     According to an exemplary embodiment, the functions of the respective configurations illustrated in  FIG. 1  are realized while the CPU  201  executes the programs stored in the RAM  202 , the ROM  203 , and the SSD  204 . According to another exemplary embodiment, the functions of the respective configurations illustrated in  FIG. 1  are realized by the GPU  207  and a circuit such as a field-programmable gate array (FPGA) (not illustrated) in addition to the CPU  201  described above. In this case, at least one of processings including an input to the GPU  207  and the FPCA (not illustrated), a setting of a parameter related to the image processing, and a reception of an output from the GPU  207  and FPGA (not illustrated) is controlled by the CPU  201 . The function of the display control unit  105  is realized by the CPU  201  and the GPU  207 . At this time, the CPU  201  instructs the GPU  207  to display predetermined information on the display unit  190  and generates display data used for the GPU  207  to perform the display on the display unit  190  to output the display data to the display unit  190 . According to another exemplary embodiment, the communication circuit  205  is realized by a single network interface card (NIC) or a plurality of NICs when appropriate. According to another exemplary embodiment, a plurality of CPUs  201  may be included in the apparatus. 
       FIG. 3  is a flow chart illustrating an example of the entirety of the processing performed by the image processing apparatus  100 .  FIGS. 4A and 4B  are explanatory diagrams for describing an example of processing in step S 301 .  FIG. 5  is a flow chart illustrating an example of the processing in step S 301 .  FIG. 6  illustrates an example of the entirety of the processing performed by the image processing apparatus  100 . Hereinafter, the processing performed by the image processing apparatus  100  will be described by appropriately referring to  FIG. 3  to  FIG. 6 . 
     &lt;Step S 300 : Obtainment of the Image Data&gt; 
     In step S 300  illustrated in  FIG. 3 , the data obtaining unit  102  obtains the first image and the second image from the data server  170 . According to the first exemplary embodiment, a case where an MRI image is obtained as the first image and a PAT image is obtained as the second image will be described as an example. Hereinafter, according to the first exemplary embodiment, the first image is denoted by I 1 , and the second image is denoted by I 2 . Herein, the first image I 1  and the second image I 2  are obtained by imaging the breast of the same object in different deformed states. The shape of the breast to be drawn varies in the first image I 1  and the second image I 2  since the object takes different body postures at the time of the MRI imaging and the time of the PAT imaging or an MRI apparatus and a PAT apparatus come to contact with the breast. It should be noted however that the first image I 1  and the second image I 2  are respectively three-dimensional images, and the image coordinate system is defined in each of the images. According to the first exemplary embodiment, a space defined by the image coordinate systems of the first image I 1  and the second image I 2  will be referred to as a real space. The processing in step S 300  is processing of obtaining a first image I 1  ( 600 ) and a second image I 2  ( 601 ) illustrated in  FIG. 6 . 
     &lt;Step S 301 : Obtainment of the Normalization Transformation&gt; 
     In step S 301  illustrated in  FIG. 3 , the normalization transformation obtaining unit  103  obtains a first normalization transformation Tn 1  and a second normalization transformation Tn 2  for respectively transforming the first image I 1  and the second image I 2  to the normalized images. That is, processing of obtaining transformations for projecting the first image I 1  and the second image I 2  having the coordinate systems defined on the real space onto the normalized space is executed. The processing in step S 301  is processing of obtaining a coordinate transformation Tn 1  ( 602 ) and a coordinate transformation Tn 2  ( 603 ) illustrated in  FIG. 6 . 
     The normalization according to the first exemplary embodiment refers to coordinate transformations for respectively transforming the first image and the second image to a first transformed image and a second transformed image on the basis of a contour of a target site of the object and a reference point on the contour. More specifically, the normalization according to the first exemplary embodiment refers to coordinate transformations for projecting positions and shapes of anatomical features in the first image and the second image corresponding to the images of the object obtained in a plurality of different deformed states onto a common space (normalized space) in which those anatomical features have substantially the same positions and shapes. The normalization transformation corresponding to the coordinate transformation of the normalization according to the first exemplary embodiment is coordinate transformation to the normalized space in which the positions and shapes of the anatomical features of the respective images have substantially the same positions and shapes on the basis of the positions and shapes of the anatomical features such as the nipple commonly drawn in the first image I 1  and the second image I 2  acquired in the different deformed states and the body surface. The first transformed image obtained by performing the normalization transformation of the first image will be referred to as the first normalized image. In addition, the second transformed image obtained by performing the normalization transformation of the second image will be referred to as the second normalized image. Hereinafter, an example of a method of obtaining the normalization transformation will be described. 
       FIG. 4A  is a schematic diagram of the first image I 1 . The first image according to the first exemplary embodiment is a three-dimensional MRI image, and a cross-sectional image  400  illustrated in  FIG. 4A  is a schematic diagram illustrating a certain cross-sectional image of the first image I 1 . In step S 300 , the normalization transformation obtaining unit  103  obtains a position coordinate P 1  of a nipple  401  of the object and shape information S 1  of a body surface  402  from the cross-sectional image  400  of the first image I 1 . These pieces of information are obtained as position information in the coordinate system of the first image I 1 , that is, position information in the real space. 
       FIG. 4B  is a schematic diagram illustrating a certain cross-sectional image of a first normalized image I 1 ′ generated in step S 302  which will be described below. In step S 300 , the normalization transformation obtaining unit  103  obtains the coordinate transformation for deforming the first image I 1  represented by the cross-sectional image  400  of  FIG. 4A  to the first normalized image I 1 ′ represented by a cross-sectional image  410  of  FIG. 4B . A nipple  411  in the cross-sectional image  410  of the first normalized image I 1  illustrated in  FIG. 4B  is located at a coordinate position P 1 ′ obtained by performing the normalization transformation obtained in step S 300  with respect to the position coordinate P 1  of the nipple  401  in the cross-sectional image  400  of the first image I 1  illustrated in  FIG. 4A . Similarly, a body surface  412  is represented by a shape S 1 ′ obtained by transforming the body surface  402  of  FIG. 4A  by the normalization transformation. The body surface  412  is a shape located on a predetermined plane in the first normalized image I 1 ′. In addition, a geodesic distance along the body surface with respect to the nipple  411  on the respective positions on the body surface  412  is set to maintain substantially the same distance before and after the normalization transformation. That is, the body surface  402  of  FIG. 4A  does not expand or contract by the normalization transformation while the nipple  401  is set as the center or is transformed to the body surface  412  of  FIG. 4B  within a small expansion and contraction range. 
     The processing executed by the normalization transformation obtaining unit  103  in step S 301  to obtain the above-described normalization transformation will be described in detail with reference to the flow chart of  FIG. 5 . 
     &lt;Step S 3010 : Obtainment of the Body Surface Shape&gt; 
     In step S 3010  illustrated in  FIG. 5 , the normalization transformation obtaining unit  103  obtains a body surface shape of the object in the first image I 1 . In this processing, for example, the image processing such as edge detection is performed on the first image I 1  so that it is possible to automatically obtain the body surface shape. Alternatively, the first image I 1  may be displayed on the display unit  190  via the display control unit  105  so that the user can observe the first image I 1 , and the body surface shape may be obtained on the basis of the operation input by the user or the like. The body surface shape obtained at this time is denoted by S 1 . According to the first exemplary embodiment, S 1  is constituted by a point group representing the body surface shape, and the point group is obtained as position information in the coordinate system of the first image I 1 , that is, the position information in the real space. 
     &lt;Step S 3011 : Obtainment of the Nipple Position&gt; 
     In step S 3011 , the normalization transformation obtaining unit  103  obtains the nipple position of the object in the first image I 1 . In this processing, for example, the nipple position can be obtained by performing the image processing such as the edge detection on the first image. Alternatively, the nipple position can be obtained while an inflection point of a curvature of the shape, a peak position, or the like may be obtained on the basis of the body surface shape S 1  obtained in step S 3010 . In addition, the nipple position may be obtained on the basis of an input operation by the user or the like. According to the first exemplary embodiment, the obtained nipple position is denoted by P 1 . The nipple position P 1  is obtained as the position information in the coordinate system of the first image, that is, the position information in the real space. 
     &lt;Step S 3012 : Obtainment of the Nipple Position and the Body Surface Shape in the Normalized Space&gt; 
     In step S 3012 , the normalization transformation obtaining unit  103  obtains the nipple position and the body surface shape in the normalized space, that is, the coordinate system of the first normalized image I 1 ′. Hereinafter, the nipple position in the normalized space is denoted by P 1 ′. According to the first exemplary embodiment, P 1 ′ is set as an origin position (x, y, z)=(0, 0, 0) in the normalized space. In addition, hereinafter, the body surface shape in the normalized space is denoted by S 1 ′. S 1 ′ is constituted by a point group in the normalized space. According to the first exemplary embodiment, S 1 ′ is located on a plane at y=0 in the normalized space. The x coordinate values and the z coordinate values of the respective points constituting S 1 ′ are set such that a change in the geodesic distance along the body surface with respect to the nipple before and after the normalization becomes small. In addition, the position of S 1 ′ in the normalized space is set such that a mutual distance of the point group constituting S 1  is maintained. The position of S 1 ′ is determined, for example, by a technique in the related art including a low-strain dimensionality reduction method such as a multi-dimensional scaling method. 
     &lt;Step S 3013 : Obtainment of the Normalization Transformation&gt; 
     In step S 3013 , the normalization transformation obtaining unit  103  obtains the first normalization transformation Tn 1  for normalizing the first image I 1  on the basis of positional relationships between P 1  and P 1 ′ and S 1  and S 1 ′. Specifically, the nomalization transformation can be obtained by using a deformation interpolation method or a deformation obtaining technique in the related art such as a free form deformation (FFD) method or a thin plate spline (TPS) method. For example, in a case where the FFD method is used, optimization of the deformation is carried out by performing iterative calculation such that the positions of the respective points constituting P 1  and S 1  of the first image I 1  corresponding to the original image come close to P 1 ′ and S 1 ′ after the normalization transformation. At this time, the normalization transformation is preferably obtained on the basis of a condition for suppressing distortion of the space, abrupt change, extreme expansion or contraction, or the like caused by the transformation. For example, the normalization transformation may also be obtained by adding a restraint for suppressing a fluctuation of the distance from the body surface caused by the normalization transformation at a position inside the breast of the object. That is, the normalization transformation refers to nonlinear transformation to the normalized space deformed such that the position and the shape of the body surface of the object are substantially matched in the first normalized image I 1 ′ and a second normalized image I 2 ′. 
     With the above-described processing, the first normalization transformation Tn 1  is obtained. Herein, Tn 1  is a transformation function for transforming a position coordinate x in the image coordinate system of the first image I 1  to a position coordinate x′ in the coordinate system of the first normalized image I 1 ′ which will be described below. According to the first exemplary embodiment, an action of the first normalization transformation Tn 1  is represented by Expression 1.
 
 x′=Tn 1( x )  (1)
 
     Furthermore, in step S 3013 , the normalization transformation obtaining unit  103  obtains an inverse transformation Tn 1 _ inv  of the first normalization transformation Tn 1  obtained in the above-described processing. At this time, an action of Tn 1 _ inv  can be represented by Expression 2.
 
 x=Tn 1_ inv ( x ′)  (2)
 
     That is, Tn 1 _ inv  is a transformation function for transforming the position coordinate x′ in the image coordinate system of the first normalized image I 1 ′ to the position coordinate x in the image coordinate system of the first image I 1 . Tn 1 _ inv  is such a transformation that an error of the inverse transformation (Σ∥x-Tn 1 _ inv (Tn 1 ( x ))∥) becomes as small as possible with respect to Tn 1 . It should be noted however that, when a certain transformation T is defined, a method for obtaining an inverse transformation T_inv thereof has been proposed in the related art, and therefore detailed descriptions will be omitted. 
     In step S 3010  to S 3013  described above, the case where the first normalization transformation Tn 1  with regard to the first image I 1  and the inverse transformation Tn 1 _ inv  thereof are obtained has been described. Furthermore, in the processing in step S 301 , the normalization transformation obtaining unit  103  according to the first exemplary embodiment executes processing similar to the processing in step S 3010  to S 3013  described above with respect to the second image I 2 . That is, a body surface shape S 2  and a nipple position P 2  are detected from the second image I 2 , and a second normalization transformation Tn 2  for normalizing the second image I 2  and an inverse transformation Tn 2 _ inv  thereof are obtained on the basis of these pieces of information. In the above-described manner, in step S 301 , the first normalization transformation Tn 1  ( 602 ) and an inverse transformation Tn 1 _ inv  ( 604 ) thereof and the second normalization transformation Tn 2  ( 603 ) and an inverse transformation Tn 2 _ inv  ( 605 ) thereof which are illustrated in  FIG. 6  are obtained. 
     &lt;Step S 302 : Generations of the Normalized Image&gt; 
     In step S 302 , the normalized image generation unit  104  generates the first normalized image I 1 ′ obtained by deforming the first image I 1  obtained in step S 300  on the basis of the first normalization transformation Tn 1  obtained in step S 301 . Similarly, the normalized image generation unit  104  generates the second normalized image I 2 ′ obtained by deforming the second image I 2  on the basis of the second normalization transformation Tn 2 . At this time, the processing of generating the deformed image obtained by deforming the image on the basis of the transformation function may be executed by using any of techniques in the related art. In addition, the inverse transformations Tn 1 _ inv  and Tn 2 _ inv  of the respective normalization transformations may be used in the image transformation. That is, in step S 302 , a first normalized image I 1 ′ ( 606 ) and a second normalized image I 2 ′ ( 607 ) illustrated in  FIG. 6  are obtained. 
     &lt;Step S 303 : Obtainment of the Corresponding Point in the Normalized Space&gt; 
     In step S 303 , the corresponding point obtaining unit  106  obtains positions of a plurality of corresponding points between the first normalized image I 1 ′ and the second normalized image I 2 ′. Hereinafter, this position will be referred to as a corresponding point position in the normalized space. A point used as the corresponding point according to the first exemplary embodiment is, for example, a feature point representing a distinctive structure of the object such as a vascular bifurcation point. That is, the corresponding point obtaining unit  106  obtains the position of the same feature point of the object such as the vascular bifurcation point commonly drawn in the first normalized image I 1 ′ and the second normalized image I 2 ′ as the corresponding point position in the normalized space. Herein, the corresponding point position in the normalized space is a group of pairs of position information in the respective coordinate systems of the first normalized image I 1 ′ and the second normalized image I 2 ′ (normalized image coordinate systems). According to the first exemplary embodiment, i-th corresponding point positions in the normalized space are denoted by C 1 _ i ′ and C 2 _ i ′. Herein, a suffix i represents an index of the corresponding point. According to the first exemplary embodiment, the number of corresponding points obtained in step S 303  is set as Nc. That is, 1 is lower than or equal to i, and also i is lower than or equal to Nc. The corresponding point positions assigned with the same index represent mutually corresponding positions. That is, it is represented that a position of a certain single (i-th) feature point in the object is drawn at a position C 1 _ i ′ in the first normalized image, and the same bifurcation point is drawn at a position C 2 _ i ′ in the second normalized image. 
     An example of the method of obtaining the corresponding point position in the normalized space includes a method of obtaining the corresponding point position by an operation input by the user. That is, the display control unit  105  displays the first normalized image I 1 ′ and the second normalized image I 2 ′ output from the normalized image generation unit  104  and cross-sectional images thereof on the display unit  190 , so that the user can observe the respective images. Subsequently, the user who has observed the respective images can find the corresponding positions between the images. In particular, when an XZ cross section of the normalized image is displayed, a cross section where a region at a position at an equal distance from the body surface is cut out is displayed. At this time, a Y coordinate on the XZ cross section represents a distance from the body surface, that is, a depth, and the user can operate the Y coordinate on the XZ cross section to be displayed. According to this configuration, since the region at the equal distance from the body surface is drawn in the same plane, an advantage is attained that it is facilitated to grasp an overall picture of the vessel or the like that runs along the body surface. In addition, since the distance (depth) from the body surface of the region in the object is hardly changed even when the object is deformed, this information can be used as reference information when the corresponding points are identified between the images. For example, by comparing the XZ cross sections in the vicinity of the same Y coordinate with each other, it is possible to efficiently perform the identification of the corresponding points. When the user inputs the corresponding positions via the operation unit  180 , the corresponding point obtaining unit  106  can obtain the information of the corresponding points. Herein, the operation unit  180  is, for example, an input device such as a mouse or a key board. As a result, the corresponding point obtaining unit  106  can obtain the corresponding point positions in the normalized space found by the user. Subsequently, when an instruction for ending the corresponding point input is received from the user, the corresponding point obtaining unit  106  transmits the corresponding point positions in the normalized space obtained up to this time to the corresponding point transformation unit  107 . Then, the flow proceeds to the processing in step S 304 . Herein, as compared with the first image I 1  and the second image I 2 , since the first normalized image I 1 ′ and the second normalized image I 2 ′ have been subjected to the normalization transformation, the common anatomical features of the object drawn in both the images are at spatially close positions. For this reason, an advantage is attained that it is more facilitated for the user to find the corresponding point positions as compared with a case where the first image I 1  and the second image I 2  are visually observed. 
     It should be noted however that the method of obtaining the corresponding point positions in the normalized space is not limited to the above-described method, and the corresponding point positions may be automatically obtained by performing the image processing on the images including the first normalized image I 1 ′ and the second normalized image I 2 ′. For example, the corresponding point obtaining unit  106  detects a plurality of feature points from the respective images and performs association on the basis of position information related to the plurality of feature points, luminance patterns of the images, or the like, so that the corresponding point positions in the normalized space can be obtained. A specific method for the processing may be executed by using any of techniques in the related art. As compared with the first image I 1  and the second image I 2 , since the first normalized image I 1 ′ and the second normalized image I 2 ′ have been subjected to the normalization transformation, the common anatomical features of the object drawn in both the images are at spatially close positions. For this reason, in a case where the method of automatically obtaining the corresponding point positions is performed instead of using the first image I 1  and the second image I 2 , an advantage is attained in terms of accuracy, stability, and calculation processing load. 
     In the above-described manner, in step S 303 , correspondence information ( 608 ) corresponding to the information of the corresponding points in the normalized space illustrated in  FIG. 6  is obtained. 
     &lt;Step S 304 : Transformation of the Corresponding Point Position to the Real Space&gt; 
     In step S 304 , the corresponding point transformation unit  107  transforms the corresponding point positions in the normalized space obtained in step S 303  on the basis of the inverse transformations Tn 1 _ inv  and Tn 2 _ inv  of the normalization transformation obtained in step S 301  to obtain the corresponding point positions in the real space. The obtainment of the corresponding point position in the real space is performed by calculation processing specifically represented by the following expression.
 
 C 1_ i=Tn 1_ inv ( C 1_ i ) (1 is lower than or equal to  i , and also  i  is lower than or equal to  Nc )  (3)
 
 C 2_ i=Tn 2_ inv ( C 2_ i ) (1 is lower than or equal to  i , and also  i  is lower than or equal to  Nc )  (4)
 
     Where C 1 _ i  (1 is lower than or equal to i, and also i is lower than or equal to Nc) denotes the corresponding point position on the first image I 1  corresponding to the i-th corresponding point position C 1 _′ on the first normalized image I 1 ′ (1 is lower than or equal to i, and also i is lower than or equal to Nc). Similarly, C 2 _ i  (1 is lower than or equal to i, and also i is lower than or equal to Nc) denotes the corresponding point position on the second image I 2  corresponding to the i-th corresponding point position C 2 _ i ′ on the second normalized image I 2 ′ (1 is lower than or equal to i, and also i is lower than or equal to Nc). According to the first exemplary embodiment, C 1 _ i  and C 2 _ i  described above (1 is lower than or equal to i, and also i is lower than or equal to Nc) are collectively referred to as the corresponding point position in the real space. 
     In the above-described manner, in step S 304 , correspondence information ( 609 ) corresponding to the information of the corresponding point in the real space illustrated in  FIG. 6  is obtained. 
     &lt;Step S 305 : Obtainment of the Deformation Transformation&gt; 
     In step S 305 , the deformation transformation obtaining unit (deformation estimation unit)  108  executes processing of obtaining the deformation transformation between the first image I 1  and the second image I 2  on the basis of the corresponding point position in the real space obtained in step S 304 . The deformation transformation is obtained on the basis of the information of the regularizing the deformation of the object in the real space. The processing of obtaining the deformation transformation may also be regarded as deformation estimation processing of estimating deformation between the images or deformation registration processing of performing deformation registration between the images. 
     Herein, any of techniques in the related art may be used for the deformation processing between the images based on the corresponding point position in the real space, and the deformation processing can be executed by using the TPS method, for example. According to this method, it is possible to obtain the deformation transformation with which C 1 _ i  and C 2 _ i  (1 is lower than or equal to i, and also i is lower than or equal to Nc) corresponding to the position coordinate values on the respective images of the first image I 1  and the second image I 2  are matched with each other, and also bending energy caused by the deformation is minimized. Specifically, each C 1 _ i  (1 is lower than or equal to i, and also i is lower than or equal to Nc) is transformed to C 2 _ i  (1 is lower than or equal to i, and also i is lower than or equal to Nc), and the spatial distortion in the real space caused by the transformation is defined as energy. The energy of the distortion can be defined as a value calculated by a spatial secondary differentiation of the deformation transformation, for example. Subsequently, the deformation transformation with which the total amount of the energy of the distortion is minimized is obtained. That is, the deformation transformation obtaining unit (deformation estimation unit)  108  obtains the deformation transformation on the basis of the information of the regularizing the deformation so as to be an appropriate deformation in the real space of the object. The obtainment of the deformation transformation based on the information of the regularization is not limited to the case based on the value calculated by the spatial secondary differentiation of the above-described deformation transformation. For example, the definition can also be made on the basis of a size of the specially high frequency component of the displacement amount based on the deformation transformation, a size of a local volume change rate based on the deformation transformation, or the like. 
     In addition, the obtainment of the deformation transformation can be executed on the basis of the information of the nipple position and the body surface shape in the first image I 1  and the second image I 2 . For example, the deformation transformation can be obtained by further adding a restraint for matching the nipple position P 1  in the first image I 1  and the nipple position P 2  in the second image I 2  with each other. Moreover, the deformation transformation can be obtained by further adding a restraint for matching the body surface shapes S 1  and S 2  with each other. At this time, since the point groups respectively constituting the body surface shapes S 1  and S 2  are not associated with each other, it is necessary to perform the registration between a curved surface of the body surface represented by the point group constituting the body surface shape S 1  and a curved surface of the body surface represented by the point group constituting the body surface shape S 2 . This processing can be executed by a technique in the related art such as, for example, an iterative closest point (ICP) method. In addition, after the deformation transformation obtained by the TPS method on the basis of the corresponding point position or the nipple position in the real space is applied to the body surface shape S 1  to obtain a deformed shape having a shape closer to the body surface shape S 2 , the association of the point group on the body surface shape may be performed between the point groups representing the two shapes by using the ICP method or the like. Subsequently, while the information of the associated point group on the body surface is added, the deformation transformation may be obtained by the TPS method again. Furthermore, the obtainment of the deformation transformation and the update of the association may be repeatedly executed until convergence is realized. 
     As a result, it is possible to obtain the deformation transformation from the first image I 1  to the second image I 2 , that is, the transformation for projecting the position x 1  in the coordinate system of the first image I 1  onto the corresponding position x 2  in the coordinate system of the second image I 2 . According to the first exemplary embodiment, the deformation transformation is denoted by T 1 _ 2 . At this time, an action of T 1 _ 2  can be represented by Expression 5.
 
 x 2= T 1_2( x 1)  (5)
 
     That is, in a case where an arbitrary position in the object is drawn at a position of the coordinate x 1  in the first image I 1  and drawn at a position of the coordinate x 2  in the second image, the deformation transformation T 1 _ 2  provides the transformation from x 1  to x 2 . It should be noted however that, in step S 305 , the deformation transformation obtaining unit (deformation estimation unit)  108  may further obtain an inverse transformation of the above-described deformation transformation T 1 _ 2 , that is, the deformation transformation from the second image I 2  to the first image I 1 . Since the obtainment of the inverse transformation is similar to the obtainment of the inverse transformation of the normalization transformation described in step S 3013 , descriptions thereof will be omitted. Herein, the inverse transformation of T 1 _ 2  is denoted by T 2 _ 1 . At this time, an action of T 2 _ 1  can be represented by Expression 6.
 
 x 1= T 2_1( x 2)  (6)
 
     With the above-described processing, the deformation transformation between the first image I 1  and the second image I 2  is obtained. In the processing in step S 305 , a deformation transformation ( 610 ) from the first image I 1  illustrated in  FIG. 6  to the second image I 2  is obtained. Even if an attempt is made to perform the registration by deforming at least one of the first normalized image I 1 ′ and the second normalized image I 2 ′, it is difficult to use the information of the regularization on which a physical validity of the deformation in the real space is reflected. According to the first exemplary embodiment, the corresponding point position in the normalized space which is deformed such that it is facilitated to identify the corresponding point is transformed to the corresponding point position in the real space, and it is possible to obtain the deformation transformation for performing the registration on which the physical validity of the deformation in the real space is reflected. 
     &lt;Step S 306 : Generation of the Deformed Image&gt; 
     In step S 306 , the deformed image generation unit  109  generates a first deformed image I 1 _ d  obtained by such a deformation that the first image I 1  is substantially matched with the second image I 2  on the basis of the deformation transformation T 1 _ 2  obtained in step S 305 . In addition, the deformed image generation unit  109  may generate a second deformed image I 2 _ d  obtained by such a deformation that the second image I 2  is substantially matched with the first image I 1  on the basis of the deformation transformation T 2 _ 1  obtained in step S 305 . It should be noted however that the method of generating the deformed image by using the deformation transformation has been proposed in the related art, and therefore detailed descriptions will be omitted. 
     &lt;Step S 307 : Display of the Deformed Image&gt; 
     In step S 307 , the display control unit  105  performs control to display the first deformed image I 1 _ d  and the second deformed image I 2 _ d  generated in step S 306  on the display unit  190 . In addition, the display control unit  105  may display the first image I 1  and the second deformed image I 2 _ d  in a manner that the images can be compared with each other on the display unit  190  in accordance with an input operation or the like by the user. Similarly, the first deformed image I 1 _ d  and the second image I 2  may be displayed in a manner that the images can be compared with each other on the display unit  190 . 
     In the above-described manner, the processing of the image processing apparatus according to the first exemplary embodiment is executed. According to the first exemplary embodiment, it is possible to provide the system where the normalized image is displayed which is easy for the user who observes the images to perform the identification of the corresponding point position, and the image is deformed on the basis of the corresponding point position specified on the image. After the corresponding point on the normalized image which is input by the user is transformed to the corresponding position on the original input image, the deformation between the images can be obtained. Therefore, it is possible to perform the transformation on which the physical restraint in the real space such as the minimization of the bending energy related to the deformation in the original input image, that is, the information of the regularization is reflected. That is, it is possible to obtain the deformation transformation in conformity to the physical actual condition with regard to the deformation of the object. In particular, in a case where the first image and the second image are the images in the real space, various algorithms related to the registration which has been proposed thus far can be used. 
     Modified Example 1-1 
     According to the first exemplary embodiment, as the processing in step S 301 , the normalization transformation for transforming the points at an equal distance from the body surface to the same plane is performed. However, the method for the coordinate transformation to facilitate the identification of the corresponding point is not limited to the above. For example, by using the technology described in PTL 2, the normalization transformation for transforming the points at an equal distance from the nipple position in the image to the same plane may be obtained. 
     In addition, as another modified example, the coordinate transformation based on the chest wall shape of the object may be used. For example, the above-described coordinate transformation can be used in a case where the MRI images or the CT images capturing a range wider than the surrounding of the breast corresponding to the object are used as the first image and the second image. In this case, the chest wall shape is also drawn in both the first image and the second image in addition to the nipple position and the body surface shape. For this reason, as the processing of obtaining the normalization transformation executed in step S 301  by the normalization transformation obtaining unit  103 , it is possible to obtain the normalization transformation based on the nipple position, the body surface shape, and the chest wall shape of the object which are drawn in the image. For example, by using the technology described in PTL 1, the normalization transformation for normalizing each of the first image and the second image can be obtained. According to this method, since it is possible to obtain the normalization transformation for projecting the deep part of the breast corresponding to the inspection target region of the object, that is, the chest wall located on an inner side with respect to the body surface of the object at a predetermined position, an advantage is attained that it is facilitated to perform the comparison between the image at the above-mentioned position. 
     In addition, according to the first exemplary embodiment, the case has been described as an example where the image of the breast is the target of the registration, but the target is not limited to the above. For example, an image of another organ such as heart or liver may also be the target of the registration. In this case, various normalization techniques in the related art for transforming the image of the target organ to the normalized space can be used. At this time, as a normalization method, similarly as in the first exemplary embodiment, the normalization based on the distance from the surface of the organ may also be used. In addition, as described in PTL 2, the normalization based on the distance from the particular landmark position may also be used. 
     Modified Example 1-2 
     According to the first exemplary embodiment, the case has been described as an example where, in step S 303 , the corresponding point position in the normalized space is obtained, and in step S 304 , the processing of transforming the corresponding point position in the normalized space to the corresponding point position in the real space is executed. However, the implementation of the present invention is not limited to the above. For example, the corresponding point position obtained in the processing in step S 304  may also include the corresponding point position directly obtained from the first image and the second image corresponding to the image in the real space in addition to the position obtained by transforming the corresponding point position in the normalized space to the real space. In this case, the display control unit  105  performs control to display the first image and the second image on the display unit  190 . Subsequently, the corresponding point obtaining unit  106  can obtain an anatomically characteristic position or the like commonly drawn in the respective images through the operation input by the user via the operation unit  180 . Then, the obtained position may be added to the corresponding point position in the above-described real space. As a result, it is possible to additionally obtain the corresponding point position that is difficult to be identified in the normalized image and also can be identified in the original input image. 
     In addition, as another modified example, the corresponding point positions may be obtained in the plurality of transformed images obtained by the various coordinate transformations illustrated in the modified example 1-1 and respectively transformed to the corresponding point positions in the real space. For example, the transformed image on which the coordinate transformation has been performed such that it is facilitated to identify the corresponding point in the vicinity of the body surface of the object and the transformed image on which the coordinate transformation has been performed such that it is facilitated to identify the corresponding point inside the object may be used in combination. As a result, it is possible to perform the obtainment of the corresponding point more accurately. 
     According to these modified examples, since the processing in step S 305  and the subsequent steps can be performed on the basis of still more corresponding point positions, an advantage is attained that the highly accurate deformation transformation can be obtained, and the deformed image can be displayed. 
     Modified Example 1-3 
     In the processing in step S 305  according to the first exemplary embodiment, in addition to the above-described processing, the following processing may be further executed. That is, the information related to the corresponding point position in the real space obtained in step S 304  and the nipple positions P 1  and P 2  and the body surface shapes S 1  and S 2  obtained in the processing in step S 302  may be saved in the data server  170 . In this case, the information is preferably saved while being associated with the first image I 1  and the second image I 2  obtained in step S 300  by the data obtaining unit  102 . In the processing in step S 300 , in addition to the above-described processing, the following processing may be further executed. That is, in a case where the data server  170  saves the information related to the corresponding point position, the nipple position, and the body surface shape in the real space which are associated with the first image I 1  and the second image I 2 , the processing of reading out and obtaining these pieces of information may be performed. Subsequently, in a case where these pieces of information are read out and obtained, in the processing in step S 301 , the processing in steps S 3010  and S 3011  can be omitted, and the processing in steps S 3012  and S 3013  can be performed on the basis of the read information. In addition, the processing in steps S 303  and S 304  may be omitted. Alternatively, the corresponding point position in the real space obtained by executing the processing in steps S 303  and S 304  may be integrated with the corresponding point position in the real space obtained by the readout from the data server  170 , and the subsequent processing may be executed. The processing for the transformation to the normalized space may be applied to the information such as the corresponding point position in the real space read from the data server  170  to be displayed on the display unit  190  together with the first normalized image I 1 ′ and the second normalized image I 2 ′. In addition, the similar processing may be performed on not only the information read out from the data server  170  but also the information obtained in step S 303  according to the first exemplary embodiment. As a result, the obtainment of the corresponding point position which uses the user operation can be executed in multiple courses, and it is possible to provide the system where the addition of the information of the corresponding point position or the like can be flexibly performed. 
     Second Embodiment: Embodiment Using the FFD Method 
     According to the first exemplary embodiment, the case has been described as an example where the deformation transformation between the images is obtained by the TPS method, but the implementation of the present invention is not limited to the above. According to a second exemplary embodiment, a case will be described where the deformation transformation between the images is obtained by using the free form deformation (FFD) method. 
     A function configuration according to the second exemplary embodiment is similar to  FIG. 1  in which the function configuration according to the first exemplary embodiment is exemplified. Here, the descriptions thereof will be omitted. In addition, the entire processing procedure according to the second exemplary embodiment can be described with reference to  FIG. 2  in which the processing procedure according to the first exemplary embodiment is exemplified. Hereinafter, a detail of the processing according to the second exemplary embodiment will be described. 
     In the processing in step S 300  to step S 304 , the processing similar to that of the first exemplary embodiment is performed. Here, the descriptions thereof will be omitted. 
     &lt;Step S 305  According to Second Embodiment&gt; 
     In step S 305 , the deformation transformation obtaining unit (deformation estimation unit)  108  executes the processing of obtaining the deformation transformation between the first image I 1  and the second image I 2  on the basis of the corresponding point positions C 1 _ i  and C 2 _ i  in the real space obtained by the transformation processing in step S 304 . According to the second exemplary embodiment, the deformation transformation for substantially matching each of the corresponding point positions C 1 _ i  and C 2 _ i  in the real space to each other is obtained by using the FFD method. The FFD method is a deformation representation method of arranging control points in a grid manner and obtaining the displacement amount in the space by using a control amount at each control point as a deformation parameter. The deformation representation obtainment based on the FFD method is a technique in the related art, and therefore detailed descriptions will be omitted. Here, the method of obtaining the control amount for determining a characteristic of the deformation transformation in the FFD method, that is, the deformation parameter will be described. 
     According to the second exemplary embodiment, the deformation transformation obtaining unit (deformation estimation unit)  108  obtains a deformation transformation Φ for minimizing a cost function represented by Expression 7.
 
 E (Φ)= E   distance (Φ)+λ smoothness   E   smoothness (Φ)+λ compression   E   compression (Φ)  (7)
 
     In Expression 7, E distance (Φ) denotes a data term in the cost function. Specifically, the distance error of the corresponding point position caused by the deformation transformation is calculated by a calculation represented by Expression 8. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       E 
                       distance 
                     
                     ⁡ 
                     
                       ( 
                       Φ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             C 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _i 
                           
                           - 
                           
                             Φ 
                             ⁡ 
                             
                               ( 
                               
                                 C 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 _i 
                               
                               ) 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Where Φ(x) represents a function for obtaining the position coordinate of the transformation result based on the deformation transformation Φ of the position coordinate x. E smoothness (Φ) in Expression 7 is a regularization term (information of the regularization) for obtaining a smooth deformation and can be calculated by the bending energy of the deformation transformation Φ obtained by a calculation represented by Expression 9, for example. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       2 
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       E 
                       smoothness 
                     
                     ⁡ 
                     
                       ( 
                       Φ 
                       ) 
                     
                   
                   = 
                   
                     ∫ 
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       ∂ 
                                       
                                         x 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       ∂ 
                                       
                                         y 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       ∂ 
                                       
                                         z 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       
                                         ∂ 
                                         x 
                                       
                                       ⁢ 
                                       
                                         ∂ 
                                         y 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       
                                         ∂ 
                                         x 
                                       
                                       ⁢ 
                                       
                                         ∂ 
                                         z 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       ⁢ 
                                       Φ 
                                     
                                     
                                       
                                         ∂ 
                                         y 
                                       
                                       ⁢ 
                                       
                                         ∂ 
                                         z 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           dxdydz 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In Expression 7, E compression (Φ) is a regularization term (information of the regularization) related to a local volume change in the spatial transformation caused by the deformation transformation Φ and can be calculated by a calculation represented by Expression 10, for example. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       E 
                       compression 
                     
                     ⁡ 
                     
                       ( 
                       Φ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       Ω 
                     
                     ⁢ 
                     
                       
                          
                         
                           log 
                           ⁡ 
                           
                             [ 
                             
                               det 
                               ⁢ 
                               
                                 { 
                                 
                                   J 
                                   ⁡ 
                                   
                                     ( 
                                     Φ 
                                     ) 
                                   
                                 
                                 } 
                               
                             
                             ] 
                           
                         
                          
                       
                       ⁢ 
                       dx 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Where J(Φ) represents a function for returning a Jacobian matrix corresponding to a spatial differentiation related to the deformation transformation Φ and a function for returning a matrix of three rows and three columns according to the second exemplary embodiment where the three-dimensional deformation is set as the target. In addition, det( ) is a function for obtaining a determinant of a matrix of an argument. Expression 10 represents a calculation for a size of the local volume change in the spatial transformation caused by the deformation transformation Φ and spatially integrating the size. 
     In addition, in Expression 7, λ smoothness  and λ compression  respectively denote the regularization parameter related to the smoothness of the deformation and the regularization parameter related to the volume change and control the respective intensities of the influences of the regularizations. 
     According to the second exemplary embodiment, the deformation transformation Φ obtained by the above-described processing is denoted by T 1 _ 2 . The deformation transformation T 1 _ 2  is the transformation function for transforming the position coordinate in the image coordinate system of the first image I 1  to corresponding position coordinate in the image coordinate system of the second image I 2 . 
     In the processing in step S 306  and step S 307 , the processing similar to that of the first exemplary embodiment is executed. Here, the descriptions thereof will be omitted. 
     With the above-described processing, the processing of the image processing apparatus according to the second exemplary embodiment is executed. According to the second exemplary embodiment, it is possible to obtain the deformation transformation in which the deformation accompanied with the local volume change of the object is suppressed as compared with the first exemplary embodiment. After the input corresponding point input by the user in the normalized space is transformed to the corresponding position on the original input image, the deformation transformation between the images can be obtained, so that it is possible to obtain the deformation transformation in which the local volume change with regard to the deformation in the original input image is suppressed. That is, with regard to the deformation of the object, it is possible to obtain the more highly accurate deformation transformation based on the information of the regularization in the real space. 
     Third Embodiment: Embodiment Using the Regularization in the Normalized Space in Combination 
     According to the first exemplary embodiment and the second exemplary embodiment, the case has been described as an example where, when the deformation transformation between the first image I 1  and the second image I 2  is obtained, the application of the regularization with regard to the deformation of the object in the real space is performed. On the other hand, it is characterized in that the image processing apparatus according to a third exemplary embodiment further uses the regularization in the normalized space in combination. 
     The function configuration and the entire processing procedure according to the image processing apparatus according to the third exemplary embodiment are similar to those according to the second exemplary embodiment. It should be noted that part of the processing performed in step S 305  by the deformation transformation obtaining unit (deformation estimation unit)  108  differs. Hereinafter, only the difference from the second exemplary embodiment will be described. 
     &lt;Step S 305  According to Third Embodiment&gt; 
     In step S 305 , the deformation transformation obtaining unit (deformation estimation unit)  108  obtains the deformation transformation between the first image and the second image on the basis of the corresponding point position in the real space which is obtained in step S 304 . At this time, the deformation transformation obtaining unit (deformation estimation unit)  108  according to the third exemplary embodiment uses a function obtained by further adding a cost term (regularization term) for performing the regularization in the normalized space to the cost function for the obtainment of the deformation transformation defined by Expression 7 according to the second exemplary embodiment as the cost function for the obtainment of the deformation transformation. That is, the deformation transformation obtained in step S 305  is configured to include the restraint of the regularization with regard to the deformation in the normalized space. Only this aspect is the difference from the processing performed in step S 305  according to the second exemplary embodiment by the deformation transformation obtaining unit (deformation estimation unit)  108 . 
     Hereinafter, a method of obtaining the cost term (regularization term) for performing the regularization in the normalized space will be described. First, the deformation transformation obtaining unit (deformation estimation unit)  108  obtains the deformation transformation T 1 _ 2 ′ between the first normalized image I 1 ′ and the second normalized image I 2 ′ on the basis of a candidate value of the deformation transformation between the first image I 1  and the second image I 2  T 1 _ 2 . Herein, the deformation transformation T 1 _ 2 ′ between the first normalized image I 1 ′ and the second normalized image I 2 ′ is a deformation function that affects in the following manner.
 
 x 2′= T 1_2′( x 1′)  (11)
 
     At this time, T 1 _ 2 ′ can be obtained by linking the transform Tn 1 _ inv  from the first normalized image I 1 ′ to the first image I 1 , the transform T 1 _ 2  from the first image I 1  to the second image I 2 , and furthermore, the transform Tn 2  from the second image I 2  to the second normalized image I 2 ′ in the stated order with one another. That is, these transformations satisfy a relationship of the following expression.
 
 x 2′= Tn 2( T 1_2( Tn 1_ inv ( x 1′)))  (12)
 
     That is, T 1 _ 2 ′ can be obtained with respect to the candidate value of the deformation transformation Φ (=T 1 _ 2 ). Therefore, similarly as in the real space, when an arbitrary regularization term in the normalized space is formulated, the value with respect to the candidate value of Φ can be obtained on the basis of T 1 _ 2 ′. As a specific example, it is possible to introduce regularization for suppressing a shear deformation between the body surface and the deep part in the normalized space. This regularization is effective to accurately estimate the deformation of the object. 
     According to the above-described method, since the body surface shape of the object corresponding to the curved surface in the first image I 1  and the second image I 2  is projected on a predetermined plane in the normalized space, an advantage is attained that the regularization in which the body surface shape of the object is used as the reference can be realized by the more simplified processing. 
     Fourth Embodiment: Embodiment in Which the Corresponding Point is Input on the Normalized Image 
     According to the first exemplary embodiment, the case has been described as an example where the information related to the corresponding point position is obtained on the normalized image, but the present invention is not limited to the above. According to the fourth exemplary embodiment, a case will be described where the information related to the corresponding point position is obtained by further using the first deformed image obtained by the registration of the first image with respect to the second image. 
       FIG. 7  is a block diagram illustrating an example of a function configuration of an image processing apparatus  700  according to the fourth exemplary embodiment. It should be noted however that configurations similar to those of  FIG. 1  are assigned with the same reference symbols, and detailed descriptions thereof will be omitted here by citing the above-described explanations. 
     The image processing apparatus  700  includes the data obtaining unit  102 , the normalization transformation obtaining unit  103 , the normalized image generation unit  104 , a display control unit  705 , a corresponding point obtaining unit  706 , a corresponding point transformation unit  707 , a deformation transformation obtaining unit (deformation estimation unit)  708 , and the deformed image generation unit  109 . 
     The display control unit  705  displays the first image, the second image, the first normalized image, the second normalized image, the first deformed image, and a first normalized deformed image which will be described below on the display unit  190  in accordance with the operation input by the user or the control based on the predetermined program. The first normalized deformed image is an image obtained by normalizing the first deformed image. 
     The corresponding point obtaining unit  706  obtains the information related to the positions of the corresponding points on the first normalized image and the second normalized image in accordance with the operation input on the operation unit  180  by the user. In addition, the corresponding point obtaining unit  706  obtains the information related to the positions of the corresponding points on the first image and the second image. Moreover, the corresponding point obtaining unit  706  obtains the information related to the positions of the corresponding points on the first deformed image and the second image. Furthermore, the corresponding point obtaining unit  706  obtains the information related to the positions of the corresponding points on the first normalized deformed image and the second normalized image. 
     The corresponding point transformation unit  707  transforms the corresponding point position from the normalized space to the real space corresponding to the space of each of the original images on the basis of the normalization transformation obtained by the normalization transformation obtaining unit  103 . In addition, the corresponding point transformation unit  707  transforms the corresponding point position on the first normalized deformed image to the position on the first deformed image on the basis of the normalization transformation. Moreover, the corresponding point transformation unit  707  transforms the corresponding point position on the first deformed image to the position on the first image on the basis of the deformation transformation obtained by the deformation transformation obtaining unit (deformation estimation unit)  708 . That is, the corresponding point position on the first deformed image is transformed to the position in the real space corresponding to the space of the original image. Furthermore, the corresponding point transformation unit  707  transforms the positions of the corresponding points in the first image, the second image, and the first deformed image to the positions of each of the first normalized image, the second normalized image, and the first normalized deformed image on the basis of the normalization transformation. The corresponding point transformation unit  707  also transforms the corresponding point position in the first image to the position in the first deformed image on the basis of the deformation transformation. 
     The deformation transformation obtaining unit (deformation estimation unit)  708  performs the deformation estimation between the first image and the second image on the basis of the corresponding point position obtained in the real space and the corresponding point positions in the real space obtained on the normalized image, the deformed image, and the normalized deformed image and transformed to the real space and obtains the deformation transformation between the images. 
     A hardware configuration of the image processing apparatus  700  is similar to the example illustrated in  FIG. 2 . Detailed descriptions thereof will be omitted here by citing the above-described explanations. 
       FIG. 8  is a flow chart illustrating an example of the entirety of the processing performed by the image processing apparatus  700 . Herein, the processing in step S 800  to step S 802  is similar to the processing in step S 300  to step S 302  illustrated in  FIG. 3 . The processing in step S 804  and step S 805  is similar to the processing in step S 303  and step S 304  illustrated in  FIG. 3  except that the flow proceeds to step S 815  after step S 805 . Detailed descriptions of the similar processing will be omitted here by citing the above-described explanations. 
     &lt;Step S 803 : Determination on Whether or Not the Corresponding Point Position on the Normalized Image is Obtained&gt; 
     In step S 803 , the corresponding point obtaining unit  706  determines whether or not the corresponding point position on the normalized image is obtained. For example, the corresponding point obtaining unit  706  obtains the information of the instruction input by the user via the operation unit  180  and performs the above-described determination in accordance with the content of the instruction. For example, the instruction is performed by the operation input for specifying the corresponding point on the normalized image displayed on the display unit  190 . Alternatively, the instruction is performed by the operation input with respect to the user interface where it is possible to select the image for inputting the corresponding point which is displayed on the display unit  190 . When it is determined that the corresponding point position on the normalized image is obtained, the flow proceeds to step S 804 . When it is not determined that the corresponding point position on the normalized image is obtained, the flow proceeds to step S 806 . 
     &lt;Step S 806 : Determination on Whether or Not the Corresponding Point Position is Obtained on the real Image&gt; 
     In step S 806 , the corresponding point obtaining unit  706  determines whether or not the corresponding point position is obtained on the real image. The real image includes the first image I 1  and the second image I 2 . For example, the corresponding point obtaining unit  706  obtains the information of the instruction input by the user via the operation unit  180  and performs the above-described determination in accordance with the content of the instruction. For example, the instruction is performed by the operation input for specifying the corresponding point on the real image displayed on the display unit  190 . Alternatively, the instruction is performed by the operation input with respect to the user interface where it is possible to select the image for inputting the corresponding point which is displayed on the display unit  190 . When it is determined that the corresponding point position is obtained on the real image, the flow proceeds to step S 807 . When it is not determined that the corresponding point position is obtained on the real image, the flow proceeds to step S 808 . 
     &lt;Step S 807 : Obtainment of the Corresponding Point on the Real Image&gt; 
     In step S 807 , the corresponding point obtaining unit  706  obtains the corresponding point position between the first image I 1  and the second image I 2 . The corresponding point obtaining unit  706  obtains the corresponding point position on the basis of the operation input with respect to each of the first image I 1  and the second image I 2  displayed on the display unit  190 . When the corresponding point is obtained, the flow proceeds to step S 815 . 
     &lt;Step S 808 : Determination on Whether or Not the Corresponding Point Position is Obtained on the Deformed Image&gt; 
     In step S 808 , the image processing apparatus  700  determines whether or not the corresponding point position is obtained on the deformed image or the normalized deformed image. The deformed image is the first deformed image I 1 _ d  obtained in step S 814 . The normalized deformed image is a first normalized deformed image I 1 _ d ′ obtained in step S 814 . For example, the corresponding point obtaining unit  706  obtains the information of the instruction input by the user via the operation unit  180  and performs the above-described determination in accordance with the content of the instruction. For example, the instruction is performed by the operation input for specifying the corresponding point on the deformed image displayed on the display unit  190 . Alternatively, the instruction is performed by the operation input with respect to the user interface where it is possible to select the image for inputting the corresponding point which is displayed on the display unit  190 . When it is determined that the corresponding point position is obtained on the deformed image, the flow proceeds to step S 809 . When it is not determined that the corresponding point position is obtained on the deformed image, the flow proceeds to step S 811 . 
     &lt;Step S 809 : Determination on Whether or Not the Deformation Transformation has been Already Obtained&gt; 
     In step S 809 , the deformation transformation obtaining unit  708  determines whether or not the deformation transformation has been already obtained. In a case where the deformation transformation T 1 _ 2  with regard to the first image I 1  and the second image I 2  has been already obtained by the deformation transformation obtaining unit  708 , the flow proceeds to step S 810 . In a case where the deformation transformation T 1 _ 2  has not been already obtained, the flow proceeds to step S 811 . 
     &lt;Step S 810 : Obtainment of the Corresponding Point Position on the Deformed Image&gt; 
     In step S 810 , the corresponding point obtaining unit  706  obtains the corresponding point on the deformed image. Hereinafter, a case where the first deformed image I 1 _ d  is generated from the first image I 1  by the deformation transformation T 1 _ 2  will be described as an example. The corresponding point obtaining unit  706  obtains the corresponding point position between the first deformed image I 1 _ d  and the second image I 2 . The obtained corresponding point position is transformed to a position in the real space by the deformation transformation T 2 _ 1 . In addition, the corresponding point obtaining unit  706  obtains the corresponding point position between the first normalized deformed image I 1 _ d ′ and the second normalized image I 2 ′. The obtained corresponding point position is transformed to a position in the real space by the inverse transformation of the normalization transformation Tn 1 _ d  and the deformation transformation T 1 _ 2 . The corresponding point obtaining unit  706  obtains information of the transformation of the corresponding point position obtained in the deformed image or the normalized deformed image to the corresponding point position in the real space. When the corresponding point is obtained, the flow proceeds to step S 815 . 
       FIG. 9  is a flow chart for describing an example of the processing in step S 810 . 
     &lt;Step S 8100 : Determination on Whether or Not the Corresponding Point Position is Obtained on the Normalized Deformed Image&gt; 
     In step S 8100 , the corresponding point obtaining unit  706  determines whether or not the corresponding point position is obtained on the normalized deformed image. The normalized deformed image refers to an image obtained by further normalizing the deformed image generated by the deformation transformation. The corresponding point obtaining unit  706  obtains the information of the instruction input by the user via the operation unit  180  and performs the above-described determination in accordance with the content of the instruction. The instruction is performed, for example, by the operation input for specifying the corresponding point on the normalized deformed image displayed on the display unit  190 . Alternatively, the instruction is performed by the operation input with respect to the user interface where it is possible to select the image for inputting the corresponding point which is displayed on the display unit  190 . When it is determined that the corresponding point position is obtained on the normalized deformed image, the flow proceeds to step S 8103 . When it is not determined that the corresponding point position is obtained on the normalized deformed image, the flow proceeds to step S 8101 . 
     &lt;Step S 8101 : Obtainment of the Corresponding Point Position on the Deformed Image&gt; 
     In step S 8101 , the corresponding point obtaining unit  706  obtains the corresponding point position between the first deformed image I 1 _ d  and the second image I 2 . The corresponding point obtaining unit  706  obtains the corresponding point position between the first deformed image I 1 _ d  and the second image I 2  in accordance with the operation input by the user via the operation unit  180 . 
     &lt;Step S 8102 : Obtainment of the Corresponding Point Position on the Normalized Deformed Image&gt; 
     In step S 8102 , the corresponding point obtaining unit  706  obtains the corresponding point position between the first normalized deformed image I 1 _ d ′ and the second normalized image I 2 ′. The corresponding point obtaining unit  706  obtains the corresponding point position between the first normalized deformed image I 1 _ d ′ and the second normalized image I 2 ′ in accordance with the operation input by the user via the operation unit  180 . 
     &lt;Step S 8103 : Transformation of the Corresponding Point Position onto the Real Image&gt; 
     In step S 8103 , the corresponding point transformation unit  707  transforms the corresponding point position obtained in the processing in step S 8101  and step S 8102  to the corresponding point position in the real space. 
     The corresponding point transformation unit  707  transforms the corresponding point position on the first deformed image I 1 _ d  obtained in step S 8101  on the basis of the deformation transformation T 2 _ 1  and obtains the corresponding point position on the first image I 1 , that is, the corresponding point position on the real image. The processing of transforming the corresponding point position in the first deformed image I 1 _ d  to the corresponding point position in the first image I 1  is represented, for example, by Expression 13.
 
 C 1= T 2_1( C 1_ d )  (13)
 
     Where C 1 _ d  denotes the corresponding point specified on the first deformed image I 1 _ d.    
     In addition, the corresponding point transformation unit  707  transforms the corresponding point position on the first normalized deformed image I 1 _ d ′ on the basis of the normalization transformation Tn 1 _ d  and the deformation transformation T 1 _ 2  obtained in step S 8102  and obtains the corresponding point position on the first image I 1 , that is, the corresponding point position on the real image. The processing of transforming the corresponding point position in the first normalized deformed image I 1 _ d ′ to the corresponding point position in the first image I 1  is represented, for example, by Expression 14.
 
 C 1= T 2_1( Tn 1_ d _ inv ( C 1_ d ))  (14)
 
     Where Tn 1 _ d _ inv  denotes the inverse transformation of Tn 1 _ d , and the inverse transformation is obtained by a related-art method. In the above-described manner, the processing of obtaining the corresponding point position on the deformed image is executed, and the flow proceeds to step S 815 . 
     &lt;Step S 811 : Determination on Whether or Not the Deformation Transformation is Obtained&gt; 
     In step S 811 , the deformation transformation obtaining unit  708  determines whether or not the deformation transformation is obtained. For example, the deformation transformation obtaining unit  708  obtains the determination instruction by the operation input by the user via the operation unit  180 . The instruction is performed by the operation input with respect to the user interface for instructing the obtainment of the deformation transformation. When it is determined that the deformation transformation is obtained, the flow proceeds to step S 812 . When it is not determined that the deformation transformation is obtained, the flow proceeds to step S 815 . 
     &lt;Step S 812 : Determination on Whether or Not the Number of Corresponding Points is Sufficient&gt; 
     In step S 812 , the deformation transformation obtaining unit  708  determines whether or not the number of corresponding points is sufficient for the obtainment of the deformation transformation. For example, the deformation transformation obtaining unit  708  determines that the number of corresponding points is sufficient in a case where the number is higher than a predetermined number. In a case where the number of corresponding points is sufficient, the flow proceeds to step S 813 . In a case where the number of corresponding points is not sufficient, the flow proceeds to step S 815 . 
     &lt;Step S 813 : Obtainment of the Deformation Transformation&gt; 
     In step S 813 , the deformation transformation obtaining unit (deformation estimation unit)  708  obtains the deformation transformation between the first image I 1  and the second image I 2  T 1 _ 2  on the basis of the corresponding point position in the real space obtained in up to step S 813 . Since the processing of obtaining the deformation transformation T 1 _ 2  is similar to the processing in step S 305 , detailed descriptions thereof will be omitted here by citing the above-described explanations. 
     &lt;Step S 814 : Generation of the Deformed Image&gt; 
     In step S 814 , the deformed image generation unit  109  generates the first deformed image I 1 _ d  obtained by such a deformation that the first image I 1  is substantially matched with the second image I 2  on the basis of the deformation transformation T 1 _ 2  obtained in step S 813 . In addition, the deformed image generation unit  109  may generate the second deformed image I 2 _ d  obtained by such a deformation that the second image I 2  is substantially matched with the first image I 1  on the basis of the deformation transformation T 2 _ 1  obtained in step S 305 . It should be noted however that the method of generating the deformed image by using the deformation transformation has been proposed in the related art, and therefore detailed descriptions will be omitted. 
     Furthermore, in step S 814 , the normalization transformation obtaining unit  103  obtains the normalization transformation Tn 1 _ d  for transforming the first deformed image I 1 _ d  to the normalized deformed image. The normalization transformation Tn 1   d  is obtained by the method similar to step S 801 , that is, the processing exemplified in  FIG. 5 . It should be noted that the normalization transformation obtaining unit  103  obtains the body surface shape or the nipple position in the first deformed image I 1 _ d  by applying the deformation transformation T 1 _ 2  to the body surface shape or the nipple position of the first image I 1  obtained in step S 801  to be transformed to the shape and the position in the first deformed image. Furthermore, in step S 814 , the normalized image generation unit  104  transforms the first deformed image I 1 _ d  on the basis of the normalization transformation Tn 1 _ d  and generates the first normalized deformed image I 1 _ d ′. It should be noted however that, since the first deformed image I 1 _ d  is the image generated by deforming the first image I 1  so as to be substantially matched with the second image I 2 , the first deformed image I 1 _ d  may be transformed on the basis of the second normalization transformation Tn 2  obtained in step S 801  to generate the first normalized deformed image I 1 _ d′.    
     &lt;Step S 815 : Display of the Image and the Corresponding Point&gt; 
     In step S 815 , the display control unit  705  displays the information indicating the corresponding point position obtained in up to step S 815  on the display unit  190 . In addition, the display control unit  705  displays the deformed image generated in step S 814  in the display unit  190 . The display control unit  705  may further display other images generated in up to step S 814  on the display unit  190 . For example, the display control unit  705  may arrange and display the images generated in up to step S 814  among the first image, the second image, the first normalized image, the second normalized image, the first deformed image, and the first normalized deformed image on the display unit  190  in a tiling manner. The display control unit  705  may display the corresponding point position in a manner that the image where the corresponding point is input can be distinguished from the others. For example, the display control unit  705  displays graphic symbols representing the corresponding point positions in different display modes (for example, different colors or shapes) in accordance with types of the images where the corresponding point is input. In addition, in a case where the deformation transformations T 1 _ 2  and T 2 _ 1  are noninvertible, the corresponding point positions that are obtained on the first deformed image or the first normalized deformed image and are not transformed are not matched with the positions obtained while those positions are transformed and further inversely transformed. Therefore, a configuration may be adopted in which the display control unit  705  does not display the graphic symbols representing the corresponding point positions that are obtained on the first deformed image or the first normalized deformed image and are not transformed on the first deformed image and the first normalized deformed image. 
     &lt;Step S 816 : Determination on the End of the Entirety of the Processing&gt; 
     In step S 816 , the image processing apparatus  700  determines whether or not the entirety of the processing is ended. For example, the image processing apparatus  700  obtains an end instruction in response to the operation input on the operation unit  180  by the user. The end instruction is performed by the operation input with respect to the user interface displayed on the display unit  190 . In step S 812 , when it is determined that the number of corresponding points is insufficient, the display control unit  705  may display a screen for notifying the user that the number of corresponding points is insufficient on the display unit  190 . When it is determined that the entirety of the processing is ended, the processing illustrated in  FIG. 8  is ended. When it is not determined that the entirety of the processing is ended, the processing returns to step S 803 . 
     In the above-described manner, the processing of the image processing apparatus according to the fourth exemplary embodiment is executed. According to the fourth exemplary embodiment, it is possible to provide the system where the images are displayed in a manner that deformation states are matched with each other (by performing the deformation registration of one image to the other image) such that it is facilitated for the user who observes the images to perform the identification of the corresponding point position, and the image is deformed on the basis of the corresponding point position specified on the image. In addition, according to the fourth exemplary embodiment, after the corresponding point input on the first normalized image and the corresponding point input on the first deformed image are transformed to the positions on the first image, it is possible to perform uniform management together with the corresponding point input on the first image. In the above-described manner, since the corresponding point is managed at the position on the first image, even after the deformation transformation is updated, the corresponding point position is transformed to the appropriate position on the first deformed image. 
     Modified Examples of Fourth Embodiment 
     Modified Example 4-1 
     According to the fourth exemplary embodiment, the case has been described where the graphic symbols representing the corresponding point positions that are obtained on the first deformed image and are not transformed are not displayed on the first deformed image. However, the implementation of the present invention is not limited to the above, and the graphic symbol representing the corresponding point position obtained on the first deformed image and transformed and inversely transformed and the graphic symbol representing the corresponding point position obtained on the first image and transformed and inversely transformed may be displayed in different display modes. 
     Modified Example 4-2 
     According to the fourth exemplary embodiment, the case has been described where the first image is registered to the second image to generate the first deformed image, and the information related to the corresponding point position on the generated first deformed image and the second image is obtained. The present invention is not limited to the above, and the information related to the corresponding point position between the second deformed image and the first image may be obtained. Alternatively, the information related to the corresponding point position between the deformed images obtained by deforming the first image and the second image may be obtained. 
     Modified Example 4-3 
     According to the fourth exemplary embodiment, the case has been described where the corresponding point is input on the original image, the deformed image, the normalized image, and the normalized deformed image. However, the implementation of the present invention is not limited to the above, and a configuration may be adopted in which the normalized image and/or the normalized deformed image is not generated, and the corresponding point is not input is not input. 
     Modified Example 4-4 
     The processing performed by the image processing apparatus according to the fourth exemplary embodiment  700  is exemplified in  FIG. 8  and  FIG. 9 , but the order of the processing is not limited to the above. For example, the processing constituted by steps S 803 , S 804 , and S 805 , the processing constituted by steps S 806  and S 807 , and the processing constituted by steps S 808 , S 809 , and S 810  may be performed in an order different from the order exemplified in  FIG. 8 . In addition, the example has been illustrated in which the flow proceeds to step S 815  when each of steps S 805 , S 807 , and S 810  is ended, but the order of the processing is not limited to the above. For example, when each of steps is ended, after the display processing of the corresponding point is performed, the flow may proceed to each of next steps (S 806  in the case of S 805 , S 808  in the case of S 807 , and S 811  in the case of S 810 ). The example has been illustrated in which the determination is performed in steps S 803 , S 806 , and S 808  in the stated order, but the order of the processing is not limited to the above. For example, the image processing apparatus  700  may stand by in a mode for standing by for an instruction of the user, and the flow may proceed to a specific step on the basis of the operation input for selecting the image where the corresponding point is input. 
     Modified Example 4-5 
     According to the fourth exemplary embodiment, the example has been described in which pieces of information of the input corresponding point positions by using various images are respectively transformed to the corresponding point positions in the real space. The image processing apparatus  700  may attach the information of the corresponding point positions in the real space to the real image, that is, at least one of the first image and the second image. According to the fourth exemplary embodiment, in step S 805  or step S 8103 , the processing of respectively transforming the corresponding point positions to the real space is performed, but the processing of transforming the corresponding point positions to the real space may be performed at a predetermined timing or performed each time the corresponding point is input. In addition, the information of the input corresponding point positions by using various images may be displayed on the display unit  190  at a predetermined timing (for example, in step S 815 ), or the corresponding point positions input thus far may be displayed on the display unit  190  in a state in which the corresponding point positions are transformed to the positions in the image displayed on the display unit  190 . 
     Modified Examples 
     According to the first exemplary embodiment to the fourth exemplary embodiment, the case has been described as an example where the deformation transformation between the images is represented by the TPS method or the FFD method, but the target is not limited to the above as long as the method is the deformation representation method in which the optimization of the deformation parameter can be performed by taking into account the regularization in the real space. For example, a radial basis function different from the TPS may be used. In addition, a large deformation diffeomorphic metric mapping (LDDMM) method of representing deformation transformation by an integral of a velocity field or the like may be used. 
     According to the first exemplary embodiment to the fourth exemplary embodiment, the example has been described in which the first transformed image and the second transformed image are obtained by applying the coordinate transformation to each of the first image and the second image, but the present invention is not limited to the above. For example, the coordinate transformation for making the shape of the object drawn in the first image close to the shape of the object drawn in the second image may be applied to obtain only the first transformed image. When the first transformed image and the shape of the object drawn in the second image are closer to each other, the identification of the corresponding points is facilitated. In this case, the corresponding point transformation unit  107  transforms the corresponding point position on the first transformed image among the information of the corresponding points included in each of the first transformed image and the second image to the position in the first image. Subsequently, the corresponding point transformation unit  107  outputs the corresponding point positions in the first image and the second image to the deformation transformation obtaining unit (deformation estimation unit)  108 . 
     The function described according to the first exemplary embodiment to the fourth exemplary embodiment may be realized, for example, by a system constituted by a server apparatus (not illustrated) and a client apparatus (not illustrated). For example, the server apparatus (not illustrated) includes the deformation transformation obtaining unit (deformation estimation unit)  108  and the deformed image generation unit  109  illustrated in  FIG. 1 , and the client apparatus (not illustrated) includes the normalization transformation obtaining unit  103 , the normalized image generation unit  104 , the display control unit  105 , the corresponding point obtaining unit  106 , and the corresponding point transformation unit  107  and is connected to the operation unit  180  and the display unit  190 . In this case, the client apparatus (not illustrated) applies the coordinate transformation to the image obtained from the data server  170  and obtains the corresponding point position in the space where the coordinate transformation has been performed to be transformed to the corresponding point position in the space in the original image. Subsequently, the client apparatus (not illustrated) outputs the plurality of images desired to be subjected to the registration and the corresponding point positions to the server apparatus (not illustrated). The server apparatus (not illustrated) obtains the deformation transformation on the basis of the plurality of images and the information of the corresponding point positions input from the client apparatus (not illustrated) and outputs the deformation transformation to the client apparatus (not illustrated). The client apparatus (not illustrated) generates the registered image on the basis of the deformation transformation input from the server apparatus (not illustrated) to be displayed on the display unit  190 . The server apparatus (not illustrated) may further include the normalization transformation obtaining unit  103 . 
     The present invention can also be realized by processing in which a program for realizing one or more of the functions according to the above-described exemplary embodiments is supplied to a system or an apparatus via a network or a storage medium, and one or more processors in a computer of the system or the apparatus reads out and executes the program. In addition, the present invention can also be realized by a circuit (for example, an application specific integrated circuit (ASIC)) that realizes one or more of the functions. 
     The image processing apparatus according to the above-described respective exemplary embodiments may be realized as a standalone apparatus or may adopt a mode in which a plurality of apparatuses are combined with each other so as to be mutually communicable to execute the above-described processing, and both of the configurations are included in the exemplary embodiments of the present invention. A common server apparatus or a server group may execute the above-described processing. It suffices when the plurality of apparatuses constituting the image processing apparatus and the image processing system can perform the communication at a predetermined communication rate, and also the plurality of apparatuses constituting does not necessarily need to exist in the same facility or the same country. 
     The exemplary embodiments of the present invention include a mode in which a program of software for realizing the functions of the above-described exemplary embodiments is supplied to the system or the apparatus, and the computer of the system or the apparatus reads out and executes a code of the supplied program. 
     Therefore, to realize the processing according to the exemplary embodiments in the computer, the program code itself installed into the computer is also one of the exemplary embodiments of the present invention. In addition, an operating system (OS) or the like running on the computer performs part or all of the actual processing on the basis of an instruction included in the program read by the computer, and the functions of the above-described exemplary embodiments may also be realized by the processing. 
     A mode in which the above-described exemplary embodiments are appropriately combined with each other is also included in the exemplary embodiments of the present invention. 
     Other Embodiments 
     Embodiment(s) of the present invention can also be realized by a computer of a system or apparatus that reads out and executes computer executable instructions (e.g., one or more programs) recorded on a storage medium (which may also be referred to more fully as a ‘non-transitory computer-readable storage medium’) to perform the functions of one or more of the above-described embodiment(s) and/or that includes one or more circuits (e.g., application specific integrated circuit (ASIC)) for performing the functions of one or more of the above-described embodiment(s), and by a method performed by the computer of the system or apparatus by, for example, reading out and executing the computer executable instructions from the storage medium to perform the functions of one or more of the above-described embodiment(s) and/or controlling the one or more circuits to perform the functions of one or more of the above-described embodiment(s). The computer may comprise one or more processors (e.g., central processing unit (CPU), micro processing unit (MPU)) and may include a network of separate computers or separate processors to read out and execute the computer executable instructions. The computer executable instructions may be provided to the computer, for example, from a network or the storage medium. The storage medium may include, for example, one or more of a hard disk, a random-access memory (RAM), a read only memory (ROM), a storage of distributed computing systems, an optical disk (such as a compact disc (CD), digital versatile disc (DVD), or Blu-ray Disc (BD)™), a flash memory device, a memory card, and the like. 
     While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions.