# Patent

## Title
Image coding method and apparatus and image decoding method and apparatus

## Abstract

An image coding and decoding technology for compressing motion pictures which makes use of critical point filter technology and that includes detection of scene changes to provide more accurate decoding. An image input unit acquires key frames. A scene change detector detects when there is a scene change between key frames, and notifies a matching processor to that effect. The matching processor then skips generation of correspondence data between those key frames. A stream generator generates a coded data stream in a manner that key frames and associated correspondence data are incorporated when there is no scene change, while key frames and associated correspondence disable data are incorporated when there is a scene change. This allows a decoder to more accurately decode the data stream and take account of scene changes.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to an image processing technology, and more particularly relates to a method and apparatus for coding and decoding images, with an emphasis on compression of moving pictures.
2. Description of the Related Art
Recently, image processing and compression methods such as those proposed by MPEG (Motion Picture Expert Group) have expanded to be used with transmission media such as network and broadcast rather than just storage media such as CDs. Generally speaking, the success of the digitization of broadcast materials has been caused at least in part by the availability of MPEG compression coding technology. In this way, a barrier that previously existed between broadcast and other types of communication has begun to disappear, leading to a diversification of service-providing businesses. Thus, we are facing a situation where it is hard to predict how the digital culture would evolve in this age of broadband.
Even in such a chaotic situation, it is clear that the direction of the compression technology of motion pictures will be to move to both higher compression rates and better image quality. It is a well-known fact that block distortion in MPEG compression is sometimes responsible for causing degraded image quality and preventing the compression rate from being improved.
SUMMARY OF THE INVENTION
The present invention has been made in view of the foregoing circumstances and an object thereof is to provide a new compression technology for image data and, more particularly, motion pictures. In the following, an image decoding technology and an image coding technology are described generally using reference to motion pictures, however, the use thereof is not limited to motion pictures but also includes image effects such as morphing and walkthrough.
A preferred embodiment according to the present invention relates to an image coding apparatus. This apparatus includes: an image input unit which receives data of key frames; a scene change detector which detects when two key frames bracket a scene change therebetween; a correspondence data generator which generates correspondence data for at least key frames which do not have a scene change between them (also referred to as “a pair of normal key frames” hereinafter); and a stream generator which generates a data stream by incorporating correspondence data for key frames which do not have a scene change between them and by incorporating disable data (also referred to as “correspondence disable data” hereinafter) indicating prohibition of correspondence for key frames (also referred to as “a pair of special key frames” hereinafter) which have a scene change between them.
Generally speaking, the term “frame” is sometimes used as a unit of an image. In this patent specification, however, the term “frame” will not be distinguished from the word “image” unless otherwise necessary. Further, the term “scene change” is not necessarily determined objectively, and it may be considered that there is a scene change in a case when there exists a predetermined difference, for example, a certain degree of difference, between images. From another standpoint, it may be stated that a scene change occurs if reproduction images obtained without going through generation of intermediate frames by interpolating key frames and so forth based on correspondence data, turn out to be more desirable than those obtained with the generation of intermediate frames by interpolating the key frames and so forth based on the correspondence data. According to this embodiment, generation of intermediate frames utilizing the correspondence data is prohibited in accordance with a decision made at a coding side, however, alternatively an indication can be made to the effect that such the generation of intermediate frames is at least not recommended.
The data stream may be formed in a manner that the correspondence data and the disable data differ in part of data values thereof, for example, data values in headers, and thus both can be identified thereby. For example, each type of data can be distinguished from the other by a bit value in a predetermined position.
The correspondence data generator may extract critical points in respective key frames, and perform a pixel-based matching computation based on a correspondence relation between the critical points. In so doing, the correspondence information generator may further multiresolutionalize respective key frames by extracting critical points thereof, and specify a correspondence relation between the critical points by performing the pixel-based matching computation in sequence starting from a coarse level of multiresolution.
Another preferred embodiment according to the present invention relates to an image coding method. This method includes: detecting when two key frames bracket a scene change therebetween (that is, have a scene change between them); generating correspondence data for at least key frames which do not bracket a scene change; and generating disable data indicating prohibition of correspondence for key frames which bracket a scene change.
Still another preferred embodiment according to the present invention relates to an image decoding apparatus. This apparatus includes: a stream input unit which receives a data stream including at least correspondence data between key frames; an intermediate image generator which generates an intermediate frame based on the correspondence data included in the input data stream and data of the key frames; and a scene change detector which detects a scene change between key frames based on the input data stream, and wherein the intermediate image generator stops generation of an intermediate frame when a scene change is detected.
The scene change detector may detect the scene change based on the presence in the data stream of correspondence disable data, which has a different file structure than the correspondence data. Moreover, the scene change detector may further detect ratio data, which may be included in the correspondence disable data, which indicate a temporal position of the scene change between two key frames which bracket the scene change.
Still further, the intermediate image generator may control in a manner such that each of the two key frames which bracket the scene change are displayed at a timing based on the ratio data. Thus, for example, if there is provided a display controller, the ratio data may be notified to the display controller such that the first key frame of the pair of the special key frames may be output repeatedly until a time indicated by the ratio and the second key frame thereof may be output repeatedly after that time until the next frame or correspondence data is handled.
Still another preferred embodiment according to the present invention relates to an image decoding method. This method includes: receiving a data steam including at least correspondence data between key frames; generating an intermediate frame based on the correspondence data and data of key frames; and detecting a scene change based on the data stream, and wherein generation of an intermediate frame is skipped when a scene change is detected. Here, a scene change may be indicated in the data stream by the inclusion of disable data which indicates that there is no correspondence data for the two adjacent key frames which bracket the scene change.
Still another preferred embodiment according to the present invention relates to an image coding apparatus. This apparatus includes: an image input unit which receives or acquires data of key frames; a scene change detector which detects a first key frame and a second key frame which bracket a scene change; and a correspondence data generator which generates correspondence data between key frames. In this apparatus, when the first key frame and the second key frame are detected, a third key frame and a fourth key frame, disposed before and after the scene change, respectively, and having an interval therebetween that is narrower than that between the first key frame and the second key frame, are input to the correspondence data generator, so that correspondence data between the first key frame and the third key frame as well as between the second key frame and the fourth key frame are generated.
In this embodiment, the correspondence data are generated with the interval being further narrowed, both before and after the scene change. Therefore, even in a case where the correspondence data may be invalid because the data cross over a scene change, more detailed correspondence data are obtained before and after the scene change, thus facilitating a more accurate reproduction of images at a decoding stage.
Further, the scene change detector may send a request to the input unit to acquire the third key frame and the fourth key frame from within the image coding apparatus, for example, a storage area, or from an external source outside the image coding apparatus, such as an external storage or a network or the like. In a particular case, the third key frame and the fourth key frame may be input only when the interval between the first key frame and the second key frame exceeds a predetermined value.
Still another preferred embodiment according to the present invention relates to an image coding method. This method includes: selecting key frames from data of moving pictures; detecting a scene change in the data of moving pictures; selecting an additional two key frames positioned before and after the scene change; and generating correspondence data between adjacent key frames after the selecting the additional two key frames.
In each of the above-described embodiments, the technology utilized to derive the correspondence data or information may be an application of the matching technology (referred to as “base technology” hereinafter) proposed by Japanese Patent No. 2927350 which is owned by the same assignee as the present patent specification.
It is to be noted that it is also possible to have replacement or substitution of the above-described structural components and elements of methods in part or whole as between method and apparatus or to add elements to either method or apparatus and also, the apparatuses and methods may be implemented by a computer program and saved on a recording medium or the like and are all effective as and encompassed by the present invention.
Moreover, this summary of the invention includes features that may not be necessary features such that an embodiment of the present invention may also be a sub-combination of these described features.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1(a) is an image obtained as a result of the application of an averaging filter to a human facial image.
FIG. 1(b) is an image obtained as a result of the application of an averaging filter to another human facial image.
FIG. 1(c) is an image of a human face at p(5,0) obtained in a preferred embodiment in the base technology.
FIG. 1(d) is another image of a human face at p(5,0) obtained in a preferred embodiment in the base technology.
FIG. 1(e) is an image of a human face at p(5,1) obtained in a preferred embodiment in the base technology.
FIG. 1(f) is another image of a human face at p(5,1) obtained in a preferred embodiment in the base technology.
FIG. 1(g) is an image of a human face at p(5,2) obtained in a preferred embodiment in the base technology.
FIG. 1(h) is another image of a human face at p(5,2) obtained in a preferred embodiment in the base technology.
FIG. 1(i) is an image of a human face at p(5,3) obtained in a preferred embodiment in the base technology.
FIG. 1(j) is another image of a human face at p(5,3) obtained in a preferred embodiment in the base technology.
FIG. 2(R) shows an original quadrilateral.
FIG. 2(A) shows an inherited quadrilateral.
FIG. 2(B) shows an inherited quadrilateral.
FIG. 2(C) shows an inherited quadrilateral.
FIG. 2(D) shows an inherited quadrilateral.
FIG. 2(E) shows an inherited quadrilateral.
FIG. 3 is a diagram showing the relationship between a source image and a destination image and that between the m-th level and the (m-1)th level, using a quadrilateral.
FIG. 4 shows the relationship between a parameter η (represented by x-axis) and energy Cf (represented by y-axis).
FIG. 5(a) is a diagram illustrating determination of whether or not the mapping for a certain point satisfies the bijectivity condition through the outer product computation.
FIG. 5(b) is a diagram illustrating determination of whether or not the mapping for a certain point satisfies the bijectivity condition through the outer product computation.
FIG. 6 is a flowchart of the entire procedure of a preferred embodiment in the base technology.
FIG. 7 is a flowchart showing the details of the process at S1 in FIG. 6.
FIG. 8 is a flowchart showing the details of the process at S10 in FIG. 7.
FIG. 9 is a diagram showing correspondence between partial images of the m-th and (m-1)th levels of resolution.
FIG. 10 is a diagram showing source hierarchical images generated in the embodiment in the base technology.
FIG. 11 is a flowchart of a preparation procedure for S2 in FIG. 6.
FIG. 12 is a flowchart showing the details of the process at S2 in FIG. 6.
FIG. 13 is a diagram showing the way a submapping is determined at the 0-th level.
FIG. 14 is a diagram showing the way a submapping is determined at the first level.
FIG. 15 is a flowchart showing the details of the process at S21 in FIG. 12.
FIG. 16 is a graph showing the behavior of energy C.f(m,s) corresponding to f(m,s) (λ=iΔλ) which has been obtained for a certain f(m,s) while varying λ.
FIG. 17 is a diagram showing the behavior of energy Cf(n) corresponding to f(n) (η=iΔη) (i=0,1, . . . ) which has been obtained while varying η.
FIG. 18 is a diagram showing a structure of an image coding apparatus according to an embodiment of the invention.
FIG. 19 shows a state in which a scene change is included in a flow of key frames.
FIG. 20 shows a structure of a data stream generated by a stream generator.
FIG. 21 shows a structure of a correspondence data file generated by a matching processor.
FIG. 22 shows a structure of a correspondence disable file generated by a matching processor or a scene change detector.
FIG. 23 is a diagram showing a structure of an image decoding apparatus according to an embodiment of the invention.
FIG. 24 illustrates the principle on which intermediate frames are generated by an intermediate image generator in the image decoding apparatus.
FIG. 25 illustrates operation of a modified example of the image coding apparatus of FIG. 23.



DETAILED DESCRIPTION OF THE INVENTION
The invention will now be described based on the preferred embodiments, which do not intend to limit the scope of the present invention, but exemplify the invention. All of the features and the combinations thereof described in the embodiment are not necessarily essential to the invention.
First, the multiresolutional critical point filter technology and the image matching processing using the technology, both of which will be utilized in the preferred embodiments, will be described in detail as “Base Technology”. Namely, the following sections [1] and [2] (below) belong to the base technology, where section [1] describes elemental techniques and section [2] describes a processing procedure. These techniques are patented under Japanese Patent No. 2927350 and owned by the same assignee of the present invention. However, it is to be noted that the image matching techniques provided in the present embodiments are not limited to the same levels. In particular, in FIGS. 18 to 25, image data coding and decoding techniques, utilizing, in part, the base technology, will be described in more detail.
Base Tecnology
[1] Detailed Description of Elemental Techniques
[1.1] Introduction
Using a set of new multiresolutional filters called critical point filters, image matching is accurately computed There is no need for any prior knowledge concerning the content of the images or objects in question. The matching of the images is computed at each resolution while proceeding through the resolution hierarchy. The resolution hierarchy proceeds from a coarse level to a fine level. Parameters necessary for the computation are set completely automatically by dynamical computation analogous to human visual systems. Thus, There is no need to manually specify the correspondence of points between the images.
The base technology can be applied to, for instance, completely automated morphing, object recognition, stereo photogrammetry, volume rendering, and smooth generation of motion images from a small number of frames. When applied to morphing, given images can be automatically transformed. When applied to volume rendering, intermediate images between cross sections can be accurately reconstructed, even when a distance between cross sections is rather large and the cross sections vary widely in shape.
[1.2] The Hierarchy of the Critical Point Filters
The multiresolutional filters according to the base technology preserve the intensity and location of each critical point included in the images while reducing the resolution. Initially, let the width of an image to be examined be N and the height of the image be M. For simplicity, assume that N=M=2n where n is a positive integer. An interval [0, N] ⊂ R is denoted by I. A pixel of the image at position (i, j) is denoted by p(i,j) where i,j ∈ I.
Here, a multiresolutional hierarchy is introduced. Hierarchized image groups are produced by a multiresolutional filter. The multiresolutional filter carries out a two dimensional search on an original image and detects critical points therefrom. The multiresolutinal filter then extracts the critical points from the original image to construct another image having a lower resolution. Here, the size of each of the respective images of the m-th level is denoted as 2m×2m (0≦m≦n). A critical point filter constructs the following four new hierarchical images recursively, in the direction descending from n.




  
    
      
        
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The above four images are referred to as subimages hereinafter. When minx≦t≦x+1 and maxx≦t≦x+1 are abbreviated to α and β, respectively, the subimages can be expressed as follows:

P(m,0)=α(x)α(y)p(m+1,0) 

P(m,1)=α(x)β(y)p(m+1,1) 

P(m,2)=β(x)α(y)p(m+1,2) 

P(m,2)=β(x)β(y)p(m,+1,3) 

Namely, they can be considered analogous to the tensor products of α and β. The subimages correspond to the respective critical points. As is apparent from the above equations, the critical point filter detects a critical point of the original image for every block consisting of 2×2 pixels. In this detection, a point having a maximum pixel value and a point having a minimum pixel value are searched with respect to two directions, namely, vertical and horizontal directions, in each block. Although pixel intensity is used as a pixel value in this base technology, various other values relating to the image may be used. A pixel having the maximum pixel values for the two directions, one having minimum pixel values for the two directions, and one having a minimum pixel value for one direction and a maximum pixel value for the other direction are detected as a local maximum point, a local minimum point, and a saddle point, respectively.
By using the critical point filter, an image (1 pixel here) of a critical point detected inside each of the respective blocks serves to represent its block image (4 pixels here) in the next lower resolution level. Thus, the resolution of the image is reduced. From a singularity theoretical point of view, α (x) α (y) preserves the local minimum point (minima point), β (x) β (y) preserves the local maximum point (maxima point), α (x) β (y) and β (x) α (y) preserve the saddle points.
At the beginning, a critical point filtering process is applied separately to a source image and a destination image which are to be matching-computed. Thus, a series of image groups, namely, source hierarchical images and destination hierarchical images are generated. Four source hierarchical images and four destination hierarchical images are generated corresponding to the types of the critical points.
Thereafter, the source hierarchical images and the destination hierarchical images are matched in a series of resolution levels. First, the minima points are matched using p(m,0). Next, the first saddle points are matched using p(m,1) based on the previous matching result for the minima points. The second saddle points are matched using p(m,2). Finally, the maxima points are matched using p(m,3).
FIGS. 1c and 1d show the subimages p(5,0) of the images in FIGS. 1a and 1b, respectively. Similarly, FIGS. 1e and 1f show the subimages p(5,1), FIGS. 1g and 1h show the subimages p(5,2), and FIGS. 1i and 1j show the subimages p(5,3). Characteristic parts in the images can be easily matched using subimages. The eyes can be matched by p(5,0) since the eyes are the minima points of pixel intensity in a face. The mouths can be matched by p(5,1) since the mouths have low intensity in the horizontal direction. Vertical lines on both sides of the necks become clear by p(5,2). The ears and bright parts of the cheeks become clear by p(5,3) since these are the maxima points of pixel intensity.
As described above, the characteristics of an image can be extracted by the critical point filter. Thus, by comparing, for example, the characteristics of an image shot by a camera with the characteristics of several objects recorded in advance, an object shot by the camera can be identified.
[1.3] Computation of Mapping Between Images
Now, for matching images, a pixel of the source image at the location (i,j) is denoted by




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and that of the destination image at (k,l) is denoted by





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      k
      ,
      l
    
    )
  
  
    (
    n
    )
  




where i, j, k, l ∈ I. The energy of the mapping between the images (described later in more detail) is then defined. This energy is determined by the difference in the intensity of the pixel of the source image and its corresponding pixel of the destination image and the smoothness of the mapping. First, the mapping f(m,0):p(m,0)→q(m,0) between p(m,0) and q(m,0) with the minimum energy is computed. Based on f(m,0), the mapping f(m,1) between p(m,1) and q(m,1) with the minimum energy is computed. This process continues until f(m,3) between p(m,3) and q(m,3) is computed. Each f(m,i) (i=0,1,2, . . . ) is referred to as a submapping. The order of i will be rearranged as shown in the following equation (3) in computing f(m,i) for reasons to be described later.

f(m,i):p(m,σ(i))→q(m,σ(i))  (3)

where σ (i) ∈ {0,1,2,3}.

[1.3.1] Bijectivity

When the matching between a source image and a destination image is expressed by means of a mapping, that mapping shall satisfy the Bijectivity Conditions (BC) between the two images (note that a one-to-one surjective mapping is called a bijection). This is because the respective images should be connected satisfying both surjection and injection, and there is no conceptual supremacy existing between these images. It is to be noted that the mappings to be constructed here are the digital version of the bijection. In the base technology, a pixel is specified by a co-ordinate point.
The mapping of the source subimage (a subimage of a source image) to the destination subimage (a subimage of a destination image) is represented by f(m,s): I/2n−m×I/2n−m→I/2n−m×I/2n−m (s=0,1, . . . ), where




  
    f
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  
  =
  
    (
    
      k
      ,
      l
    
    )
  




means that





  p
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




of the source image is mapped to





  q
  
    (
    
      k
      ,
      l
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




of the destination image. For simplicity, when f(i,j)=(k,l) holds, a pixel q(k,l) is denoted by qf(i,j).

When the data sets are discrete as image pixels (grid points) treated in the base technology, the definition of bijectivity is important. Here, the bijection will be defined in the following manner, where i, j, k and l are all integers. First, a square region R defined on the source image plane is considered




  
    
      
        
          p
          
            (
            
              i
              ,
              j
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          p
          
            (
            
              
                i
                +
                1
              
              ,
              j
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          p
          
            (
            
              
                i
                +
                1
              
              ,
              
                j
                +
                1
              
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          p
          
            (
            
              i
              ,
              
                j
                +
                1
              
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
      
    
    
      
        (
        4
        )
      
    
  




where i=0, . . . , 2m−1, and j=0, . . . , 2m−1. The edges of R are directed as follows:





  
    
      
        
          
            
              p
              
                (
                
                  i
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          →
        
        ,
        
          
            
              p
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          →
        
        ,
        
          
            
              
                p
                
                  (
                  
                    
                      i
                      +
                      1
                    
                    ,
                    
                      j
                      +
                      1
                    
                  
                  )
                
                
                  (
                  
                    m
                    ,
                    s
                  
                  )
                
              
              ⁢
              
                p
                
                  (
                  
                    i
                    ,
                    
                      j
                      +
                      1
                    
                  
                  )
                
                
                  (
                  
                    m
                    ,
                    s
                  
                  )
                
              
            
            →
          
          ⁢
          
            
          
          ⁢
          
            and
          
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          
            
              p
              
                (
                
                  i
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  i
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          →
        
        ⁢
        
          
        
      
    
    
      
        
      
    
  




This square region R will be mapped by f to a quadrilateral on the destination image plane:




  
    
      
        
          q
          
            f
            ⁡
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                i
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
      
    
    
      
        (
        6
        )
      
    
  




This mapping f(m,s)(R), that is,





  
    
      
        f
        
          (
          
            m
            ,
            s
          
          )
        
      
      ⁡
      
        (
        R
        )
      
    
    =
    
      
        
          f
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁡
        
          (
          
            
              p
              
                (
                
                  i
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              p
              
                (
                
                  i
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          )
        
      
      =
      
        
          q
          
            f
            ⁡
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            f
            ⁡
            
              (
              
                i
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
      
    
  
  )




should satisfy the following bijectivity conditions(referred to as BC hereinafter):

    1. The edges of the quadrilateral f(m,s)(R) should not intersect one another.
    2. The orientation of the edges of f(m,s)(R) should be the same as that of R (clockwise in the case shown in FIG. 2, described below).
    3. As a relaxed condition, a retraction mapping is allowed.


Without a certain type of a relaxed condition as in, for example, condition 3 above, there would be no mappings which completely satisfy the BC other than a trivial identity mapping. Here, the length of a single edge of f(m,s)(R) may be zero. Namely, f(m,s)(R) may be a triangle. However, f(m,s)(R) is not allowed to be a point or a line segment having area zero. Specifically speaking, if FIG. 2R is the original quadrilateral, FIGS. 2A and 2D satisfy the BC while FIGS. 2B, 2C and 2E do not satisfy the BC.
In actual implementation, the following condition may be further imposed to easily guarantee that the mapping is surjective. Namely, each pixel on the boundary of the source image is mapped to the pixel that occupies the same location at the destination image. In other words, f(i,j)=(i,j) (on the four lines of i=0, i=2m−1, j=0, j=2m−1). This condition will be hereinafter referred to as an additional condition.
[1.3.2] Energy of Mapping
[1.3.2.1] Cost Related to the Pixel Intensity
The energy of the mapping f is defined. An objective here is to search a mapping whose energy becomes minimum. The energy is determined mainly by the difference in the intensity between the pixel of the source image and its corresponding pixel of the destination image. Namely, the energy C(i,j)(m,s) of the mapping f(m,s) at (i,j) is determined by the following equation (7).




  
    
      
        
          C
          
            (
            
              i
              ,
              j
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            
            
              
                V
                ⁡
                
                  (
                  
                    p
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                    
                      (
                      
                        m
                        ,
                        s
                      
                      )
                    
                  
                  )
                
              
              -
              
                V
                ⁡
                
                  (
                  
                    q
                    
                      f
                      ⁡
                      
                        (
                        
                          i
                          ,
                          j
                        
                        )
                      
                    
                    
                      (
                      
                        m
                        ,
                        s
                      
                      )
                    
                  
                  )
                
              
            
            
          
          2
        
      
    
    
      
        (
        7
        )
      
    
  




where





  
    V
    ⁡
    
      (
      
        p
        
          (
          
            i
            ,
            j
          
          )
        
        
          (
          
            m
            ,
            s
          
          )
        
      
      )
    
  
  ⁢
  
    
  
  ⁢
  
    and
  
  ⁢
  
    
  
  ⁢
  
    V
    ⁡
    
      (
      
        q
        
          f
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        
          (
          
            m
            ,
            s
          
          )
        
      
      )
    
  




are the intensity values of the pixels





  
    
      p
      
        (
        
          i
          ,
          j
        
        )
      
      
        (
        
          m
          ,
          s
        
        )
      
    
    ⁢
    
      
    
    ⁢
    
      and
    
    ⁢
    
      
    
    ⁢
    
      q
      
        f
        ⁡
        
          (
          
            i
            ,
            j
          
          )
        
      
      
        (
        
          m
          ,
          s
        
        )
      
    
  
  ,




respectively. The total energy C(m,s) of f is a matching evaluation equation, and can be defined as the sum of C(i,j)(m,s) as shown in the following equation (8).





  
    
      
        
          C
          f
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            ∑
            
              i
              =
              0
            
            
              i
              =
              
                
                  2
                  m
                
                -
                1
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                j
                =
                
                  
                    2
                    m
                  
                  -
                  1
                
              
            
            ⁢
            
              
            
            ⁢
            
              C
              
                (
                
                  i
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




[1.3.2.2] Cost Related to the Locations of the Pixel for Smooth Mapping

In order to obtain smooth mappings, another energy Df for the mapping is introduced. The energy Df is determined by the locations of




  
    
      p
      
        (
        
          i
          ,
          j
        
        )
      
      
        (
        
          m
          ,
          s
        
        )
      
    
    ⁢
    
      
    
    ⁢
    
      and
    
    ⁢
    
      
    
    ⁢
    
      q
      
        f
        ⁡
        
          (
          
            i
            ,
            j
          
          )
        
      
      
        (
        
          m
          ,
          s
        
        )
      
    
    ⁢
    
      
    
    ⁢
    
      (
      
        
          i
          =
          0
        
        ,
        1
        ,
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        ,
        
          
            2
            m
          
          -
          1
        
        ,
        
          j
          =
          0
        
        ,
        1
        ,
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        ,
        
          
            2
            m
          
          -
          1
        
      
      )
    
  
  ,




regardless of the intensity of the pixels. The energy D(i,j)(m,s) of the mapping f(m,s) at a point (i,j) is determined by the following equation (9).





  
    
      
        
          D
          
            (
            
              i
              ,
              j
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            η
            ⁢
            
              
            
            ⁢
            
              E
              
                0
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          +
          
            E
            
              1
              ⁢
              
                
                  
                    
                  
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




where the coefficient parameter η which is equal to or greater than 0 is a real number. And we have





  
    
      
        
          
        
        ⁢
        
          
            E
            
              0
              ⁢
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
          =
          
            
              
              
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
                -
                
                  
                    f
                    
                      (
                      
                        m
                        ,
                        s
                      
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
              
              
            
            2
          
        
      
    
    
      
        (
        10
        )
      
    
  
  
    
      
        
          E
          
            1
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            ∑
            
              
                i
                ′
              
              =
              
                i
                -
                1
              
            
            i
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                
                  j
                  ′
                
                =
                
                  j
                  -
                  1
                
              
              j
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                  
                    
                      (
                      
                        
                          
                            f
                            
                              (
                              
                                m
                                ,
                                s
                              
                              )
                            
                          
                          ⁡
                          
                            (
                            
                              i
                              ,
                              j
                            
                            )
                          
                        
                        -
                        
                          (
                          
                            i
                            ,
                            j
                          
                          )
                        
                      
                      )
                    
                    -
                    
                      (
                      
                        
                          
                            f
                            
                              (
                              
                                m
                                ,
                                s
                              
                              )
                            
                          
                          ⁡
                          
                            (
                            
                              
                                i
                                ′
                              
                              ,
                              
                                j
                                ′
                              
                            
                            )
                          
                        
                        -
                        
                          (
                          
                            
                              i
                              ′
                            
                            ,
                            
                              j
                              ′
                            
                          
                          )
                        
                      
                      )
                    
                  
                  
                
                2
              
              /
              4
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where

∥(x,y)∥=√{square root over (x2+y2)}  (12),

i′ and j′ are integers and f(i′,j′) is defined to be zero for i′<0 and j′<0. E0 is determined by the distance between (i,j) and f(i,j). Eoprevents a pixel from being mapped to a pixel too far away from it. However, as explained below, E0 can be replaced by another energy function. E1 ensures the smoothness of the mapping. E1 represents a distance between the displacement of p(i,j) and the displacement of its neighboring points. Based on the above consideration, another evaluation equation for evaluating the matching, or the energy Df is determined by the following equation:





  
    
      
        
          D
          f
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            ∑
            
              i
              =
              0
            
            
              i
              =
              
                
                  2
                  m
                
                -
                1
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                j
                =
                
                  
                    2
                    m
                  
                  -
                  1
                
              
            
            ⁢
            
              
            
            ⁢
            
              D
              
                (
                
                  i
                  ,
                  j
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




[1.3.2.3] Total Energy of the Mapping

The total energy of the mapping, that is, a combined evaluation equation which relates to the combination of a plurality of evaluations, is defined as λCf(m,s)+Df(m,s), where λ≦0 is a real number. The goal is to detect a state in which the combined evaluation equation has an extreme value, namely, to find a mapping which gives the minimum energy expressed by the following:




  
    
      
        
          min
          f
        
        ⁢
        
          {
          
            
              λ
              ⁢
              
                
              
              ⁢
              
                C
                f
                
                  (
                  
                    m
                    ,
                    s
                  
                  )
                
              
            
            +
            
              D
              f
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          }
        
      
    
    
      
        (
        14
        )
      
    
  




Care must be exercised in that the mapping becomes an identity mapping if λ=0 and η=0 (i.e., f(m,s)(i,j)=(i,j) for all i=0,1, . . . , 2m−1 and j=0,1, . . . , 2m−1). As will be described later, the mapping can be gradually modified or transformed from an identity mapping since the case of λ=0 and η=0 is evaluated at the outset in the base technology. If the combined evaluation equation is defined as Cf(m,s)+λDf(m,s) where the original position of λ is changed as such, the equation with λ=0 and η=0 will be Cf(m,s) only. As a result thereof, pixels would randomly matched to each other only because their pixel intensities are close, thus making the mapping totally meaningless. Transforming the mapping based on such a meaningless mapping makes no sense. Thus, the coefficient parameter is so determined that the identity mapping is initially selected for the evaluation as the best mapping.
Similar to this base technology, differences in the pixel intensity and smoothness are considered in a technique called “optical flow” that is known in the art. However, the optical flow technique cannot be used for image transformation since the optical flow technique takes into account only the local movement of an object. However, global correspondence can also be detected by utilizing the critical point filter according to the base technology.
[1.3.3] Determining the Mapping with Multiresolution
A mapping fmin which gives the minimum energy and satisfies the BC is searched by using the multiresolution hierarchy. The mapping between the source subimage and the destination subimage at each level of the resolution is computed. Starting from the top of the resolution hierarchy (i.e., the coarsest level), the mapping is determined at each resolution level, and where possible, mappings at other levels are considered. The number of candidate mappings at each level is restricted by using the mappings at an upper (i.e., coarser) level of the hierarchy. More specifically speaking, in the course of determining a mapping at a certain level, the mapping obtained at the coarser level by one is imposed as a sort of constraint condition.
We thus define a parent and child relationship between resolution levels. When the following equation (15) holds,




  
    
      
        
          
            (
            
              
                i
                ′
              
              ,
              
                j
                ′
              
            
            )
          
          =
          
            (
            
              
                ⌊
                
                  i
                  2
                
                ⌋
              
              ,
              
                ⌊
                
                  j
                  2
                
                ⌋
              
            
            )
          
        
        ,
      
    
    
      
        (
        15
        )
      
    
  




where └x┘ denotes the largest integer not exceeding x,





  
    p
    
      (
      
        
          i
          ′
        
        ,
        
          j
          ′
        
      
      )
    
    
      (
      
        
          m
          -
          1
        
        ,
        s
      
      )
    
  
  ⁢
  
    
  
  ⁢
  
    and
  
  ⁢
  
    
  
  ⁢
  
    q
    
      (
      
        
          i
          ′
        
        ,
        
          j
          ′
        
      
      )
    
    
      (
      
        
          m
          -
          1
        
        ,
        s
      
      )
    
  




are respectively called the parents of





  
    
      p
      
        (
        
          i
          ,
          j
        
        )
      
      
        (
        
          m
          ,
          s
        
        )
      
    
    ⁢
    
      
    
    ⁢
    
      and
    
    ⁢
    
      
    
    ⁢
    
      q
      
        (
        
          i
          ,
          j
        
        )
      
      
        (
        
          m
          ,
          s
        
        )
      
    
  
  ,
  .




Conversely,





  
    p
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  
  ⁢
  
    
  
  ⁢
  
    and
  
  ⁢
  
    
  
  ⁢
  
    q
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  




are the child of





  p
  
    (
    
      
        i
        ′
      
      ,
      
        j
        ′
      
    
    )
  
  
    (
    
      
        m
        -
        1
      
      ,
      s
    
    )
  




and the child of





  
    q
    
      (
      
        
          i
          ′
        
        ,
        
          j
          ′
        
      
      )
    
    
      (
      
        
          m
          -
          1
        
        ,
        s
      
      )
    
  
  ,




respectively. A function parent (i,j) is defined by the following equation (16):





  
    
      
        
          parent
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          (
          
            
              ⌊
              
                i
                2
              
              ⌋
            
            ,
            
              ⌊
              
                j
                2
              
              ⌋
            
          
          )
        
      
    
    
      
        (
        16
        )
      
    
  




Now, a mapping between




  
    p
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  
  ⁢
  
    
  
  ⁢
  
    and
  
  ⁢
  
    
  
  ⁢
  
    q
    
      (
      
        k
        ,
        l
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  




is determined by computing the energy and finding the minimum thereof. The value of f(m,s)(i,j)=(k,l) is determined as follows using f(m−1,s) (m=1,2, . . . , n). First of all, a condition is imposed that





  q
  
    (
    
      k
      ,
      l
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




should lie inside a quadrilateral defined by the following definitions (17) and (18). Then, the applicable mappings are narrowed down by selecting ones that are thought to be reasonable or natural among them satisfying the BC.





  
    
      
        
          q
          
            
              g
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁡
            
              (
              
                
                  i
                  -
                  1
                
                ,
                
                  j
                  -
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            
              g
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁡
            
              (
              
                
                  i
                  -
                  1
                
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            
              g
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        ⁢
        
          q
          
            
              g
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁡
            
              (
              
                
                  i
                  +
                  1
                
                ,
                
                  j
                  -
                  1
                
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
      
    
    
      
        (
        17
        )
      
    
  




where

g(m,s)(i,j)=f(m−1,s)(parent(i,j))+f(m−1,s)(parent(i,j)+(1,1))  (18)

The quadrilateral defined above is hereinafter referred to as the inherited quadrilateral of




  
    p
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  
  .




The pixel minimizing the energy is sought and obtained inside the inherited quadrilateral.

FIG. 3 illustrates the above-described procedures. The pixels A, B, C and D of the source image are mapped to A′, B′, C′ and D′ of the destination image, respectively, at the (m−1)th level in the hierarchy. The pixel




  p
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




should be mapped to the pixel





  q
  
    
      f
      
        (
        m
        )
      
    
    ⁡
    
      (
      
        i
        ,
        j
      
      )
    
  
  
    (
    
      m
      ,
      s
    
    )
  




which exists inside the inherited quadrilateral A′B′C′D′. Thereby, bridging from the mapping at the (m−1)th level to the mapping at the m-th level is achieved.

The energy E0 defined above may now be replaced by the following equations (19) and (20):

E0(i,j)=∥f(m,0)(i,j)−g(m)(i,j)∥2  (19)

E0(i,j)=∥f(m,s)(i,j)−f(m,s−1)(i,j)∥2,(1≦i)  (20)

for computing the submapping f(m,0) and the submapping f(m,s) at the m-th level, respectively.

In this manner, a mapping which maintains a low energy of all the submappings is obtained. Using the equation (20) makes the submappings corresponding to the different critical points associated to each other within the same level in order that the subimages can have high similarity. The equation (19) represents the distance between f(m,s)(i,j) and the location where (i,j) should be mapped when regarded as a part of a pixel at the (m−1)the level.
When there is no pixel satisfying the BC inside the inherited quadrilateral A′B′C′D′, the following steps are taken. First, pixels whose distance from the boundary of A′B′C′D′ is L (at first, L=1) are examined. If a pixel whose energy is the minimum among them satisfies the BC, then this pixel will be selected as a value of f(m,s)(i,j). L is increased until such a pixel is found or L reaches its upper bound




  
    L
    max
    
      (
      m
      )
    
  
  .
  
    
  
  ⁢
  
    L
    max
    
      (
      m
      )
    
  




is fixed for each level m. If no pixel is found at all, the third condition of the BC is ignored temporarily and such mappings that caused the area of the transformed quadrilateral to become zero (a point or a line) will be permitted so as to determine f(m,s)(i,j). If such a pixel is still not found, then the first and the second conditions of the BC will be removed.

Multiresolution approximation is essential to determining the global correspondence of the images while preventing the mapping from being affected by small details of the images. Without the multiresolution approximation, it is impossible to detect a correspondence between pixels whose distances are large. In the case where the multiresolution approximation is not available, the size of an image will generally be limited to a very small size, and only tiny changes in the images can be handled. Moreover, imposing smoothness on the mapping usually makes it difficult to find the correspondence of such pixels. That is because the energy of the mapping from one pixel to another pixel which is far therefrom is high. On the other hand, the multiresolution approximation enables finding the approximate correspondence of such pixels. This is because the distance between the pixels is small at the upper (coarser) level of the hierarchy of the resolution.
[1.4] Automatic Determination of the Optimal Parameter Values
One of the main deficiencies of the existing image matching techniques lies in the difficulty of parameter adjustment. In most cases, the parameter adjustment is performed manually and it is extremely difficult to select the optimal value. However, according to the base technology, the optimal parameter values can be obtained completely automatically.
The systems according to this base technology include two parameters, namely, λ and η, where λ and η represent the weight of the difference of the pixel intensity and the stiffness of the mapping, respectively. In order to automatically determine these parameters, the are initially set to 0. First, λ is gradually increased from λ=0 while η is fixed at 0. As λ becomes larger and the value of the combined evaluation equation (equation (14)) is minimized, the value of Cf(m,s) for each submapping generally becomes smaller. This basically means that the two images are matched better. However, if λ exceeds the optimal value, the following phenomena occur:

    1. Pixels which should not be corresponded are erroneously corresponded only because their intensities are close.
    2. As a result, correspondence between images becomes inaccurate, and the mapping becomes invalid.
    3. As a result, Df(m,s) in equation (14) tends to increase abruptly.
    4. As a result, since the value of equation (14) tends to increase abruptly, f(m,s) changes in order to suppress the abrupt increase of Df(m,s). As a result, Cf(m,s) increases.


Therefore, a threshold value at which Cf(m,s) turns to an increase from a decrease is detected while a state in which equation (14) takes the minimum value with λ being increased is kept. Such λ is determined as the optimal value at η=0. Next, the behavior of Cf(m,s) is examined while η is increased gradually, and η will be automatically determined by a method described later. λ will then again be determined corresponding to such an automatically determined η.
The above-described method resembles the focusing mechanism of human visual systems. In the human visual systems, the images of the respective right eye and left eye are matched while moving one eye. When the objects are clearly recognized, the moving eye is fixed.
[1.4.1] Dynamic Determination of λ
Initially, λ is increased from 0 at a certain interval, and a subimage is evaluated each time the value of λ changes. As shown in equation (14), the total energy is defined by λCf(m,s)+Df(m,s).




  D
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




in equation (9) represents the smoothness and theoretically becomes minimum when it is the identity mapping. E0 and E1 increase as the mapping is further distorted. Since E1 is an integer, 1 is the smallest step of Df(m,s). Thus, it is impossible to change the mapping to reduce the total energy unless a changed amount (reduction amount) of the current





  λ
  ⁢
  
    
  
  ⁢
  
    C
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  




is equal to or greater than 1. Since Df(m,s) increases by more than 1 accompanied by the change of the mapping, the total energy is not reduced unless





  λ
  ⁢
  
    
  
  ⁢
  
    C
    
      (
      
        i
        ,
        j
      
      )
    
    
      (
      
        m
        ,
        s
      
      )
    
  




is reduced by more than 1.

Under this condition, it is shown that




  C
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




decreases in normal cases as λ increases. The histogram of





  C
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




is denoted as h(l), where h(l) is the number of pixels whose energy





  C
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




is l2. In order that λl2≧1 for example, the case of l2=1/λ is considered. When λ varies from λ1 to λ2, a number of pixels (denoted A) expressed by the following equation (21):





  
    
      
        A
        =
        
          
            
              
                ∑
                
                  l
                  =
                  
                    ⌈
                    
                      1
                      
                        λ
                        2
                      
                    
                    ⌉
                  
                
                
                  ⌊
                  
                    1
                    
                      λ
                      1
                    
                  
                  ⌋
                
              
              ⁢
              
                
              
              ⁢
              
                h
                ⁡
                
                  (
                  l
                  )
                
              
            
            ≅
            
              
                ∫
                
                  l
                  =
                  
                    1
                    
                      λ
                      2
                    
                  
                
                
                  1
                  
                    λ
                    1
                  
                
              
              ⁢
              
                
                  h
                  ⁡
                  
                    (
                    l
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  l
                
              
            
          
          =
          
            
              -
              
                
                  ∫
                  
                    λ
                    2
                  
                  
                    λ
                    1
                  
                
                ⁢
                
                  
                    h
                    ⁡
                    
                      (
                      l
                      )
                    
                  
                  ⁢
                  
                    1
                    
                      λ
                      
                        3
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    λ
                  
                
              
            
            =
            
              
                ∫
                
                  λ
                  1
                
                
                  λ
                  2
                
              
              ⁢
              
                
                  
                    h
                    ⁡
                    
                      (
                      l
                      )
                    
                  
                  
                    λ
                    
                      3
                      /
                      2
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  λ
                
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




changes to a more stable state having the energy shown in equation (22)





  
    
      
        
          
            C
            f
            
              (
              
                m
                ,
                s
              
              )
            
          
          -
          
            l
            2
          
        
        =
        
          
            C
            f
            
              (
              
                m
                ,
                s
              
              )
            
          
          -
          
            
              1
              λ
            
            .
          
        
      
    
    
      
        (
        22
        )
      
    
  




Here, it is assumed that the energy of these pixels is approximated to be zero. This means that the value of




  C
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




changes by:





  
    
      
        
          ∂
          
            C
            f
            
              (
              
                m
                ,
                s
              
              )
            
          
        
        =
        
          -
          
            A
            λ
          
        
      
    
    
      
        (
        23
        )
      
    
  




As a result, equation (24) holds.





  
    
      
        
          
            ∂
            
              C
              f
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          
            ∂
            λ
          
        
        =
        
          -
          
            
              h
              ⁡
              
                (
                l
                )
              
            
            
              λ
              
                5
                /
                2
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Since h(l)>0 , Cf(m,s) decreases in the normal case. However, when λ exceeds the optimal value, the above phenomenon, that is, an increase in Cf(m,s) occurs. The optimal value of λ is determined by detecting this phenomenon.

When




  
    
      
        
          h
          ⁡
          
            (
            l
            )
          
        
        =
        
          
            H
            ⁢
            
              
            
            ⁢
            
              l
              k
            
          
          =
          
            H
            
              λ
              
                k
                /
                2
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




is assumed, where both H(H>0) and k are constants, the equation (26) holds:





  
    
      
        
          
            ∂
            
              C
              f
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          
            ∂
            λ
          
        
        =
        
          -
          
            H
            
              λ
              
                
                  5
                  /
                  2
                
                +
                
                  k
                  /
                  2
                
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




Then, if k≠−3, the following equation (27) holds:




  
    
      
        
          C
          f
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          C
          +
          
            H
            
              
                (
                
                  
                    3
                    /
                    2
                  
                  +
                  
                    k
                    /
                    2
                  
                
                )
              
              ⁢
              
                λ
                
                  
                    3
                    /
                    2
                  
                  +
                  
                    k
                    /
                    2
                  
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




The equation (27) is a general equation of Cf(m,s) (where C is a constant).

When detecting the optimal value of λ, the number of pixels violating the BC may be examined for safety. In the course of determining a mapping for each pixel, the probability of violating the BC is assumed as a value p0 here. In this case, since




  
    
      
        
          
            ∂
            A
          
          
            ∂
            λ
          
        
        =
        
          
            h
            ⁡
            
              (
              l
              )
            
          
          
            λ
            
              3
              /
              2
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




holds, the number of pixels violating the BC increases at a rate of:





  
    
      
        
          B
          0
        
        =
        
          
            
              h
              ⁡
              
                (
                l
                )
              
            
            ⁢
            
              p
              0
            
          
          
            λ
            
              3
              /
              2
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




Thus,





  
    
      
        
          
            
              B
              0
            
            ⁢
            
              λ
              
                3
                /
                2
              
            
          
          
            
              p
              0
            
            ⁢
            
              h
              ⁡
              
                (
                l
                )
              
            
          
        
        =
        1
      
    
    
      
        (
        30
        )
      
    
  




is a constant. If it is assumed that h(l)=Hlk, the following equation (31), for example,

B0λ3/2+k/2=p0H  (31)

becomes a constant. However, when λ exceeds the optimal value, the above value of equation (31) increases abruptly. By detecting this phenomenon, i.e. whether or not the value of B0λ3/2+k/2 /2m exceeds an abnormal value B0 thres, the optimal value of λ can be determined. Similarly, whether or not the value of B1λ3/2+k/2 /2m exceeds an abnormal value B1 thres can be used to check for an increasing rate B1 of pixels violating the third condition of the BC. The reason why the factor 2m is introduced here will be described at a later stage. This system is not sensitive to the two threshold values B0 thres and B1 thres. The two threshold values B0 thres and B1 thres can be used to detect excessive distortion of the mapping which may not be detected through observation of the energy Cf(m,s).

In the experimentation, when λ exceeded 0.1 the computation of f(m,s) was stopped and the computation of f(m,s+1) was started. That is because the computation of submappings is affected by a difference of only 3 out of 255 levels in pixel intensity when λ>0.1 and it is then difficult to obtain a correct result.
[1.4.2] Histogram h(l)
The examination of Cf(m,s) does not depend on the histogram h(l), however, the examination of the BC and its third condition may be affected by h(l). When (λ, Cf(m,s)) is actually plotted, k is usually close to 1. In the experiment, k=1 is used, that is, B0λ2 and B1λ2 are examined. If the true value of k is less than 1, B0λ2 and B1λ2 are not constants and increase gradually by a factor of λ(1−k)/2. If h(l) is a constant, the factor is, for example, λ1/2. However, such a difference can be absorbed by setting the threshold B0 thres appropriately.
Let us model the source image by a circular object, with its center at(x0,y0) and its radius r, given by:




  
    
      
        
          p
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          {
          
            
              
                
                  
                  ⁢
                  
                    
                      255
                      r
                    
                    ⁢
                    
                      c
                      (
                      
                        
                          
                            
                              (
                              
                                i
                                -
                                
                                  x
                                  0
                                
                              
                              )
                            
                            2
                          
                          +
                          
                            
                              (
                              
                                j
                                -
                                
                                  y
                                  0
                                
                              
                              )
                            
                            2
                          
                        
                      
                      ⁢
                      
                        
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ...
                      ⁡
                      
                        (
                        
                          
                            
                              
                                
                                  (
                                  
                                    i
                                    -
                                    
                                      x
                                      0
                                    
                                  
                                  )
                                
                                2
                              
                              +
                              
                                
                                  (
                                  
                                    j
                                    -
                                    
                                      y
                                      0
                                    
                                  
                                  )
                                
                                2
                              
                            
                          
                          ⁢
                          
                            
                          
                          ≤
                          r
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    ...
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        o
                        ⁢
                        
                          
                        
                        ⁢
                        t
                        ⁢
                        
                          
                        
                        ⁢
                        h
                        ⁢
                        
                          
                        
                        ⁢
                        e
                        ⁢
                        
                          
                        
                        ⁢
                        r
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        s
                        ⁢
                        
                          
                        
                        ⁢
                        e
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




and the destination image given by:





  
    
      
        
          q
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          {
          
            
              
                
                  
                  ⁢
                  
                    
                      255
                      r
                    
                    ⁢
                    
                      c
                      (
                      
                        
                          
                            
                              (
                              
                                i
                                -
                                
                                  x
                                  1
                                
                              
                              )
                            
                            2
                          
                          +
                          
                            
                              (
                              
                                j
                                -
                                
                                  y
                                  1
                                
                              
                              )
                            
                            2
                          
                        
                      
                      ⁢
                      
                        
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ...
                      ⁡
                      
                        (
                        
                          
                            
                              
                                
                                  (
                                  
                                    i
                                    -
                                    
                                      x
                                      1
                                    
                                  
                                  )
                                
                                2
                              
                              +
                              
                                
                                  (
                                  
                                    j
                                    -
                                    
                                      y
                                      1
                                    
                                  
                                  )
                                
                                2
                              
                            
                          
                          ⁢
                          
                            
                          
                          ≤
                          r
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    ...
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        o
                        ⁢
                        
                          
                        
                        ⁢
                        t
                        ⁢
                        
                          
                        
                        ⁢
                        h
                        ⁢
                        
                          
                        
                        ⁢
                        e
                        ⁢
                        
                          
                        
                        ⁢
                        r
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        s
                        ⁢
                        
                          
                        
                        ⁢
                        e
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




with its center at(x1,y1) and radius r. In the above, let c(x) have the form of c(x)=xk. When the centers (x0,y0) and (x1,y1) are sufficiently far from each other, the histogram h(l) is then in the form:

h(l)∝rlk(k≠0)  (34)

When k=1, the images represent objects with clear boundaries embedded in the background. These objects become darker toward their centers and brighter toward their boundaries. When k=−1, the images represent objects with vague boundaries. These objects are brightest at their centers, and become darker toward their boundaries. Without much loss of generality, it suffices to state that objects in images are generally between these two types of objects. Thus, choosing k such that −1≦k≦1 can cover most cases and the equation (27) is generally a decreasing function for this range.
As can be observed from the above equation (34), attention must be directed to the fact that r is influenced by the resolution of the image, that is, r is proportional to 2m. This is the reason for the factor 2m being introduced in the above section [1.4.1].
[1.4.3] Dynamic Determination of η
The parameter η can also be automatically determined in a similar manner. Initially, η is set to zero, and the final mapping f(n) and the energy Cf(n) at the finest resolution are computed. Then, after η is increased by a certain value Δη, the final mapping f(n) and the energy Cf(n) at the finest resolution are again computed. This process is repeated until the optimal value of η is obtained. η represents the stiffness of the mapping because it is a weight of the following equation (35):




  
    
      
        
          E
          
            0
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            
            
              
                
                  f
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              -
              
                
                  f
                  
                    (
                    
                      m
                      ,
                      
                        s
                        -
                        1
                      
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
            
          
          2
        
      
    
    
      
        (
        35
        )
      
    
  




If η is zero, Df(n) is determined irrespective of the previous submapping, and the present submapping may be elastically deformed and become too distorted. On the other hand, if η is a very large value, Df(n) is almost completely determined by the immediately previous submapping. The submappings are then very stiff, and the pixels are mapped to almost the same locations. The resulting mapping is therefore the identity mapping. When the value of η increases from 0, Cf(n) gradually decreases as will be described later. However, when the value of η exceeds the optimal value, the energy starts increasing as shown in FIG. 4. In FIG. 4, the x-axis represents η, and y-axis represents Cf.
The optimum value of η which minimizes Cf(n) can be obtained in this manner. However, since various elements affect this computation as compared to the case of λ, Cf(n) changes while slightly fluctuating. This difference is caused because a submapping is re-computed once in the case of λ whenever an input changes slightly, whereas all the submappings must be re-computed in the case of η. Thus, whether the obtained value of Cf(n) is the minimum or not cannot be determined as easily. When candidates for the minimum value are found, the true minimum needs to be searched by setting up further finer intervals.
[1.5] Supersampling
When deciding the correspondence between the pixels, the range of f(m,s) can be expanded to R×R (R being the set of real numbers) in order to increase the degree of freedom. In this case, the intensity of the pixels of the destination image is interpolated, to provide f(m,s) having an intensity at non-integer points:




  
    
      
        V
        ⁡
        
          (
          
            q
            
              
                f
                
                  (
                  
                    m
                    ,
                    s
                  
                  )
                
              
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
          )
        
      
    
    
      
        (
        36
        )
      
    
  




That is, supersampling is performed. In an example implementation, f(m,s) may take integer and half integer values, and





  
    
      
        V
        ⁡
        
          (
          
            q
            
              
                (
                
                  i
                  ,
                  j
                
                )
              
              +
              
                (
                
                  0.5
                  ,
                  0.5
                
                )
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
          )
        
      
    
    
      
        (
        37
        )
      
    
  




is given by





  
    
      
        
          (
          
            
              V
              ⁡
              
                (
                
                  q
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                )
              
            
            +
            
              V
              ⁡
              
                (
                
                  q
                  
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                    +
                    
                      (
                      
                        1
                        ,
                        1
                      
                      )
                    
                  
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                )
              
            
          
          )
        
        /
        2
      
    
    
      
        (
        38
        )
      
    
  




[1.6] Normalization of the Pixel Intensity of Each Image

When the source and destination images contain quite different objects, the raw pixel intensity may not be used to compute the mapping because a large difference in the pixel intensity causes excessively large energy Cf(m,s) and thus making it difficult to obtain an accurate evaluation.
For example, a matching between a human face and a cat's face is computed as shown in FIGS. 20(a) and 20(b). The cat's face is covered with hair and is a mixture of very. bright pixels and very dark pixels. In this case, in order to compute the submappings of the two faces, subimages are normalized. That is, the darkest pixel intensity is set to 0 while the brightest pixel intensity is set to 255, and other pixel intensity values are obtained using linear interpolation.
[1.7] Implementation
In an example implementation, a heuristic method is utilized wherein the computation proceeds linearly as the source image is scanned. First, the value of f(m,s) is determined at the top leftmost pixel (i,j)=(0,0). The value of each f(m,s)(i,j) is then determined while i is increased by one at each step. When i reaches the width of the image, j is increased by one and i is reset to zero. Thereafter, f(m,s)(i,j) is determined while scanning the source image. Once pixel correspondence is determined for all the points, it means that a single mapping f(m,s) is determined.
When a corresponding point qf(i,j) is determined for p(i,j), a corresponding point qf(i,j+1) of p(i,j+1) is determined next. The position of qf(i,j+l) is constrained by the position of qf(i,j) since the position of qf(i,j+1) satisfies the BC. Thus, in this system, a point whose corresponding point is determined earlier is given higher priority. If the situation continues in which (0,0) is always given the highest priority, the final mapping might be unnecessarily biased. In order to avoid this bias, f(m,s) is determined in the following manner in the base technology.
First, when (s mod 4) is 0, f(m,s) is determined starting from (0,0) while gradually increasing both i and j. When (s mod 4) is 1, f(m,s) is determined starting from the top rightmost location while decreasing i and increasing j. When (s mod 4) is 2, f(m,s) is determined starting from the bottom rightmost location while decreasing both i and j. When (s mod 4) is 3, f(m,s) is determined starting from the bottom leftmost location while increasing i and decreasing j. Since a concept such as the submapping, that is, a parameter s, does not exist in the finest n-th level, f(m,s) is computed continuously in two directions on the assumption that s=0 and s=2.
In this implementation, the values of f(m,s)(i,j) (m=0, . . . , n) that satisfy the BC are chosen as much as possible from the candidates (k,l) by imposing a penalty on the candidates violating the BC. The energy D(k,l) of a candidate that violates the third condition of the BC is multiplied by ø and that of a candidate that violates the first or second condition of the BC is multiplied by ψ. In this implementation, ø=2 and ψ=100000 are used.
In order to check the above-mentioned BC, the following test may be performed as the procedure when determining (k,l)=f(m,s)(i,j). Namely, for each grid point (k,l) in the inherited quadrilateral of f(m,s)(i,j), whether or not the z-component of the outer product of

W={right arrow over (A)}×{right arrow over (B)}  (39)

is equal to or greater than 0 is examined, where





  
    
      
        
          A
          →
        
        =
        
          
            
              q
              
                
                  f
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    t
                    ,
                    
                      j
                      -
                      1
                    
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              q
              
                
                  f
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      i
                      +
                      1
                    
                    ,
                    
                      j
                      -
                      1
                    
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          →
        
      
    
    
      
        (
        40
        )
      
    
  
  
    
      
        
          b
          →
        
        =
        
          
            
              q
              
                
                  f
                  
                    (
                    
                      m
                      ,
                      s
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    i
                    ,
                    
                      j
                      -
                      1
                    
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
            ⁢
            
              q
              
                (
                
                  k
                  ,
                  l
                
                )
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          →
        
      
    
    
      
        (
        41
        )
      
    
  




Here, the vectors are regarded as 3D vectors and the z-axis is defined in the orthogonal right-hand coordinate system. When W is negative, the candidate is imposed with a penalty by multiplying





  D
  
    (
    
      k
      .
      l
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




by ψ so that it is not as likely to be selected.

FIGS. 5(a) and 5(b) illustrate the reason why this condition is inspected. FIG. 5(a) shows a candidate without a penalty and FIG. 5(b) shows one with a penalty. When determining the mapping f(m,s)(i,j+1) for the adjacent pixel at (i,j+1), there is no pixel on the source image plane that satisfies the BC if the z-component of W is negative because then




  q
  
    (
    
      k
      ,
      l
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




passes the boundary of the adjacent quadrilateral.

[1.7.1] The Order of Submappings

In this implementation, σ(0)=0, σ(1)=1, σ(2)=2, σ(3)=3, σ(4)=0 are used when the resolution level is even, while σ(0)=3, σ(1)=2, σ(2)=1, σ(3)=0, σ(4)=3 are used when the resolution level is odd. Thus, the submappings are shuffled to some extent. It is to be noted that the submappings are primarily of four types, and s may be any of 0 to 3. However, a processing with s=4 is used in this implementation for a reason to be described later.
[1.8] Interpolations
After the mapping between the source and destination images is determined, the intensity values of the corresponding pixels are interpolated. In the implementation, trilinear interpolation is used. Suppose that a square p(i,j)p(i+1,j)p(i+1,j+1)p(i,j+1) on the source image plane is mapped to a quadrilateral qf(i,j)qf(i+1,j)qf(i+1,j+1)qf(i,j+1) on the destination image plane. For simplicity, the distance between the image planes is assumed to be 1. The intermediate image pixels r(x,y,t) (0≦x≦N−1, 0≦y≦M−1) whose distance from the source image plane is t (0≦t≦1) are obtained as follows. First, the location of the pixel r(x,y,t), where x,y,t∈R, is determined by equation (42):




  
    
      
        
          (
          
            x
            ,
            y
          
          )
        
        =
        
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                t
              
              )
            
            ⁢
            
              (
              
                i
                ,
                j
              
              )
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
              
              )
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              f
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            
              x
              ⁡
              
                (
                
                  1
                  -
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    y
                  
                
                )
              
            
            ⁢
            
              (
              
                1
                -
                t
              
              )
            
            ⁢
            
              (
              
                
                  i
                  +
                  1
                
                ,
                j
              
              )
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            
              x
              ⁡
              
                (
                
                  1
                  -
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    y
                  
                
                )
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              f
              ⁡
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  j
                
                )
              
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            
              y
              ⁡
              
                (
                
                  1
                  -
                  t
                
                )
              
            
            ⁢
            
              (
              
                i
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            y
            ⁢
            
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              f
              ⁡
              
                (
                
                  i
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            
              y
              ⁡
              
                (
                
                  1
                  -
                  t
                
                )
              
            
            ⁢
            
              (
              
                
                  i
                  +
                  1
                
                ,
                
                  j
                  +
                  1
                
              
              )
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            y
            ⁢
            
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              f
              ⁡
              
                (
                
                  
                    i
                    +
                    1
                  
                  ,
                  
                    j
                    +
                    1
                  
                
                )
              
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




The value of the pixel intensity at r(x,y,t) is then determined by equation (43):





  
    
      
        
          V
          ⁡
          
            (
            
              r
              ⁡
              
                (
                
                  x
                  ,
                  y
                  ,
                  t
                
                )
              
            
            )
          
        
        =
        
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                t
              
              )
            
            ⁢
            
              V
              ⁡
              
                (
                
                  p
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
                )
              
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
              
              )
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  q
                  
                    f
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            
              x
              ⁡
              
                (
                
                  1
                  -
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    y
                  
                
                )
              
            
            ⁢
            
              (
              
                1
                -
                t
              
              )
            
            ⁢
            
              V
              ⁡
              
                (
                
                  p
                  
                    (
                    
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                    )
                  
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            
              x
              ⁡
              
                (
                
                  1
                  -
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    y
                  
                
                )
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  q
                  
                    f
                    ⁡
                    
                      (
                      
                        
                          i
                          +
                          1
                        
                        ,
                        j
                      
                      )
                    
                  
                
                )
              
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            
              y
              ⁡
              
                (
                
                  1
                  -
                  t
                
                )
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  p
                  
                    (
                    
                      i
                      ,
                      
                        j
                        +
                        1
                      
                    
                    )
                  
                
                )
              
            
          
          +
          
            
              (
              
                1
                -
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              )
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            y
            ⁢
            
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  q
                  
                    f
                    ⁡
                    
                      (
                      
                        i
                        ,
                        
                          j
                          +
                          1
                        
                      
                      )
                    
                  
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            
              y
              ⁡
              
                (
                
                  1
                  -
                  t
                
                )
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  p
                  
                    (
                    
                      
                        i
                        +
                        1
                      
                      ,
                      
                        j
                        +
                        1
                      
                    
                    )
                  
                
                )
              
            
          
          +
          
            d
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            d
            ⁢
            
              
            
            ⁢
            y
            ⁢
            
              
            
            ⁢
            t
            ⁢
            
              
            
            ⁢
            
              V
              ⁡
              
                (
                
                  q
                  
                    f
                    ⁡
                    
                      (
                      
                        
                          i
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          1
                        
                      
                      )
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




where dx and dy are parameters varying from 0 to 1.

[1.9] Mapping to which Constraints are Imposed

So far, the determination of a mapping in which no constraints are imposed has been described. However, if a correspondence between particular pixels of the source and destination images is provided in a predetermined manner, the mapping can be determined using such correspondence as a constraint.
The basic idea is that the source image is roughly deformed by an approximate mapping which maps the specified pixels of the source image to the specified pixels of the destination image and thereafter a mapping f is accurately computed.
First, the specified pixels of the source image are mapped to the specified pixels of the destination image, then the approximate mapping that maps other pixels of the source image to appropriate locations are determined. In other words, the mapping is such that pixels in the vicinity of a specified pixel are mapped to locations near the position to which the specified one is mapped. Here, the approximate mapping at the m-th level in the resolution hierarchy is denoted by F(m).
The approximate mapping F is determined in the following manner. First, the mappings for several pixels are specified. When ns pixels

p(i0,j0),p(i1,j1), . . . , p(ins−1,jns−1)  (44)

of the source image are specified, the following values in the equation (45) are determined.





  
    
      
        
          
            
              
                
                  
                    F
                    
                      (
                      n
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        i
                        0
                      
                      ,
                      
                        j
                        0
                      
                    
                    )
                  
                
                =
                
                  (
                  
                    
                      k
                      0
                    
                    ,
                    
                      l
                      0
                    
                  
                  )
                
              
              ,
            
          
        
        
          
            
              
                
                  
                    F
                    
                      (
                      n
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        i
                        1
                      
                      ,
                      
                        j
                        1
                      
                    
                    )
                  
                
                =
                
                  (
                  
                    
                      k
                      1
                    
                    ,
                    
                      l
                      1
                    
                  
                  )
                
              
              ,
              ...
              ⁢
              
                
              
              ,
            
          
        
        
          
            
              
                
                  F
                  
                    (
                    n
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      i
                      
                        
                          n
                          s
                        
                        -
                        1
                      
                    
                    ,
                    
                      j
                      
                        
                          n
                          s
                        
                        -
                        1
                      
                    
                  
                  )
                
              
              =
              
                (
                
                  
                    k
                    
                      
                        n
                        s
                      
                      -
                      1
                    
                  
                  ,
                  
                    l
                    
                      
                        n
                        s
                      
                      -
                      1
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




For the remaining pixels of the source image, the amount of displacement is the weighted average of the displacement of p(ih,jh) (h=0, . . . , ns−1). Namely, a pixel p(i,j) is mapped to the following pixel (expressed by the equation (46)) of the destination image.




  
    
      
        
          
            F
            
              (
              m
              )
            
          
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          
            
              (
              
                i
                ,
                j
              
              )
            
            +
            
              
                ∑
                
                  h
                  =
                  0
                
                
                  h
                  =
                  
                    
                      n
                      s
                    
                    -
                    1
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      
                        k
                        h
                      
                      -
                      
                        i
                        h
                      
                    
                    ,
                    
                      
                        l
                        h
                      
                      -
                      
                        j
                        h
                      
                    
                  
                  )
                
                ⁢
                
                  
                    weight
                    h
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
              
            
          
          
            2
            
              n
              -
              m
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




where





  
    
      
        
          
            weight
            h
          
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          
            1
            /
            
              
                
                
                  (
                  
                    
                      
                        i
                        h
                      
                      -
                      i
                    
                    ,
                    
                      
                        j
                        h
                      
                      -
                      j
                    
                  
                  )
                
                
              
              2
            
          
          
            total_weight
            ⁢
            
              (
              
                i
                ,
                j
              
              )
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




where





  
    
      
        
          total_weight
          ⁢
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          
            ∑
            
              h
              =
              0
            
            
              h
              =
              
                
                  n
                  s
                
                -
                1
              
            
          
          ⁢
          
            
          
          ⁢
          
            1
            /
            
              
                
                
                  (
                  
                    
                      
                        i
                        h
                      
                      -
                      i
                    
                    ,
                    
                      
                        j
                        h
                      
                      -
                      j
                    
                  
                  )
                
                
              
              2
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




Second, the energy




  D
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




of the candidate mapping f is changed so that a mapping f similar to F(m) has a lower energy. Precisely speaking,





  D
  
    (
    
      i
      ,
      j
    
    )
  
  
    (
    
      m
      ,
      s
    
    )
  




is expressed by the equation (49):





  
    
      
        
          D
          
            (
            
              i
              ,
              j
            
            )
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          
            E
            
              0
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
          +
          
            η
            ⁢
            
              
            
            ⁢
            
              E
              
                l
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
          +
          
            κ
            ⁢
            
              
            
            ⁢
            
              E
              
                2
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              
                (
                
                  m
                  ,
                  s
                
                )
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




where





  
    
      
        
          E
          
            2
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
        =
        
          {
          
            
              
                
                  0
                  ,
                  
                    
                      if
                      ⁢
                      
                        
                          
                          
                            
                              
                                F
                                
                                  (
                                  m
                                  )
                                
                              
                              ⁡
                              
                                (
                                
                                  i
                                  ,
                                  j
                                
                                )
                              
                            
                            -
                            
                              
                                f
                                
                                  (
                                  
                                    m
                                    ,
                                    s
                                  
                                  )
                                
                              
                              ⁡
                              
                                (
                                
                                  i
                                  ,
                                  j
                                
                                )
                              
                            
                          
                          
                        
                        2
                      
                    
                    ≤
                    
                      ⌊
                      
                        
                          ρ
                          2
                        
                        
                          2
                          
                            2
                            ⁢
                            
                              (
                              
                                n
                                -
                                m
                              
                              )
                            
                          
                        
                      
                      ⌋
                    
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    
                      
                        
                        
                          
                            
                              F
                              
                                (
                                m
                                )
                              
                            
                            ⁡
                            
                              (
                              
                                i
                                ,
                                j
                              
                              )
                            
                          
                          -
                          
                            
                              f
                              
                                (
                                
                                  m
                                  ,
                                  s
                                
                                )
                              
                            
                            ⁡
                            
                              (
                              
                                i
                                ,
                                j
                              
                              )
                            
                          
                        
                        
                      
                      2
                    
                    ,
                    otherwise
                  
                
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




where κ, ρ≧0. Finally, the resulting mapping f is determined by the above-described automatic computing process.

Note that




  E
  
    2
    
      (
      
        i
        ,
        j
      
      )
    
  
  
    (
    
      m
      ,
      s
    
    )
  




becomes 0 if f(m,s) (i,j) is sufficiently close to F(m)(i,j) i.e., the distance therebetween is equal to or less than





  
    
      
        ⌊
        
          
            ρ
            2
          
          
            2
            
              2
              ⁢
              
                (
                
                  n
                  -
                  m
                
                )
              
            
          
        
        ⌋
      
    
    
      
        (
        51
        )
      
    
  




This has been defined in this way because it is desirable to determine each value f(m,s)(i,j) automatically to fit in an appropriate place in the destination image as long as each value f(m,s)(i,j) is close to F(m) (i,j). For this reason, there is no need to specify the precise correspondence in detail to have the source image automatically mapped so that the source image matches the destination image.

[2] Concrete Processing Procedure

The flow of a process utilizing the respective elemental techniques described in [1] will now be described.
FIG. 6 is a flowchart of the overall procedure of the base technology. Referring to FIG. 6, a source image and destination image are first processed using a multiresolutional critical point filter (S1). The source image and the destination image are then matched (S2). As will be understood, the matching (S2) is not required in every case, and other processing such as image recognition may be performed instead, based on the characteristics of the source image obtained at S1.
FIG. 7 is a flowchart showing details of the process S1 shown in FIG. 6. This process is performed on the assumption that a source image and a destination image are matched at S2. Thus, a source image is first hierarchized using a critical point filter (S10) so as to obtain a series of source hierarchical images. Then, a destination image is hierarchized in the similar manner (S11) so as to obtain a series of destination hierarchical images. The order of S10 and S11 in the flow is arbitrary, and the source image and the destination image can be generated in parallel. It may also be possible to process a number of source and destination images as required by subsequent processes.
FIG. 8 is a flowchart showing details of the process at S10 shown in FIG. 7. Suppose that the size of the original source image is 2n×2n. Since source hierarchical images are sequentially generated from an image with a finer resolution to one with a coarser resolution, the parameter m which indicates the level of resolution to be processed is set to n (S100). Then, critical points are detected from the images p(m,0), p(m,1), p(m,2) and p(m,3) of the m-th level of resolution, using a critical point filter (S101), so that the images p(m−1,0), p(m−1,1), p(m−1,2) and p(m−1,3) of the (m−1)th level are generated (S102). Since m=n here, p(m,0) =p(m,1) =p(m,2) =p(m,3) =p(n) holds and four types of subimages are thus generated from a single source image.
FIG. 9 shows correspondence between partial images of the m-th and those of (m−1)th levels of resolution. Referring to FIG. 9, respective numberic values shown in the figure represent the intensity of respective pixels. p(m,s) symbolizes any one of four images p(m,0) through p(m,3), and when generating p(m−1,0), p(m,0) is used from p(m,s). For example, as for the block shown in FIG. 9, comprising four pixels with their pixel intensity values indicated inside, images p(m−1,0), p(m−1,1), p(m−1,2) and p(m−1,3) acquire “3”, “8”, “6” and “10”, respectively, according to the rules described in [1.2]. This block at the m-th level is replaced at the (m−1)th level by respective single pixels thus acquired. Therefore, the size of the subimages at the (m−1)th level is 2m−1×2m−1.
After m is decremented (S103 in FIG. 8), it is ensured that m is not negative (S104). Thereafter, the process returns to S101, so that subimages of the next level of resolution, i.e., a next coarser level, are generated. The above process is repeated until subimages at m=0 (0-th level) are generated to complete the process at S10. The size of the subimages at the 0-th level is 1×1.
FIG. 10 shows source hierarchical images generated at S10 in the case of n=3. The initial source image is the only image common to the four series followed. The four types of subimages are generated independently, depending on the type of critical point. Note that the process in FIG. 8 is common to S11 shown in FIG. 7, and that destination hierarchical images are generated through a similar procedure. Then, the process at S1 in FIG. 6 is completed.
In this base technology, in order to proceed to S2 shown in FIG. 6 a matching evaluation is prepared. FIG. 11 shows the preparation procedure. Referring to FIG. 11, a plurality of evaluation equations are set (S30). The evaluation equations may include the energy Cf(m,s) concerning a pixel value, introduced in [1.3.2.1], and the energy Df(m,s) concerning the smoothness of the mapping introduced in [1.3.2.2]. Next, by combining these evaluation equations, a combined evaluation equation is set (S31). Such a combined evaluation equation may be




  
    λ
    ⁢
    
      
    
    ⁢
    
      C
      
        (
        
          i
          ,
          j
        
        )
      
      
        (
        
          m
          ,
          s
        
        )
      
    
  
  +
  
    
      D
      f
      
        (
        
          m
          ,
          s
        
        )
      
    
    .
  




Using η introduced in [1.3.2.2], we have





  ∑
  
    ∑
    
      (
      
        
          λ
          ⁢
          
            
          
          ⁢
          
            C
            
              (
              
                i
                ,
                j
              
              )
            
            
              (
              
                m
                ,
                s
              
              )
            
          
        
        +
        
          η
          ⁢
          
            
          
          ⁢
          
            E
            
              0
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
            
              (
              
                m
                ,
                s
              
              )
            
          
        
        +
        
          E
          
            1
            
              (
              
                i
                ,
                j
              
              )
            
          
          
            (
            
              m
              ,
              s
            
            )
          
        
      
      )
    
  




In the equation (52) the sum is taken for each i and j where i and j run through 0, 1, . . . , 2m−1. Now, the preparation for matching evaluation is completed.

FIG. 12 is a flowchart showing the details of the process of S2 shown in FIG. 6. As described in [1], the source hierarchical images and destination hierarchical images are matched between images having the same level of resolution. In order to detect global correspondence correctly, a matching is calculated in sequence from a coarse level to a fine level of resolution. Since the source and destination hierarchical images are generated using the critical point filter, the location and intensity of critical points are stored clearly even at a coarse level. Thus, the result of the global matching is superior to conventional methods.
Referring to FIG. 12, a coefficient parameter η and a level parameter m are set to 0 (S20). Then, a matching is computed between the four subimages at the m-th level of the source hierarchical images and those of the destination hierarchical images at the m-th level, so that four types of submappings f(m,s) (s=0, 1, 2, 3) which satisfy the BC and minimize the energy are obtained (S21). The BC is checked by using the inherited quadrilateral described in [1.3.3]. In that case, the submappings at the m-th level are constrained by those at the (m−1)th level, as indicated by the equations (17) and (18). Thus, the matching computed at a coarser level of resolution is used in subsequent calculation of a matching. This is called a vertical reference between different levels. If m=0, there is no coarser level and this exceptional case will be described using FIG. 13.
A horizontal reference within the same level is also performed. As indicated by the equation (20) in [1.3.3], f(m,3), f(m,2) and f(m,1) are respectively determined so as to be analogous to f(m,2), f(m,1) and f(m,0) This is because a situation in which the submappings are totally different seems unnatural even though the type of critical points differs so long as the critical points are originally included in the same source and destination images. As can been seen from the equation (20), the closer the submappings are to each other, the smaller the energy becomes, so that the matching is then considered more satisfactory.
As for f(m,0), which is to be initially determined, a coarser level by one may be referred to since there is no other submapping at the same level to be referred to as shown in the equation (19). In this base technology, however, a procedure is adopted such that after the submappings were obtained up to f(m,3), f(m,0) is recalculated once utilizing the thus obtained subamppings as a constraint. This procedure is equivalent to a process in which s=4 is substituted into the equation (20) and f(m,4) is set to f(m,0) anew. The above process is employed to avoid the tendency in which the degree of association between f(m,0) and f(m,3) becomes too low. This scheme actually produced a preferable result. In addition to this scheme, the submappings are shuffled in the experiment as described in [1.7.1], so as to closely maintain the degrees of association among submappings which are originally determined independently for each type of critical point. Furthermore, in order to prevent the tendency of being dependent on the starting point in the process, the location thereof is changed according to the value of s as described in [1.7].
FIG. 13 illustrates how the submapping is determined at the 0-th level. Since at the 0-th level each sub-image is consitituted by a single pixel, the four submappings f(0,s) are automatically chosen as the identity mapping. FIG. 14 shows how the submappings are determined at the first level. At the first level, each of the sub-images is constituted of four pixels, which are indicated by solid lines. When a corresponding point (pixel) of the point (pixel) x in p(1,s) is searched within q(1,s), the following procedure is adopted:

    1. An upper left point a, an upper right point b, a lower left point c and a lower right point d with respect to the point x are obtained at the first level of resolution.
    2. Pixels to which the points a to d belong at a coarser level by one, i.e., the 0-th level, are searched. In FIG. 14, the points a to d belong to the pixels A to D, respectively. However, the pixels A to C are virtual pixels which do not exist in reality.
    3. The corresponding points A′ to D′ of the pixels A to D, which have already been defined at the 0-th level, are plotted in q(1,s). The pixels A′ to C′ are virtual pixels and regarded to be located at the same positions as the pixels A to C.
    4. The corresponding point a′ to the point a in the pixel A is regarded as being located inside the pixel A′, and the point a′ is plotted. Then, it is assumed that the position occupied by the point a in the pixel A (in this case, positioned at the lower right) is the same as the position occupied by the point a′ in the pixel A′.
    5. The corresponding points b′ to d′ are plotted by using the same method as the above 4 so as to produce an inherited quadrilateral defined by the points a′ to d′.
    6. The corresponding point x′ of the point x is searched such that the energy becomes minimum in the inherited quadrilateral. Candidate corresponding points x′ may be limited to the pixels, for instance, whose centers are included in the inherited quadrilateral. In the case shown in FIG. 14, the four pixels all become candidates.


The above described is a procedure for determining the corresponding point of a given point x. The same processing is performed on all other points so as to determine the submappings. As the inherited quadrilateral is expected to become deformed at the upper levels (higher than the second level), the pixels A′ to D′ will be positioned apart from one another as shown in FIG. 3.
Once the four submappings at the m-th level are determined in this manner, m is incremented (S22 in FIG. 12). Then, when it is confirmed that m does not exceed n (S23), return to S21. Thereafter, every time the process returns to S21, submappings at a finer level of resolution are obtained until the process finally returns to S21 at which time the mapping f(n) at the n-th level is determined. This mapping is denoted as f(n) (η=0) because it has been determined relative to η=0.
Next, to obtain the mapping with respect to other different η, η is shifted by Δη and m is reset to zero (S24). After confirming that new η does not exceed a predetermined search-stop value ηmax (S25), the process returns to S21 and the mapping f(n) (η=Δη) relative to the new η is obtained. This process is repeated while obtaining f(n) (η=i Δη) (i=0,1, . . . ) at S21. When η exceeds ηmax, the process proceeds to S26 and the optimal η=ηopt is determined using a method described later, so as to let f(n) (η=ηopt) be the final mapping f(n).
FIG. 15 is a flowchart showing the details of the process of S21 shown in FIG. 12. According to this flowchart, the submappings at the m-th level are determined for a certain predetermined η. In this base technology, when determining the mappings, the optimal λ is defined independently for each submapping.
Referring to FIG. 15, s and λ are first reset to zero (S210). Then, obtained is the submapping f(m,s) that minimizes the energy with respect to the then λ (and, implicitly, η) (S211), and the thus obtained submapping is denoted as f(m,s) (λ=0). In order to obtain the mapping with respect to other different λ, λ is shifted by Δλ. After confirming that the new λ does not exceed a predetermined search-stop value λmax (S213), the process returns to S211 and the mapping f(m s) (λ=Δλ) relative to the new λ is obtained. This process is repeated while obtaining f(m,s) (λ=iΔ80 )(i=0,1, . . . ). When λ exceeds λmax, the process proceeds to S214 and the optimal λ=λopt is determined , so as to let f(n) (λ=λopt) be the final mapping f(m,s) (S214).
Next, in order to obtain other submappings at the same level, λ is reset to zero and s is incremented (S215). After confirming that s does not exceed 4 (S216), return to S211. When s=4, f(m,0) is renewed utilizing f(m,3) as described above and a submapping at that level is determined.
FIG. 16 shows the behavior of the energy Cf(m,s) corresponding to f(m,s) (λ=iΔλ) (i=0,1, . . . ) for a certain m and s while varying λ. As described in [1.4], as λ increases, Cf(m,s) normally decreases but changes to increase after λ exceeds the optimal value. In this base technology, λ in which Cf(m,s) becomes the minima is defined as λopt. As observed in FIG. 16, even if Cf(m,s) begins to decrease again in the range λ>λopt, the mapping will not be as good. For this reason, it suffices to pay attention to the first occurring minima value. In this base technology, λopt is independently determined for each submapping including f(n).
FIG. 17 shows the behavior of the energy Cf(n) corresponding to f(n) (η=iΔη) (i=0,1, . . . ) while varying η. Here too, Cf(n) normally decreases as η increases, but Cf(n) changes to increase after η exceeds the optimal value. Thus, η in which Cf(n) becomes the minima is defined as ηopt. FIG. 17 can be considered as an enlarged graph around zero along the horizontal axis shown in FIG. 4. Once ηopt is determined, f(n) can be finally determined.
As described above, this base technology provides various merits. First, since there is no need to detect edges, problems in connection with the conventional techniques of the edge detection type are solved. Furthermore, prior knowledge about objects included in an image is not necessitated, thus automatic detection of corresponding points is achieved. Using the critical point filter, it is possible to preserve intensity and locations of critical points even at a coarse level of resolution, thus being extremely advantageous when applied to object recognition, characteristic extraction, and image matching. As a result, it is possible to construct an image processing system which significantly reduces manual labor.
Some further extensions to or modifications of the above-described base technology may be made as follows:

    (1) Parameters are automatically determined when the matching is computed between the source and destination hierarchical images in the base technology. This method can be applied not only to the calculation of the matching between the hierarchical images but also to computing the matching between two images in general.


For instance, an energy E0 relative to a difference in the intensity of pixels and an energy E1 relative to a positional displacement of pixels between two images may be used as evaluation equations, and a linear sum of these equations, i.e., Etot=αE0+E1, may be used as a combined evaluation equation. While paying attention to the neighborhood of the extrema in this combined evaluation equation, α is automatically determined. Namely, mappings which minimize Et,t are obtained for various α's. Among such mappings, α at which Etot takes the minimum value is defined as an optimal parameter. The mapping corresponding to this parameter is finally regarded as the optimal mapping between the two images.
Many other methods are available in the course of setting up evaluation equations. For instance, a term which becomes larger as the evaluation result becomes more favorable, such as 1/E1 and 1/E2, may be employed. A combined evaluation equation is not necessarily a linear sum, but an n-powered sum (n=2, 1⁄2, −1, −2, etc.), a polynomial or an arbitrary function may be employed when appropriate.
The system may employ a single parameter such as the above α, two parameters such as η and λ as in the base technology, or more than two parameters. When there are more than three parameters used, they may be determined while changing one at a time.

    (2) In the base technology, a parameter is determined in a two-step process. That is, in such a manner that a point at which Cf(m,s) takes the minima is detected after a mapping such that the value of the combined evaluation equation becomes minimum is determined. However, instead of this two-step processing, a parameter may be effectively determined, as the case may be, in a manner such that the minimum value of a combined evaluation equation becomes minimum. In this case, αE0+βE1, for example, may be used as the combined evaluation equation, where α+β=1 may be imposed as a constraint so as to equally treat each evaluation equation. The automatic determination of a parameter is effective when determining the parameter such that the energy becomes minimum.
    (3) In the base technology, four types of submappings related to four types of critical points are generated at each level of resolution. However, one, two, or three types among the four types may be selectively used. For instance, if there exists only one bright point in an image, generation of hierarchical images based solely on f(m,3) related to a maxima point can be effective to a certain degree. In this case, no other submapping is necessary at the same level, thus the amount of computation relative on s is effectively reduced.
    (4) In the base technology, as the level of resolution of an image advances by one through a critical point filter, the number of pixels becomes 1⁄4. However, it is possible to suppose that one block consists of 3×3 pixels and critical points are searched in this 3×3 block, then the number of pixels will be 1/9 as the level advances by one.
    (5) In the base technology, if the source and the destination images are color images, they would generally first be converted to monochrome images, and the mappings then computed. The source color images may then be transformed by using the mappings thus obtained. However, as an alternate method, the submappings may be computed regarding each RGB component.

Preferred Embodiments for Image Coding and Decoding



In the above-described base technology, the correspondence data are generated by computing a matching between key frames and, based on this correspondence information, an intermediate frame is generated. As such, this technology can be used for the compression of motion or moving pictures; in fact, experiments are beginning to show evidence of both picture quality and compression rate (also termed ‘compressibility’, hereinafter) that are superior to those of MPEG. An image coding and decoding technology utilizing the base technology will now be described with reference to FIGS. 18 to 25. This image coding and decoding technology, which takes scene changes of moving pictures into consideration, will prove to be an important elemental technique for the base technology and such other technologies which can reproduce excellent images even when a certain considerable length of time interval exists between key frames.
(Coding Side)
FIG. 18 shows a structure of an image coding apparatus 10 according to an embodiment. The image coding apparatus 10 includes an image input unit 12 which receives data on key frames KFi, a matching processor 14 which generates correspondence data files Ci,j by computing a matching between key frames KFi and KFj and a stream generator 16 which generates a coded data stream CBS (Coded Bit Stream) (sometimes simply referred to as a data stream hereinafter), by incorporating the key frame data and the correspondence data file or files. The data stream may be stored in a storage unit 18 as needed. A scene change detector 20 detects a scene change, if any, between key frames, and notifies the matching processor 14 and the stream generator 16 of the scene change.
While there are a variety of methods for detecting scene changes that may be applied to the embodiments of the invention, the method considered herein as an example simply determines the presence of a scene change when there is a substantially large difference between two key frames. The difference is determined, for example, by comparing a calculated value, such as the absolute value or the total sum of squares of the difference between pixel values of corresponding points in two key frames, with a predetermined threshold value.
Key frames are defined, for instance, by extracting frames of motion pictures at predetermined intervals, such as of 0.5 seconds, and intermediate frames, which are the frames between key frames, are generated by an interpolation computation based on the correspondence data file at a decoding stage. It is to be appreciated here that the image input unit 12 may either receive already existing key frames from an external storage device (not shown), network (not shown), or the like or may be an image photographing device which captures images, such as a digital camera or the like.
The matching processor 14 performs a pixel-by-pixel matching between two key frames, based on critical points or otherwise by applying the base technology or some other arbitrary technology. Unlike the case with the base technology, upon receipt of notification of the detection of a scene change from the scene change detector 20, generation of correspondence data is skipped for a “special pair of key frames”, which is a pair made up of key frames before and after the scene change. This is because the generation of an interpolated image from the frames before and after a scene change sometimes produces a generally meaningless morphing image although there may be some aesthetic effect.
FIG. 19 shows a flow of key frames which have a scene change therein. Here, defining nonnegative integers represented by i, j, a key frame is denoted by KFi, and a correspondence data file obtained between KFi and KFj is denoted by Ci,j. In the example shown in FIG. 19, there is a scene change between KF2 and KF3, and the generation of C2,3 is canceled. Moreover, it is shown that the scene change takes place at a position that divides the time interval between KF2 and KF3 at a ratio of s:(1−s)
FIG. 20 shows a structure of a data stream CBS generated by the stream generator 16 taking the condition in FIG. 19 into consideration. This stream is, to be concrete, KF0, KF1, C0,1, KF2, C1,2, KF3, D2,3, KF4, C3,4, . . . A “correspondence disable file” “D2,3” is a file for use with a scene change that is inserted in place of the normal correspondence data file C2,3.
It is noted that, in a data stream, for more efficient use of memory during decoding, it is advantageous if the key frame data and the correspondence data file, when they are related to each other, are closer to each other. This is because after either of the key frame data and the correspondence data file is obtained, the data must be held in a buffer memory or the like while the other is waited for. However, it will be understood that the arrangement of these data within the data stream has a certain degree of freedom.
FIG. 21 and FIG. 22 show formats of the correspondence data file Ci,j and the correspondence disable file Di,j, respectively. First, a leading bit, which in this example is assigned as “0” and “1” respectively, is structured such that the correspondence data file or the correspondence disable file can be identified. Of course, this leading part of a file generally corresponds to a header portion that contains a certain amount of information, and this part may be made in a manner that it is possible to identify the file type by another method, such as the difference in data within the header or the like. Next, in the case of the correspondence data file Ci,j, the correspondence data between key frames are described, for instance, for each pixel. Then added thereto is the time data for reproduction or display of KFi and KFj. This time data is not necessary when key frames are selected at fixed intervals. The time data is, however, generally required when the intervals vary. This time data may serve its purpose if it indicates time elapsed from the starting frame KF0 or the previous key frame (the key frame immediately before) and so forth. It is to be noted that the time data may alternatively be arranged at the head of a data stream CBS or be positioned together with or apart from the key frame data.
In the correspondence disable file Di,j, there is a description of a parameter “s” which indicates a relative position of the above-described scene change. Alternatively, this parameter can be omitted, and in such a case, on decoding, the parameter may be interpreted as s=0, 1⁄2, or 1, and so forth. Further, where this parameter is omitted, the correspondence disable file itself may be omitted, so that data on the subsequent key frame (that is, the one that comes right after), such as KFj+1, may be placed closer in the data stream, and at the same time the header information of the key frame may indicate that it is the data on a key frame, thus showing implicitly the omission of the correspondence disable file. In this case, too, the correspondence disable instruction can be clearly understood, thus in this description, this case is interpreted that the “correspondence disable data” is inserted.
(Decoding Side)
FIG. 23 shows a structure of an image decoding apparatus 40 according to an embodiment of the invention. The image decoding apparatus 40 includes a stream input unit 42 which receives a coded data stream CBS, an intermediate image generator 44 which generates intermediate frames from the data stream CBS by interpolation based on, for example, the base technology, and a display controller 50 which performs a processing to allow display of the key frames and intermediate frames as moving pictures. Display data generated by the display controller 50 are output to a display device where the images are reproduced. The intermediate image generator 44 is also provided with a scene change detector 46.
FIG. 24 shows the principle on which intermediate frames are generated by the intermediate image generator 44, based on the processing of the base technology. Namely, when there are two key frames KF5 and KF6 and there is a correspondence data file C5,6 therefor, the correspondence data file C5,6 indicates that a point p0 (x0, y0) on a key frame KF5 corresponds to a point p1 (x1, y1) on the other key frame KF6. In particular, a point may be specified by a coordinate (xi, yi) and a pixel value pi. Thus, a point pt (xt, yt) on an intermediate frame IF5,6(t) may be obtained by interpolating the coordinate and the pixel value on a time axis representing a time interval between the key frames KF5 and KF6.
Referring back to FIG. 23, the scene change detector 46 detects correspondence disable files within the data stream CBS. When a correspondence disable file is detected, the intermediate image generator 44 operates differently from when a correspondence data file is received, and may skip or stop generation of intermediate frames. In the case of the data stream shown in FIG. 20, a correspondence disable file D2,3 is detected between KF2 and KF3. Thus, a scene change is to be provided between these key frames. Namely, when D2,3 is as shown in FIG. 22, the intermediate image generator 44 divides the interval between KF2 and KF3 at a ratio of s:(1−s) and outputs KF2 continuously in the first half and KF3 continuously in the second half. This accomplishes a reasonable reproduction of a scene change, thus making it possible to avoid an incorrect or meaningless morphing image.
The present invention has been described based on the preferred embodiments. It is to be noted that the present invention is not limited to these embodiments, and various modifications thereto are also effective as the present invention.
As one such example, a modification of the image coding apparatus 10 is now described. The structure of this image coding apparatus 10 is essentially the same as shown in FIG. 18. However, in this modification, when a scene change is detected, the scene change detector 20 sends a request to the image input unit 12 to acquire a new key frame or frames. The image input unit 12 communicates the request to an external image data supply source (not shown) and obtains the requested key frame. Alternatively, the image input unit 12 may have buffered a predetermined number of frames within an internal storage (not shown) or the like.
FIG. 25 illustrates a processing for this modification. Here, the special pair of key frames, where the scene change is detected, are called a “first KF” and a “second KF”. If the image input unit 12 acquires key frames at fixed intervals of, for example, 0.5 seconds as a default setting, then this restriction also applies to the interval between the special pair of key frames. Now, in the method described above, in this interval between the special pair of key frames, there is no generation of intermediate frames by interpolation and instead images before and after the scene change will be displayed as still pictures of the first KF and second KF, respectively.
However, in this modification, in order to reduce or eliminate the duration of display using only still pictures, a third KF and a fourth KF, which are located before and after the scene change at a narrower interval than that of the first KF and the second KF, are acquired based on the request sent by the scene change detector 46. It will be understood that, depending on the application, this procedure may only be useful or effective when the interval between the first KF and the second KF is greater than a predetermined value.
After the new frames third KF and fourth KF are received, the matching processor 14 computes respective correspondence data between the first KF and the third KF and between the second KF and the fourth KF, so that the stream generator 16 forms a data stream by incorporating results of the correspondence data computed. In this case, however, the stream generator 16, when generating time data, generates the time data taking the presence of the third KF and the fourth KF into consideration.
Moreover, as another modification, the image coding apparatus 10 may be configured such that key frames are selected from a plurality of frames or complete motion pictures stream that is acquired. In this modification, for example, the scene change detector 20 may detect scene changes in a preprocess and then select pairs of key frames as close to before and after the scene changes as possible.
Although the present invention has been described by way of exemplary embodiments, it should be understood that many changes and substitutions may be made by those skilled in the art without departing from the scope of the present invention which is defined by the appended claims.

## Claims

1. An image coding apparatus, comprising:
an image input unit which receives data of key frames;
a scene change detector which detects a first key frame and a second key frame which bracket a scene change; and
a correspondence data generator which generates correspondence data between key frames,
wherein, when the first key frame and the second key frame are detected, a third key frame and a fourth key frame, disposed before and after the scene change, respectively, and having an interval therebetween that is narrower than that between the first key frame and the second key frame are input to said correspondence data generator, so that correspondence data between the first key frame and the third key frame as well as between the second key frame and the fourth key frame are generated.

2. An image coding apparatus according to claim 1, wherein said correspondence data generator extracts critical points in respective key frames, and performs a pixel-based matching computation based on a correspondence relation between the critical points.

3. An image coding apparatus according to claim 2, wherein said correspondence data generator further multiresolutionalizes respective key frames by extracting critical points thereof, and specifies a correspondence relation between the critical points by performing the pixel-based matching computation in sequence starting from a coarse level of multiresolution.

4. An image coding apparatus according to claim 1, wherein said correspondence data generator computes a matching between key frames using a multiresolutional critical point filter.

5. An image coding apparatus according to claim 1, wherein said scene change detector sends a request to the input unit to acquire the third key frame and the fourth key frame from within the image coding apparatus or from an external source outside the image coding apparatus.

6. An image coding apparatus according to claim 1, wherein the third key frame and the fourth key frame are input only when the interval between the first key frame and the second key frame exceeds a predetermined value.

7. An image coding apparatus according to claim 1, wherein said correspondence data generator generates the correspondence data to include time data which indicate a timing at which key frames are to be reproduced.

8. An image coding apparatus according to claim 1, wherein said correspondence data generator generates a data stream incorporating correspondence data for key frames which do not have a scene change between them and incorporating disable data indicating prohibition of correspondence for key frames which have a scene change between them.

9. An image coding apparatus according to claim 8, wherein the disable data further includes ratio data that indicates a temporal positional relation of the scene change between the key frames which have the scene change between them.

10. An image coding apparatus according to claim 9, wherein the disable data indicates that the third key frame is to be output repeatedly from an intended timing of the third key frame until the temporal position of the scene change and the fourth key frame is to be output repeatedly after the temporal position until an intended timing of the fourth key frame, the third key frame being a key frame that comes before the scene change and the fourth key frame being the other of the key frames that bracket the scene change.