Abstract:
Apparatuses, computer media, and methods for altering a camera image, in which the source image may be angularly displaced from a camera image. A plurality of points on the camera image is located and a mesh is generated. Compensation information based on the displacement is determined, and a reshaped image is rendered from the mesh, the compensation information, and the camera image. The camera image is reshaped by relocating a proper subset of the points on the camera image. Deformation vectors are applied to corresponding points on the mesh using the compensation information. A correction factor is obtained from an angular displacement and a translation displacement of the source image from the camera image. The deformation factor is multiplied by the compensation factor to faun a deformation vector to compensate for angular and translational displacements of the source image from the camera image.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 11/674,255, filed on Feb. 13, 2007, which is a continuation-in-part of U.S. application Ser. No. 11/625,937, filed on Jan. 23, 2007, the disclosures of which are incorporated by reference herein in their entirety. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to altering a camera image. More particularly, the invention applies to a source image being angularly displaced from the camera image plane. 
       BACKGROUND OF THE INVENTION 
       [0003]    Excessive body weight is a major cause of many medical illnesses. With today&#39;s life style, people are typically exercising less and eating more. Needless to say, this life style is not conducive to good health. For example, it is acknowledged that type-2 diabetes is trending to epidemic proportions. Obesity appears to be a major contributor to this trend. 
         [0004]    On the other hand, a smaller proportion of the population experiences from being underweight. However, the effects of being underweight may be even more divesting to the person than to another person being overweight. In numerous related cases, people eat too little as a result of a self-perception problem. Anorexia is one affliction that is often associated with being grossly underweight. 
         [0005]    While being overweight or underweight may have organic causes, often such afflictions are the result of psychological issues. If one can objectively view the effect of being underweight or underweight, one may be motivated to change one&#39;s life style, e.g., eating in a healthier fashion or exercising more. Viewing a predicted image of one&#39;s body if one continues one&#39;s current life style may motivate the person to live in a healthier manner. 
       BRIEF SUMMARY OF THE INVENTION 
       [0006]    Embodiments of invention provide apparatuses, computer media, and methods for altering a camera image, in which the source image may be angularly displaced from a camera image. 
         [0007]    With an aspect of the invention, a plurality of points on the camera image is located and a mesh is generated. The mesh is superimposed on the camera image and associated with corresponding texture information of the camera image. Compensation information based on the displacement is determined, and a reshaped image is rendered from the mesh, the compensation information, and the camera image. 
         [0008]    With another aspect of the invention, the camera image is reshaped by relocating a proper subset of the points on the camera image. Deformation vectors are applied to corresponding points on the mesh using the compensation information. A deformation vector may comprise a product of factors, including a weight value factor (A), a scale factor (s), a deformation factor (w), and a direction vector ({right arrow over (u)}). 
         [0009]    With another aspect of the invention, a correction factor is obtained from an angular displacement and a translation displacement of the source image from the camera image. The deformation factor is multiplied by the compensation factor to form a deformation vector to compensate for angular and translational displacements of the source image from the camera image. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    The present invention is illustrated by way of example and not limited in the accompanying figures in which like reference numerals indicate similar elements and in which: 
           [0011]      FIG. 1  shows a mesh that is superimposed in a face image in accordance with an embodiment of the image. 
           [0012]      FIG. 2  shows a set of points for altering a face image in accordance with an embodiment of the invention. 
           [0013]      FIG. 3  shows controlling points for face alteration in accordance with an embodiment of the invention. 
           [0014]      FIG. 4  shows visual results for altering a face image in accordance with an embodiment of the invention. 
           [0015]      FIG. 5  shows additional visual results for altering a face image in accordance with an embodiment of the invention. 
           [0016]      FIG. 6  shows additional visual results for altering a face image in accordance with an embodiment of the invention. 
           [0017]      FIG. 7  shows additional visual results for altering a face image in accordance with an embodiment of the invention. 
           [0018]      FIG. 8  shows a flow diagram for altering a face image in accordance with an embodiment of the invention. 
           [0019]      FIG. 9  shows an architecture of a computer system used in altering a face image in accordance with an embodiment of the invention. 
           [0020]      FIG. 10  shows a schema of a reference system and camera model for an adaptive process for processing an image in accordance with an embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0021]      FIG. 1  shows a mesh that is superimposed in a face image in accordance with an embodiment of the image. As will be discussed, an algorithm fattens or thins the face image in accordance with an embodiment of the invention. Points along the face, neck, and image boundary are determined in order to form the mesh. As will be further discussed, the algorithm alters the facial contour and then reshapes the area around the neck. (Points  136 - 145  will be discussed in a later discussion.) The altered image is rendered by using the points as vertices of the mesh. 
         [0022]    This mesh is associated to its corresponding texture from the picture where the alteration is taking place. The corners and four points along each side of the picture (as shown in  FIG. 1 ) are also considered as part of the mesh. Computer graphics software API (Application Programming Interface) is used to render the altered image (e.g., as shown in  FIGS. 4-7 ). OpenGL API is an example of computer graphics software that may be used to render the altered image. 
         [0023]      FIG. 2  shows a set of points (including points  200 ,  206 ,  218 , and  231  which will be discussed in further detail) for altering a face image in accordance with an embodiment of the invention. (Please note that  FIG. 2  shows a plurality of points, which correspond to the vertices of the mesh.) Points  200 ,  206 ,  218 , and  231  are only some of the plurality of points. An embodiment of the invention uses the search function of a software technique called Active Appearance Model (AAM), which utilizes a trained model. (Information about AAM is available at http://www2.1 mm.dtu.dk/˜aam and has been utilized by other researchers.) However, points  200 ,  206 ,  218 , and  231  may be determined with other approaches, e.g., a manual process that is performed by medical practitioner manually entering the points. With an embodiment of the invention, the trained model is an AMF file, which is obtained from the training process. For the training the AAM, a set of images with faces is needed. These images may belong to the same person or different people. Training is typically dependent on the desired degree of accuracy and the degree of universality of the population that is covered by the model. With an exemplary embodiment, one typically processes at least five images with the algorithm that is used. During the training process, the mesh is manually deformed on each image. Once all images are processed, the AAM algorithms are executed over the set of points and images, and a global texture/shape model is generated and stored in an AMF file. The AMF file permits an automatic search in future images not belonging to the training set. With an exemplary embodiment, one uses the AAM API to generate Appearance Model Files (AMF), Embodiments of the invention also support inputting the plurality of points through an input device as entered by a user. A mesh is superimposed on the image at points (e.g., the set of points shown in  FIG. 2 ) as determined by the trained process. 
         [0024]      FIG. 2  also shows the orientation of the x and y coordinates of the points as shown in  FIGS. 1-3 . 
         [0025]      FIG. 3  shows controlling points  306 - 331  for face alteration in accordance with an embodiment of the invention. (Points  306 ,  318 , and  331  correspond to points  206 ,  218 , and  231  respectively as shown in  FIG. 2 .) Points  306 - 331 , which correspond to points around the cheeks and chin of the face, are relocated (transformed) for fattening or thinning a face image to a desired degree. With an embodiment of the invention, only a proper subset (points  306 - 331 ) of the plurality of points (as shown in  FIG. 2 ) are relocated. (With a proper subset, only some, and not all, of the plurality points are included.) 
         [0026]    In the following discussion that describes the determination of the deformation vectors for reshaping the face image, index i=6 to index i=31 correspond to points  306  to points  331 , respectively. The determined deformation vectors are added to points  306  to points  331  to re-position the point, forming a transformed mesh. A reshaped image is consequently rendered using the transformed mesh. 
         [0027]    In accordance with embodiments of the invention, deformation vector correspond to a product of four elements (factors): 
         [0000]        {right arrow over (v)}   s   ={right arrow over (u)}·s·w·A   (EQ. 1)
 
         [0000]    where A is the weight value factor, s is the scale factor, w is the deformation factor, and {right arrow over (u)} is the direction vector. In accordance with an embodiment of the invention:
       Weight value factor [A]: It determines the strength of the thinning and fattening that we wan to apply.       
 
         [0000]      A&gt;0 fattening  (EQ. 2A)
 
         [0000]      A&lt;0 thinning  (EQ. 2B)
 
         [0000]      A=0 no change  (EQ. 2C)
       Scale factor [s]. It is the value of the width of the face divided by B. One uses this factor to make this vector calculation independent of the size of the head we are working with. The value of B will influence how the refined is the scale of the deformation. It will give the units to the weight value that will be applied externally.       
 
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             Deformation factor [w]. It is calculated differently for different parts of cheeks and chin. One uses a different equation depending on which part of the face one is processing: 
           
         
       
     
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             Direction vector [{right arrow over (u)}]: It indicates the sense of the deformation. One calculates the direction vector as the ratio between: the difference (for each coordinate) between the center and our point, and the absolute distance between this center and our point. One uses two different centers in this process: center C 2  (point  253  as shown in  FIG. 2 ) for the points belonging to the jaw and center C 1  (point  251  as shown in  FIG. 2 ) for the points belonging to the cheeks. 
           
         
       
     
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         [0032]    Neck point-coordinates x i  are based on the lower part of the face, where 
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         [0000]    where y 18  and y 0  are the y-coordinates of points  218  and  200 , respectively, as shown in  FIG. 2 . Referring back to  FIG. 1 , index i=36 to i=45 correspond to points  136  to  145 , respectively. Index j=14 to j=23 correspond to points  314  to  323 , respectively, (as shown in  FIG. 3 ) on the lower part of the face, from which points  136  to  145  on the neck are determined. (In an embodiment of the invention, points  136  to  145  are determined from points  314  to  323  before points  314  to  323  are relocated in accordance with EQs. 1-5.) 
         [0033]    The deformation vector ({right arrow over (v)} d     —     neck ) applied at points  136  to  145  has two components: 
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         [0034]    The Appendix provides exemplary software code that implements the above algorithm. 
         [0035]      FIG. 4  shows visual results for altering a face image in accordance with an embodiment of the invention. Images  401  to  411  correspond to A=+100 to A=+50, respectively, which correspond to decreasing degrees of fattening. 
         [0036]    With an embodiment of the invention, A=+100 corresponds to a maximum degree of fattening and A=−100 corresponds to a maximum degree of thinning. The value of A is selected to provide the desired degree of fattening or thinning. For example, if a patient were afflicted anorexia, the value of A would have a negative value that would depend on the degree of affliction and on the medical history and body type of the patient. As another example, a patient may be over-eating or may have an unhealthy diet with many empty calories. In such a case, A would have a positive value. A medical practitioner may be able to gauge the value of A based on experience. However, embodiments of invention may support an automated implementation for determining the value of A. For example, an expert system may incorporate knowledge based on information provided by experienced medical practitioners. 
         [0037]      FIG. 5  shows additional visual results for altering a face image in accordance with an embodiment of the invention. Images  501 - 511 , corresponding to A=+40 to A=−10, show the continued reduced sequencing of the fattening. When A=0 (image  509 ), the face is shown as it really appears. With A=−10 (image  511 ), the face is shows thinning. As A becomes more negative, the effects of thinning is increased. 
         [0038]      FIG. 6  shows additional visual results for altering a face image in accordance with an embodiment of the invention. Images  601 - 611  continue the sequencing of images with increased thinning (i.e., A becoming more negative). 
         [0039]      FIG. 7  shows additional visual results for altering a face image in accordance with an embodiment of the invention. Images  701 - 705  complete the sequencing of the images, in which the degree of thinning increases. 
         [0040]      FIG. 8  shows flow diagram  800  for altering a face image in accordance with an embodiment of the invention. In step  801 , points are located on the image of the face and neck in order form a mesh. Points may be determined by a trained process or may be entered through an input device by a medical practitioner. In step  803 , reshaping parameters (e.g., a weight value factor A) are obtained. The reshaping factors may be entered by the medical practitioner or may be determined by a process (e.g. an expert system) from information about the person associated with the face image. 
         [0041]    In step  805  deformation vectors are determined and applied to points (e.g. points  306 - 331  as shown in  FIG. 3 ) on the face. For example, as discussed above, EQs. 1-5. are used to determine the relocated points. In step  807  deformation vectors are determined (e.g., using EQs. 6-9) and applied to points (e.g., points  136 - 145  as shown in  FIG. 1 ) on the neck. A transformed mesh is generated from which a reshaped image is rendered using computer graphics software in step  809 . 
         [0042]      FIG. 9  shows computer system  1  that supports an alteration of a face image in accordance with an embodiment of the invention. Elements of the present invention may be implemented with computer systems, such as the system  1 . Computer system  1  includes a central processor  10 , a system memory  12  and a system bus  14  that couples various system components including the system memory  12  to the central processor unit  10 . System bus  14  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The structure of system memory  12  is well known to those skilled in the art and may include a basic input/output system (BIOS) stored in a read only memory (ROM) and one or more program modules such as operating systems, application programs and program data stored in random access memory (RAM). 
         [0043]    Computer  1  may also include a variety of interface units and drives for reading and writing data. In particular, computer  1  includes a hard disk interface  16  and a removable memory interface  20  respectively coupling a hard disk drive  18  and a removable memory drive  22  to system bus  14 . Examples of removable memory drives include magnetic disk drives and optical disk drives. The drives and their associated computer-readable media, such as a floppy disk  24  provide nonvolatile storage of computer readable instructions, data structures, program modules and other data for computer  1 . A single hard disk drive  18  and a single removable memory drive  22  are shown for illustration purposes only and with the understanding that computer  1  may include several of such drives. Furthermore, computer  1  may include drives for interfacing with other types of computer readable media. 
         [0044]    A user can interact with computer  1  with a variety of input devices.  FIG. 7  shows a serial port interface  26  coupling a keyboard  28  and a pointing device  30  to system bus  14 . Pointing device  28  may be implemented with a mouse, track ball, pen device, or similar device. Of course one or more other input devices (not shown) such as a joystick, game pad, satellite dish, scanner, touch sensitive screen or the like may be connected to computer  1 . 
         [0045]    Computer  1  may include additional interfaces for connecting devices to system bus  14 .  FIG. 7  shows a universal serial bus (USB) interface  32  coupling a video or digital camera  34  to system bus  14 . An IEEE 1394 interface  36  may be used to couple additional devices to computer  1 . Furthermore, interface  36  may configured to operate with particular manufacture interfaces such as FireWire developed by Apple Computer and i.Link developed by Sony. Input devices may also be coupled to system bus  114  through a parallel port, a game port, a PCI board or any other interface used to couple and input device to a computer. 
         [0046]    Computer  1  also includes a video adapter  40  coupling a display device  42  to system bus  14 . Display device  42  may include a cathode ray tube (CRT), liquid crystal display (LCD), field emission display (FED), plasma display or any other device that produces an image that is viewable by the user. Additional output devices, such as a printing device (not shown), may be connected to computer  1 . 
         [0047]    Sound can be recorded and reproduced with a microphone  44  and a speaker  66 . A sound card  48  may be used to couple microphone  44  and speaker  46  to system bus  14 . One skilled in the art will appreciate that the device connections shown in  FIG. 7  are for illustration purposes only and that several of the peripheral devices could be coupled to system bus  14  via alternative interfaces. For example, video camera  34  could be connected to IEEE 1394 interface  36  and pointing device  30  could be connected to USB interface  32 . 
         [0048]    Computer  1  can operate in a networked environment using logical connections to one or more remote computers or other devices, such as a server, a router, a network personal computer, a peer device or other common network node, a wireless telephone or wireless personal digital assistant. Computer  1  includes a network interface  50  that couples system bus  14  to a local area network (LAN)  52 . Networking environments are commonplace in offices, enterprise-wide computer networks and home computer systems. 
         [0049]    A wide area network (WAN)  54 , such as the Internet, can also be accessed by computer  1 .  FIG. 7  shows a modem unit  56  connected to serial port interface  26  and to WAN  54 . Modem unit  56  may be located within or external to computer  1  and may be any type of conventional modem such as a cable modem or a satellite modem. LAN  52  may also be used to connect to WAN  54 .  FIG. 7  shows a router  58  that may connect LAN  52  to WAN  54  in a conventional manner. 
         [0050]    It will be appreciated that the network connections shown are exemplary and other ways of establishing a communications link between the computers can be used. The existence of any of various well-known protocols, such as TCP/IP, Frame Relay, Ethernet, FTP, HTTP and the like, is presumed, and computer  1  can be operated in a client-server configuration to permit a user to retrieve web pages from a web-based server. Furthermore, any of various conventional web browsers can be used to display and manipulate data on web pages. 
         [0051]    The operation of computer  1  can be controlled by a variety of different program modules. Examples of program modules are routines, programs, objects, components, and data structures that perform particular tasks or implement particular abstract data types. The present invention may also be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCS, minicomputers, mainframe computers, personal digital assistants and the like. Furthermore, the invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. 
         [0052]    In an embodiment of the invention, central processor unit  10  obtains a face image from digital camera  34 . A user may view the face image on display device  42  and enter points (e.g., points  206 - 231  as shown in  FIG. 2 ) to form a mesh that is subsequently altered by central processor  10  as discussed above. The user may identify the points with a pointer device (e.g. mouse  30 ) that is displayed on display device  42 , which overlays the mesh over the face image. With embodiments of the invention, a face image may be stored and retrieved from hard disk drive  18  or removable memory drive  22  or obtained from an external server (not shown) through LAN  52  or WAN  54 . 
       Adaptation of Deformation Factor for Pose Angularly Offset 
       [0053]      FIG. 10  shows a schema of a reference system and camera model for an adaptive process for processing an image in accordance with an embodiment of the invention. Schema  1000  establishes a relationship of source point  1001  (x n ,y n ,z n ) and corresponding projected point  1003  (x p ,y p ) on camera image plane  1005 . A source image consists of a collection of source points, and the corresponding camera consists of a collection of projected points. (In  FIG. 10 , the source image is an image of a person&#39;s head or face. The source image may be an actual object or a visual representation of the actual object.) 
         [0054]    The camera is characterized by optical center  1007  and focal length (F)  1009 . The axis orientation of the camera is characterized by angles α  1011 , β  1013 , and γ  1015  corresponding to the x, y, and z axes, respectively. The origin of the axis orientation is located at the center of the camera image plane of the projected section that is shown in  FIG. 10 . Projected point  1003  (x p ,y p ) is related to the corresponding source point  1001  by the following relationship: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         x 
                         p 
                       
                       , 
                       
                         y 
                         p 
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           F 
                           · 
                           
                             x 
                             n 
                           
                         
                         
                           F 
                           - 
                           
                             z 
                             n 
                           
                         
                       
                       , 
                       
                         
                           F 
                           · 
                           
                             y 
                             n 
                           
                         
                         
                           F 
                           - 
                           
                             z 
                             n 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                     . 
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where F is the focal length of the camera. 
         [0055]    With embodiments of the invention, one may assume that the face of the person is perpendicular to the axis orientation of the camera. Taking into account the 3D observation model detailed, as will be discussed, a direct pose occurs when α=β=γ=0. 
         [0056]    Embodiments of the invention support image poses in which the pose is angularly offset. The correction factor for such a situation adapts the deformation factor w applied to the deformation vector of each vertex (e.g., as the vertices shown in  FIG. 1 ) that is moved during the reshaping of the image (e.g., the face of a person). With an embodiment of the invention, the correction factor may be obtained from an angular displacement and a translation displacement of the source image from the camera image. The translation and the displacement may be determined from the difference from the 3D face pose in a frontal position (from which one has previously computed the weights) and the 3D pose of the face that one has actually taken the picture of. 
         [0057]    The observation model utilized to relate the head in its neutral pose (source image facing the camera) and its projected representation taking into account the rigid motion (translations and rotations) of the head observed from reference origin  1017  and the projection due to the camera. Although the acquisition camera is not calibrated because one does not control the nature of the input sequences, one can still consider that it obtains a perspective projection and not an orthogonal projection. 
         [0058]    Reference origin  1017  is situated along the optical axis of the camera at the center of camera image plane  1005 . Camera image plane  1005  represents the video image where the face is focused. Focal distance F  1009 , represents the distance from camera image plane  1005  to the optical center of the camera. To describe the rigid motion of the head, one may specify three translations, along the X, Y and Z-axes, and three rotations, around the X, Y, and Z axes.  FIG. 10  presents the graphical interpretation of the model and the orientation of the reference axes. 
         [0059]    One may describe points using their homogenous coordinates to be able to describe a perspective transform linearly and derive the relationship between 3D neutral coordinates and 2D projections. 
         [0060]    (x, y, z,o) T  A vector corresponds to a homogenous point if at least one of its elements is not 0. (o is the coordinate that is added to convert the coordinates to homogenous coordinates. Homogeneous coordinates allow affine transformations to be easily represented by a matrix. Also, homogeneous coordinates make calculations possible in projective space just as Cartesian coordinates do in Euclidean space. The homogeneous coordinates of a point of projective space of dimension n are typically written as (x: y: z: . . . :o), a row vector of length n+1, other than (0:0:0: . . . : 0)). If a is a real number and is not 0, (x, y, z, or and (ax,ay,az,ao) T  represent the same homogenous point. The relationship between a point in 3D or 2D Euclidean space and its homogenous representation is: 
         [0000]      ( x,y,z ) 3D →( x,y,z, 1) 3D  and ( x,y ) 2D →( x,y, 0,1) 2D  
 
         [0000]    One can obtain the Euclidean representation of a homogenous point only if o≠0: 
         [0000]      ( x,y,z,o ) H →( x/o,y/o,z/o ) 3D  and ( x,y,o ) H →( x/o,y/o ) 2D  
 
         [0000]    As an example of projective space in three dimensions, there are corresponding homogeneous coordinates (x: y: z: o). The plane at infinity is typically identified with the set of points with o=0. Away from this plane, one can denote (x/o, y/o, z/o) as an ordinary Cartesian system; therefore, the affine space complementary to the plane at infinity is assigned coordinates in a similar way, with a basis corresponding to (1:0:0:1), (0:1:0:1), (0:0:1:1). 
         [0061]    The following matrices represent different transformations that describe rigid motion, where s α , =sin(α), c α =cos(α), s β =sin(β), c β =cos(β), s γ =sin(γ), and c γ =cos(γ).
       Translation following vector (t x ,t y ,t z ) T         
 
         [0000]    
       
         
           
             
               T 
               
                 ( 
                 
                   
                     t 
                     X 
                   
                   , 
                   
                     t 
                     Y 
                   
                   , 
                   
                     t 
                     Z 
                   
                 
                 ) 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         t 
                         x 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       
                         t 
                         y 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       
                         t 
                         z 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
       
         
           
             Rotation by an angle of α radians around the X-axis: 
           
         
       
     
         [0000]    
       
         
           
             
               R 
               
                 α 
                 , 
                 X 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         c 
                         α 
                       
                     
                     
                       
                         - 
                         
                           s 
                           α 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         s 
                         α 
                       
                     
                     
                       
                         c 
                         α 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
       
         
           
             Rotation by an angle of β radians around the Y-axis: 
           
         
       
     
         [0000]    
       
         
           
             
               R 
               
                 β 
                 , 
                 Y 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         c 
                         β 
                       
                     
                     
                       0 
                     
                     
                       
                         s 
                         β 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         - 
                         
                           s 
                           β 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         c 
                         β 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
       
         
           
             Rotation by an angle of γ radians around the Z-axis: 
           
         
       
     
         [0000]    
       
         
           
             
               R 
               
                 γ 
                 , 
                 Z 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         c 
                         γ 
                       
                     
                     
                       
                         - 
                         
                           s 
                           γ 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         s 
                         γ 
                       
                     
                     
                       
                         c 
                         γ 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
         [0066]    The final location of the head regarding reference origin  1017  is obtained applying the translation and rotation matrices upon the coordinates of the head in its neutral pose. 
         [0000]    
       
      
       x 
       trans 
       T 
       =G·x 
       n 
       T  
      
     
         [0000]      where 
         [0000]    
       
      
       G=T 
       (t 
       
         X 
       
       ,t 
       
         Y 
       
       ,T 
       
         Z 
       
       ) 
       ·R 
       α,x 
       ·R 
       β,y 
       ·R 
       γ,z  
      
     
         [0067]    Then, the position “head is facing the camera” is defined when (t x ,t y ,t z ) T =(0,0,0) α=0, β=0 and γ=0. The observed projection on camera image plane  1005  is: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   x 
                   p 
                   T 
                 
                 = 
                 
                   
                     P 
                     F 
                   
                   · 
                   
                     T 
                     
                       ( 
                       
                         0 
                         , 
                         0 
                         , 
                         
                           - 
                           F 
                         
                       
                       ) 
                     
                   
                   · 
                   
                     x 
                     trans 
                     T 
                   
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               where 
             
           
         
       
       
         
           
             
               
                 P 
                 F 
               
               · 
               
                 T 
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     
                       - 
                       F 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         F 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         F 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           
                             - 
                             2 
                           
                            
                           
                               
                           
                            
                           F 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           - 
                           F 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       F 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       F 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         F 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       F 
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    represents the complete projection from the combination of the perspective projection matrix, P F , whose origin is located on the optical center of the camera and the translation −F along the Z-axis, and T (0,0,-F) , which relocates the origin of the reference axis on the image plane (just as with the observation model shown in  FIG. 10 ). One obtains the following expression to relate the homogenous coordinates of the points belonging to the head in its neutral pose and their observed equivalent representation on camera image plane  1005 : 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       x 
                       p 
                     
                   
                 
                 
                   
                     
                       y 
                       p 
                     
                   
                 
                 
                   
                     
                       z 
                       p 
                     
                   
                 
                 
                   
                     
                       O 
                       p 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       
                         
                           Fc 
                           β 
                         
                          
                         
                           c 
                           γ 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             Fc 
                             β 
                           
                         
                          
                         
                           s 
                           γ 
                         
                       
                     
                     
                       
                         Fs 
                         β 
                       
                     
                     
                       
                         Ft 
                         X 
                       
                     
                   
                   
                     
                       
                         F 
                          
                         
                           ( 
                           
                             
                               
                                 c 
                                 α 
                               
                                
                               
                                 s 
                                 γ 
                               
                             
                             + 
                             
                               
                                 s 
                                 α 
                               
                                
                               
                                 s 
                                 β 
                               
                                
                               
                                 c 
                                 γ 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         F 
                          
                         
                           ( 
                           
                             
                               
                                 c 
                                 α 
                               
                                
                               
                                 c 
                                 γ 
                               
                             
                             - 
                             
                               
                                 s 
                                 α 
                               
                                
                               
                                 s 
                                 β 
                               
                                
                               
                                 s 
                                 γ 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         F 
                          
                         
                           ( 
                           
                             
                               - 
                               
                                 s 
                                 α 
                               
                             
                              
                             
                               c 
                               β 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         Ft 
                         Y 
                       
                     
                   
                   
                     
                       
                         
                           
                             c 
                             α 
                           
                            
                           
                             s 
                             β 
                           
                            
                           
                             c 
                             γ 
                           
                         
                         - 
                         
                           
                             s 
                             α 
                           
                            
                           
                             c 
                             γ 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             
                               c 
                               α 
                             
                           
                            
                           
                             s 
                             β 
                           
                            
                           
                             c 
                             γ 
                           
                         
                         - 
                         
                           
                             s 
                             α 
                           
                            
                           
                             c 
                             γ 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             c 
                             α 
                           
                         
                          
                         
                           c 
                           β 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             t 
                             Z 
                           
                         
                         - 
                         F 
                       
                     
                   
                   
                     
                       
                         
                           
                             c 
                             α 
                           
                            
                           
                             s 
                             β 
                           
                            
                           
                             c 
                             γ 
                           
                         
                         - 
                         
                           
                             s 
                             α 
                           
                            
                           
                             c 
                             γ 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             
                               c 
                               α 
                             
                           
                            
                           
                             s 
                             β 
                           
                            
                           
                             c 
                             γ 
                           
                         
                         - 
                         
                           
                             s 
                             α 
                           
                            
                           
                             c 
                             γ 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             c 
                             α 
                           
                         
                          
                         
                           c 
                           β 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             t 
                             Z 
                           
                         
                         + 
                         F 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       
                         x 
                         n 
                       
                     
                   
                   
                     
                       
                         y 
                         n 
                       
                     
                   
                   
                     
                       
                         z 
                         n 
                       
                     
                   
                   
                     
                       
                         O 
                         n 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0068]    After transforming the homogenous coordinates to Euclidean space coordinates (o=1 and z p  is not taken into account), the observation (x p ,y p ) 2D   T  on the image plane of a given point (x n , y n , z n ) 3D   T  belonging to the face in its neutral pose is: 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         x 
                         p 
                       
                     
                   
                   
                     
                       
                         y 
                         p 
                       
                     
                   
                 
                 ] 
               
               
                 2 
                  
                 
                     
                 
                  
                 D 
               
             
             = 
             
               
                 F 
                 N 
               
                
               
                 [ 
                 
                   
                     
                       
                         
                           
                             c 
                             β 
                           
                            
                           
                             c 
                             γ 
                           
                            
                           
                             x 
                             n 
                           
                         
                         - 
                         
                           
                             c 
                             β 
                           
                            
                           
                             s 
                             γ 
                           
                            
                           
                             y 
                             n 
                           
                         
                         + 
                         
                           
                             s 
                             β 
                           
                            
                           
                             z 
                             n 
                           
                         
                         + 
                         
                           t 
                           X 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   s 
                                   α 
                                 
                                  
                                 
                                   s 
                                   β 
                                 
                                  
                                 
                                   c 
                                   γ 
                                 
                               
                               + 
                               
                                 
                                   c 
                                   α 
                                 
                                  
                                 
                                   s 
                                   γ 
                                 
                               
                             
                             ) 
                           
                            
                           
                             x 
                             n 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   s 
                                   α 
                                 
                                  
                                 
                                   s 
                                   β 
                                 
                                  
                                 
                                   s 
                                   γ 
                                 
                               
                               - 
                               
                                 
                                   c 
                                   α 
                                 
                                  
                                 
                                   c 
                                   γ 
                                 
                               
                             
                             ) 
                           
                            
                           
                             y 
                             n 
                           
                         
                         - 
                         
                           
                             s 
                             α 
                           
                            
                           
                             c 
                             β 
                           
                            
                           
                             z 
                             n 
                           
                         
                         + 
                         
                           t 
                           Y 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               N 
               = 
               
                 
                   
                     ( 
                     
                       
                         
                           c 
                           α 
                         
                          
                         
                           s 
                           β 
                         
                          
                         
                           c 
                           γ 
                         
                       
                       - 
                       
                         
                           s 
                           α 
                         
                          
                         
                           s 
                           γ 
                         
                       
                     
                     ) 
                   
                    
                   
                     x 
                     n 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             c 
                             α 
                           
                         
                          
                         
                           s 
                           β 
                         
                          
                         
                           s 
                           γ 
                         
                       
                       - 
                       
                         
                           s 
                           α 
                         
                          
                         
                           c 
                           γ 
                         
                       
                     
                     ) 
                   
                    
                   
                     y 
                     n 
                   
                 
                 - 
                 
                   
                     c 
                     α 
                   
                    
                   
                     c 
                     β 
                   
                    
                   
                     z 
                     n 
                   
                 
                 - 
                 
                   t 
                   z 
                 
                 + 
                 F 
               
             
           
         
       
     
         [0069]    For each of the vertices i to be moved during the reshaping of the face (referring to  FIG. 2 ) according to the new deformation factor w new . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                       
                         i 
                         = 
                       
                       new 
                     
                     = 
                     
                       
                         w 
                         i 
                       
                       · 
                       
                         ( 
                         
                           
                             
                               r 
                               i 
                             
                              
                             
                               ( 
                               
                                 α 
                                 , 
                                 β 
                                 , 
                                 
                                   t 
                                   z 
                                 
                               
                               ) 
                             
                           
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   
                     EQ 
                     . 
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       i 
                     
                      
                     
                       ( 
                       
                         α 
                         , 
                         β 
                         , 
                         
                           t 
                           z 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             α 
                             · 
                             
                               ( 
                               
                                 
                                   y 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             β 
                             · 
                             
                               ( 
                               
                                 
                                   x 
                                   i 
                                 
                                 - 
                                 
                                   x 
                                   18 
                                 
                               
                               ) 
                             
                           
                         
                         E 
                       
                       ) 
                     
                     + 
                     
                       
                         t 
                         z 
                       
                       G 
                     
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                     . 
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    x i  and y i  are the 2D coordinates of the vertices on the i th  image as the have been determined on the mesh and not on the 3D observation model. With embodiments of the invention, x 18  and y c1  refer to point  218  and point  251 , respectively, as shown in  FIG. 2 . One should note that the Y-axis of the observation model and the Y-axis of the reference system for the mesh are inverted; thus, the consideration of one system or the other does change how the adaptation should be treated. E and G are scale values that are determined empirically in each system that uses this approach. E controls the amount of deformation due to the rotations and G controls the influence of the distance of the person to the camera. Once the “neutral” position of a face on a picture is determined for a concrete instance of the system (neutral meaning α=β=γ=tz=t y =t z =0), E and G are chosen so that correction function r stays within reasonable limits. (For most implementations that would be from 0 to 1.) E scales down the units from the image vertices coordinates (x,y) and sets how much influence the angles have with respect to the face translation. G scales down the units from the z-translation on the 3D model used and also sets the influence of this parameter in the rectifying factor. For example, E takes a value of the order of magnitude of the face coordinate units (e.g., (y c1 -y i )&amp;(x 1 -x 18 )max value=1500, E˜2000*2*3.1415˜12000) and the same applies to G regarding t z  (e.g., t 2  max value 60, G˜100*2˜200). In the given example, E and G would have approximately equivalent influence accounting for half of the influence in the final rectification. 
         [0070]    From EQs. 11 and 12, a deformation factor w (e.g., as determined with EQs. 4A-4D) is multiplied by a correction factor r(α,⊕,t z )+1 in order obtain a new (corrected) deformation factor w new  From EQs. 1-5B, a corrected deformation vector is determined. Each deformation vector is applied to a corresponding vertex to obtain a transformed mesh. Experimental data using EQs. 11-12 have been obtained for angular displacement α  1011  varying between ±0.087 radians and angular displacement β  1013  varying between ±0.17 radians. 
         [0071]    As can be appreciated by one skilled in the art, a computer system (e.g., computer  1  as shown in  FIG. 9 ) with an associated computer-readable medium containing instructions for controlling the computer system may be utilized to implement the exemplary embodiments that are disclosed herein. The computer system may include at least one computer such as a microprocessor, a cluster of microprocessors, a mainframe, and networked workstations. 
         [0072]    While the invention has been described with respect to specific examples including presently preferred modes of carrying out the invention, those skilled in the art will appreciate that there are numerous variations and permutations of the above described systems and techniques that fall within the spirit and scope of the invention as set forth in the appended claims. 
         [0000]    
       
         
               
             
               
             
           
               
                 APPENDIX 
               
               
                   
               
               
                 EXEMPLARY CODE FOR THE ALGORITHM 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 void CAAMUtil::PTS2ASF_SKIN( const CString &amp;path, const CString 
               
               
                 &amp;path_image, const CString &amp;path_out ) { 
               
               
                 using namespace std; 
               
               
                 //read the ASF model 
               
               
                 char name[ ]=“C:\\hello\\Picture_001.bmp”; 
               
               
                 double original_width=0; 
               
               
                 double original_height=0; 
               
               
                 CDMultiBand&lt;TAAMPixel&gt; texture; 
               
               
                 CAAMShape shape; 
               
               
                 CAAMShape shape_dummy; 
               
               
                 int www=0; 
               
               
                 int hhh=0; 
               
               
                 long sss=0; 
               
               
                 BYTE* buffer; 
               
               
                 buffer = LoadBMP(&amp;www,&amp;hhh,&amp;sss,path_image); 
               
               
                 double W = double(www); 
               
               
                 double H = double(hhh); 
               
               
                 bool ok = shape.ReadASF(path); 
               
               
                 //change from relative coordinates to absolute ones, cos the 
               
               
                 algorithm works with those 
               
               
                 shape.Rel2Abs(W,H); 
               
               
                 original_width = shape.Width( ); 
               
               
                 original_height = shape.Height( ); 
               
               
                 //cout&lt;&lt;original_width&lt;&lt;endl; 
               
               
                 //cout&lt;&lt;original_height&lt;&lt;endl; 
               
               
                 //get the number of points 
               
               
                 int n = shape.NPoints( ); 
               
               
                 //cout&lt;&lt;“number of points: ”&lt;&lt;n&lt;&lt;endl; 
               
               
                 int k=0; 
               
               
                 //create the userfields with a resize 
               
               
                 shape.UserField(1).resize(n); 
               
               
                 shape.UserField(2).resize(n); 
               
               
                 shape.UserField(3).resize(n); 
               
               
                 //create all the variables needed for the manipulation 
               
               
                 // data needed for deformation computation 
               
               
                 double scale = (shape.MaxX( ) − shape.MinX( ))/1000.0; 
               
               
                 double center1_x = (shape.MaxX( ) + shape.MinX( ))/2.0; 
               
               
                 double center1_y = (shape.MaxY( ) + shape.MinY( ))/2.0; 
               
               
                 double center2_x = center1_x; 
               
               
                 double center2_y = center1_y+(shape.MaxY( ) − 
               
               
                 shape.MinY( ))/4.0; //!!watch out signs 
               
               
                 double x1, y1, x2, y2; 
               
               
                 //esto tan hardcode no puede ser bueno... 
               
               
                 shape.GetPoint(6,x1,y1); 
               
               
                 //cout&lt;&lt;“point 6 :”&lt;&lt;x1&lt;&lt;“,”&lt;&lt;y1&lt;&lt;endl; //aqui hem dona 
               
               
                 x1=0,y1=0 ?? 
               
               
                 shape.GetPoint(13,x2,y2); 
               
               
                 double dAB = sqrt(pow(x1−x2,2)+pow(y1−y2,2)); 
               
               
                 //mirar que sea el 18 seguro 
               
               
                 shape.GetPoint(18,x1,y1); 
               
               
                 double dBC = sqrt(pow(x1−x2,2)+pow(y1−y2,2)); 
               
               
                 shape.GetPoint(18,x1,y1); 
               
               
                 shape.GetPoint(24,x2,y2); 
               
               
                 double dCD = sqrt(pow(x1−x2,2)+pow(y1−y2,2)); 
               
               
                 shape.GetPoint(31,x1,y1); 
               
               
                 double dDE = sqrt(pow(x1−x2,2)+pow(y1−y2,2)); 
               
               
                 //bucle to modify the interesting points and modify the 
               
               
                 interesting points 
               
               
                 for(k=0;k&lt;n;k++){ 
               
               
                 if(k==6∥k==7∥k==8∥k==9∥k==10∥k==11∥k==12∥k==13∥k==24∥ 
               
               
                 k==25∥k==26∥k==27∥k==28∥k==29∥k==30∥k==31) 
               
               
                 { 
               
               
                 shape.UserField(1)[k]=1.0; 
               
               
                 shape.GetPoint(k,x1,y1); 
               
               
                 //cout&lt;&lt;“point ”&lt;&lt;k&lt;&lt;“ :”&lt;&lt;x1&lt;&lt;“,”&lt;&lt;y1&lt;&lt;endl; 
               
               
                 double weight = 1.; 
               
               
                 if(6&lt;=k&amp;&amp;k&lt;=13) 
               
               
                 { 
               
               
                 shape.GetPoint(6,x2,y2); 
               
               
                 double distance = sqrt(pow(x2 − 
               
               
                 x1,2)+pow(y2 − y1,2)); 
               
               
                 weight = 
               
               
                 2.0/3.0*(1.0/dAB)*distance+1./3.; 
               
               
                 } 
               
               
                 if(24&lt;=k&amp;&amp;k&lt;=31) 
               
               
                 { 
               
               
                 shape.GetPoint(31,x2,y2); 
               
               
                 double distance = sqrt(pow(x2 − 
               
               
                 x1,2)+pow(y2 − y1,2)); 
               
               
                 weight = 
               
               
                 2.0/3.0*(1.0/dDE)*distance+1./3.; 
               
               
                 } 
               
               
                 double vector_x = (x1− 
               
               
                 center1_x)/sqrt(pow(center1_x − x1,2)+pow(center1_y − y1,2)); 
               
               
                 double vector_y = (y1− 
               
               
                 center1_y)/sqrt(pow(center1_x − x1,2)+pow(center1_y − y1,2)); 
               
               
                 shape.UserField(2)[k]=vector_x*scale*weight; 
               
               
                 shape.UserField(3)[k]=vector_y*scale*weight; 
               
               
                 } 
               
               
                 if(k==14∥k==15∥k==16∥k==17∥k==18∥k==19∥k==20∥k==21∥k==22| 
               
               
                 |k==23) 
               
               
                 { 
               
               
                 shape.UserField(1)[k]=1.0; 
               
               
                 shape.GetPoint(k,x1,y1); 
               
               
                 double weight = 1.; 
               
               
                 if(12&lt;=k&amp;&amp;k&lt;=17) 
               
               
                 { 
               
               
                 shape.GetPoint(13,x2,y2); 
               
               
                 double distance = sqrt(pow(x2 − 
               
               
                 x1,2)+pow(y2 − y1,2)); 
               
               
                 weight = − 
               
               
                 (1.0/pow(dBC,2))*pow(distance,2)+1.; 
               
               
                 } 
               
               
                 if(18&lt;=k&amp;&amp;k&lt;=23) 
               
               
                 { 
               
               
                 shape.GetPoint(24,x2,y2); 
               
               
                 double distance = sqrt(pow(x2 − 
               
               
                 x1,2)+pow(y2 − y1,2)); 
               
               
                 weight = − 
               
               
                 (1.0/pow(dCD,2))*pow(distance,2)+1.; 
               
               
                 } 
               
               
                 double vector_x = (x1− 
               
               
                 center2_x)/sqrt(pow(center2_x − x1,2)+pow(center2_y − y1,2)); 
               
               
                 double vector_y = (y1− 
               
               
                 center2_y)/sqrt(pow(center2_x − x1,2)+pow(center2_y − y1,2)); 
               
               
                 shape.UserField(2)[k]=vector_x*weight*scale; 
               
               
                 shape.UserField(3)[k]=vector_y*weight*scale; 
               
               
                 } 
               
               
                 } 
               
               
                 CAAMShape shape2; 
               
               
                 //change the size of the shape2−&gt; 10 points for the neck 
               
               
                 shape2.Resize(20,0); 
               
               
                 //when resize the fields for the users, we are creating the space for 
               
               
                 them 
               
               
                 shape2.UserField(1).resize(n); 
               
               
                 shape2.UserField(2).resize(n); 
               
               
                 shape2.UserField(3).resize(n); 
               
               
                 //filling the fields 
               
               
                 //first we obtain the distance of the face, and will displace the 
               
               
                 interesting points a third of this distance to the bottom of the image 
               
               
                 double desp_y =0; 
               
               
                 double desp_x =0; 
               
               
                 double xa,ya,xb,yb,xc,yc; 
               
               
                 desp_y = shape.Height( ); 
               
               
                 desp_y = desp_y/3.0; 
               
               
                 //cout&lt;&lt;“distance of the neck: ”&lt;&lt;desp_y&lt;&lt;endl; 
               
               
                 //we also need the distance between the extrems of the neck, we can do 
               
               
                 it like this 
               
               
                 shape.GetPoint(14,xb,yb); 
               
               
                 shape.GetPoint(23,xc,yc); 
               
               
                 desp_x = (xc−xb)/2; 
               
               
                 shape.GetPoint(18,xc,yb); 
               
               
                 //then we take the interesting points, the x will be the same, and the 
               
               
                 y will be desplaced desp_y 
               
               
                 double neck[10]; 
               
               
                 double dist; 
               
               
                 for(k=14;k&lt;24;k++){ 
               
               
                 shape.GetPoint(k,xa,ya); 
               
               
                 ya=ya−desp_y; 
               
               
                 shape2.SetPoint(k−14,xa,ya); 
               
               
                 dist=xa−xc; 
               
               
                 if(k&lt;18) 
               
               
                 neck[k−14]=−(((dist*dist)/(10*desp_x*desp_x))); 
               
               
                 else neck[k−14]=(((dist*dist)/(10*desp_x*desp_x))); 
               
               
                 }