Method for the determination of motion vector fields from digital image sequences
This is a continuation-in-part, of application Ser. No. 202,150, filed 6/2/88 now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention is directed to a method for the determination of motion vector fields from digital image sequences, in which a motion vector field is calculated from two successive images, said motion vector field relating every picture element of an image to a picture element of the other image, whereby the relation is respectively defined by a motion vector that reproduces the relative shift of the picture elements relative to one another, and whereby all picture elements in a square or rectangular block of picture elements receive the same motion vector. 
     It is necessary for various applications, for example, image data compression or machine vision (e.g. robots and automated scene analysis), to automatically acquire the shifts of the image contents from image frame to image frame in a digital image sequence that result from object movements or from camera movements. These shifts of the local image contents can be represented by motion vector fields that indicate, for example, for every picture element of an image, by how much the image content has shifted at this location in comparison to the preceding image frame. 
     In, for example, image data compression for the purpose of transmitting digital images with low data rates, the motion vector fields can be used to predict the next image frame that has not yet been transmitted from image frames that have already been transmitted. The data rate that is required for the transmission of the new image frame is all the lower the better this prediction can be made. 
     A further application of the motion vector fields is the reconstruction of missing image frames from an image sequence that was temporally subsampled for the purpose of data compression. For example, this means that only every third image frame of the sequence is available and the two missing image frames of the sequence are to be interpolated as optimally &#34;correct in motion&#34; as possible between two respectively existing images frames (the &#34;point of reference images&#34;), so that the motion of subjects in the reconstructed scene are executed as uniformly as in the original. Motion vector fields are required for this purpose, these indicating which picture elements are to be used in the two appertaining reference images for the reconstruction of every picture element of an image frame to be interpolated. 
     In every instance, a motion vector that describes the local motion with two components, namely, the horizontal and the vertical motion commponent, is allocated in the motion vector fields, for every picture element of an image frame or to a respective group of neighboring picture elements. 
     One problem in the determination of such motion vector fields results because the movements present in an image frame sequence are usually dependent on the location of the picture elements, so that a plurality of different motion vectors can occur in a small picture detail, particularly at the edges of moving subjects. For determining a motion vector for a specific picture element, only this picture element itself should actually be considered. On the other hand, a motion vector cannot be determined from a single picture element for the reason that the motion vector contains two components and every individual picture element defines only one equation for these two unknowns, cf., for example, B. K. P. Horn, B. G. Schunck, &#34;Determining Optical Flow&#34;, Artificial Intelligence 17, Pages 185-203, 1981. Even in a small environment or surround around the picture element, however, the image content is often structured to such a slight degree that the motion at the location of the appertaining picture element cannot be unambiguously identified. This produces the difficulty that, first, only small environments or surrounds are to be used for the calculation of a motion vector in regions having motion vectors that are highly dependent on location and second, large environments or surrounds are required in regions having image contents that are not clearly structured, such being required in order to be able to unambiguously recognize the motion. It is therefore necessary to vary the size of the respective environments, and an assumption of a defined smoothness of the motion vector field must also be utilized in order to obtain motion vectors useable for the above applications even in the use of grainy noise-infested image frames, and in picture details that have little differentiation. 
     Essentially three different approaches to motion vector estimation have been previously investigated, cf., for example, H. G. Musmann, P. Pirsch, H.-J. Gallert, &#34;Advances in Picture Coding&#34;, Proc. IEEE 73 (1985) 4, Pages 523-548, namely, 
     (1) Block matching method, 
     (2) Differential method, 
     (3) Methods that work with distinctive points. 
     The operations of these methods shall be set forth briefly below for the case in which the shift of the picture contents in comparison to the predecessor picture (Picture  A) is to be identified for a picture (for example Picture B of a picture sequence). 
     BLOCK MATCHING 
     In block matching methods, the picture for which the motion vectors are to be determined is subdivided into square or rectangular blocks having a constant size, i.e. having a prescribed plurality of picture elements (frequently 16×16 or 8×8), cf., for example, C. M. Lin, S. C. Kwatra, &#34;Motion Compensated Interframe Color Image Coding&#34;, Proc. Int. Conf. On Communications, 1984, Vol. 1, Pages 516-520; and H. Brusewitz, P. Weiss, &#34;A Video-Conference System At 384 kbit/s&#34;, Picture Coding Symposium, Tokyo, Abstracts, Page 212, 1986. The same motion vector is determined for all picture elements in a block, namely, in accord with the assumption that the motion in the small picture detail that corresponds to a block is approximately constant. 
     The motion vector for a block in image frame B is thereby determined in that that for a plurality of possible motion vectors in a prescribed value range, the respective block in image frame A that contains the picture elements from image frame A, displaced by the motion vector, is respectively extracted and one block from the plurality of these blocks is selected that exhibits the least difference in comprison to the given block in image frame B. The difference between two blocks in image frame A and image frame B is thereby expressed with a suitable distancing dimension, namely, for example, the sum of the squares (L2-norm) or the sum of the absolute values (L1-Norm) of the picture element differences. That motion vector for which the two blocks from image frame A and image frame B comprise the least distance, is accepted into the motion vector field sought. 
     As initially set forth, the problem is in selecting the suitable block size: given excessively large blocks, the motion vector field becomes too course and imprecise because the assumption of constant motion in the individual blocks no longer applies; given excessively small blocks, the picture content is frequently too undifferentiated in order to allow the correct subject motion to be recognized. The publication by G. Kummerfeldt, F. May, W. Wolf, &#34;Coding Television Signals At 320 and 64 Kbit/s&#34;, Image Coding, M. Kunt, T. S. Huang, Editors, Proc. SPIE 594, Pages 119-128, 1985 makes the attempt to resolve the problem of incorrectly estimated vectors in blocks having ambivalent image content by subsequent combination and smoothing of motion vectors of a plurality of blocks that are classified as belonging to one subject. The results, however, have shown that this approach effects an improvement of the motion vector field only in cases of overall motion of the image content which is extremely simple to describe, such as &#34;camera zoom&#34; without additional subject movements--in this case, for utilization in motion-adaptive prediction in picture data compression. 
     DIFFERENTIAL 
     In the differential methods, cf., for example, P. Robert, C. Cafforio, F. Rocca, &#34;Time/Space Recursions For Differential Motion Estimation&#34;, Image Coding, M. Kunt, T. S. Huang, Editors, Proc. Spie 594, Pages 175-185, 1986, the assumption of a constant motion for a block of neighboring picture elements is abandoned and a separate motion vector is determined for every picture element instead. To this end, specific model parameters are calculated for every picture element, these describing the local evolution of the picture signal in the environment of the picture element; and conclusions regarding the underlying motion, i.e. the shift of the image contents relative to one another, are drawn from these parameters as well as from the difference in the picture contents between image frame A and B at the location of the picture element under consideration. Since the estimate of the motion is usually only an approximative solution at first, the procedure is iteratively continued until no further improvement of the motion vector can be made. 
     One problem in this method is that the description of the picture content with model parameters is valid only within narrow limits; and fails, for example, in the case of great shifts between image frame A and image frame B. Further, the environment or surround of the picture element is co-employed for the calculation of the model parameters for a picture element, whereby the uniformity of the motion in this environment or surround is again assumed, so that the selection of the size of this environment or surround raises the same problems as in the selection of the block size in block-matching methods. In that the environments of the picture elements mutually overlap, vector fields are produced that change only little from one picture element to the next, and therefore do not correctly reproduce discontinuities in the motion such as occur at subject boundaries. 
     The publication by B. K. P. Horn, B. G. Schunck, &#34;Determining Optical Flow&#34;, Artificial Intelligence 17, pages 185-203, 1981, discusses the question how these differential methods can see to it that meaningful motion vector fields largely agreeing with reality can also be determined in uniform picture regions that do not allow any unambiguous motion recognition. It is proposed that a term be incorporated into the objective function that is to be minimized, this term expressing the unsmoothness of the arising motion vector field. Due to the configuration of this component of the performance function measuring the unsmoothness of the motion vector field--the quadratic norm of what is referred to as the &#34;Laplacian&#34; of the vector field is measured, particularly because this yields an analytic function that is simple to mathematically manipulate, problems arise at subject boundaries in this method in that discontinuities in the motion vector field are erroneously suppressed. 
     DISTINCTIVE POINTS 
     In the third type of method, an attempt is made to avoid the problem that the actual subject motion can frequently not be unambiguously recognized from the local picture content. Distinctive points (&#34;gray scale corners&#34;) or lines (brightness edges) are first sought in the picture and a motion vector is determined only for these points or along the lines, cf., for example, R. Lenz, &#34;Estimation of 2-D General Motion Parameters in TV Scenes&#34;, Proc. 7th Int. Conf. Pattern Rec., Montreal, Canada, 30 July through 2 Aug. 1984, Vol. 1, pages 546-548; and C. J. Radford, &#34;Optical Flow Fields in Hough Transform Space&#34;, Pattern Recognition Letters 4, pages 293 through 303, 1986. The motion vector field for the remainder of the picture elements must then be interpolated with suitable means from the established motion vectors. What is problematical in this method is the reliable locating of the distinctive points or lines for which the motion vectors are determined first, as is the segmenting of the picture into regions having uniform motion that can be determined from the given vectors of the distinctive points or lines by interpolation. Due to the difficulties of these sub-tasks, these methods are suitable practically only for image sequences having rigid bodies such as, for example, vehicles, but are not suitable for processing scenes having moving persons as frequently occur in image data compression. 
     Smoothing operators have also been developed for these methods, cf., for example, H. H. Nagel, W. Enkelmann, &#34;An Investigation Of Smoothness Constraints For the Estimation Of Displacement Vector Fields from Image Sequences&#34;, IEEE Trans. Pattern Analysis and Mach. Intell., PAMI-8/5, Pages 565 through 593, September 1986; namely, again based on the quadratic norm of an unsmoothness function derived from the motion vector field. Since the known problems at subject boundaries thereby arise, this article proposed that a &#34;Directed Smoothness Demand&#34; be erected that is intended to effect a smoothing of the motion vector field, only in a direction perpendicular to the gradient of the luminance in the appertaining picture. The method resulting therefrom, however, is extremely involved. 
     SUMMARY OF THE INVENTION 
     The principal object of the present invention is to create a new method of the species initially cited, by means of which motion vector fields can be determined from an established image sequence, whereby special measures accomplish the result that the motion vector fields optimally reproduce the motion actually present in the picture. 
     This object is achieved by a method for the determination of motion vector fields from digital image sequences that determines a motion vector field from two successive image frames, and relates a picture element of the other image frame to every picture element of the one image frame, whereby the relation is defined by a motion vector that reproduces the displacement of the picture elements relative to one another, and whereby all picture elements in a square or rectangular block of picture elements receive the same motion vector. Such method is inventively chracterized in that a determination of the motion vectors is implemented by minimization of a composite objective function that, first, takes into consideration the differences in the luminance values of the picture elements to one another in the two given image frames and, second, evaluates or weights the differences between adjacent or neighboring motion vectors, evaluating or weighting these with the assistance of a smoothing measure; and in that the minimization of this performance function is implemented such that, first, the motion vectors minimizing the performance function are determined given the restriction that the motion vectors are constant in blocks that are larger than the blocks ultimately desired, and that, subsequently, each of these blocks is subdivided into smaller blocks that are preferably of the same size until the desired block size is achieved, whereby the performance function is again minimized by variation of the motion vectors after every dimminution of the blocks. 
     The method of the invention proceeds on the basis of the principle of block matching set forth above, c.f. H. G. Musmann et al, as recited above, i.e. a motion vector is respectively determined for a block of picture elements by evaluating a performance function for a variety of possible vectors, and by a search for that motion vector that supplies the optimum of the performance function. 
     In order to overcome the problems set forth above, especially for the block matching method, the following new principles have been incorporated into the method of determining motion vectors of the present invention: 
     (1) In order to obtain a motion vector field with high resolution that also describes the actual motion at subject boundaries with high precision, the image frame is subdivided into small blocks of, for example, 4*4 picture elements (pixels), for each of which a motion vector is determined. For the purpose of overcoming the disadvantages of small blocks (viz., possibly ambivalent pictuire content that is not sufficiently characteristic), however, a determination of motion vectors for considerably larger blocks (for example, 16*16 or 32*32 picture elements) is first carried out in a first step of the method. These large blocks are subdivided into smaller blocks in further steps of the method, a separate motion vector being determined for respectively each of these, whereby the motion vectors of the large blocks serve as a starting point for decreasing block size. 
     (2) Instead of the usual objective functions of block matching methods, that only express the differences in the picture element values of the successive images frames, the method of the invention uses objective functions that also take the &#34;smoothness&#34; of the motion vector field into consideration on the basis of suitable auxiliary terms. In the first step of the method, upon initialization of the motion vector field having a large block size, the length of the individual motion vectors is first co-involved in the objective function to be minimized. In the sequential steps of the method, the differences of &#34;neighboring&#34; motion vectors, i.e. the motion vectors of neighboring blocks, are introduced into the objective function. A smoothing of the motion vector field and a suppression of determination errors that appear as &#34;mavericks&#34; can be effected by this type of objective function, namely, even in the determination of the motion of subjects that change in shape, such as, for example, persons. 
     (3) The problems at object edges where discontinuities in the motion vector field are possible, these problems arising in the other methods (differential methods and the methods working with distinctive points) in combination with smoothing measures, are avoided in the block matching method of the invention in that the smoothness function that measures the differences of neighboring motion vectors is not based on the quadratic norm but on the absolute value norm of the differences. The effect is similar to a median filtering of the motion vector field, in which pronounced discontinuities in the motion vector field are likewise preserved, and only &#34;mavericks&#34; are suppressed. This smoothness measure therefor also enables the correct determination of motion vector fields at subject edges. 
     (4) The method can be specifically employed in such fashion in picture coding for the picture data compression that only larger blocks are initially used at the transmission side and the appertaining motion vectors are transmitted eg. over a transmission line, to the receiver. These motion vectors are used for motion-compensating prediction of images in the transmitter (coder) and receiver (decoder). At the receiver side, in addition, the refining steps that are still missing for the motion vector field are iteratively executed with the assistance of the received pictures, until the desired, smallest block size is achieved. This motion vector field can then be used to interpolate missing frames of the image sequence that had been skipped at the transmitter for the purpose of data reduction. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     The invention shall be set forth in detail below with reference to a number of drawings, in which: 
     FIG. 1 shows a schematic illustration of the execution of the method for determining motion vectors for an ultimate block size 4*4 and a starting block size of 16*16; 
     FIG. 2 shows a schematic illustration of a block subdivision to be carried out; 
     FIG. 3 shows an illustration of a motion vector X16(m,n) with the motion vectors of the four neighboring blocks; 
     FIG. 4 shows a flow chart for the overall execution of the method; 
     FIG. 5 shows a first sub-flow chart that illustrates the execution of the initialization of the method; 
     FIG. 6 shows a second sub-flow chart that illustrates the execution of the iteration steps within the method; 
     FIG. 7 shows a third sub-flow chart that illustrates the execution of an optimization procedure within the method; 
     FIG. 8 shows an illustration of the method for the determination of motion vectors that is similar to a block circuit diagram; and 
     FIG. 9 shows a block circuit diagram for an image sequence transmission arrangement that contains apparatus for the implementation of the method of the invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The method shall be set forth below with reference to a preferred exemplary embodiment for that case in which a motion vector field is determined from two images viz., frame A and frame B, whereby a uniform motion vector field is defined for respective blocks of 4*4 picture elements (pixels). The method initially begins with larger blocks that contain a plurality of small blocks. In the present example, blocks having the size 16*16 are used at the beginning. 
     The overall method execution then follows a pattern as shown in FIG. 1. 
     The method steps shall be set forth below. 
     Block-Matching 16*16 (INIT16) 
     The givens are the two successive frames A and B of a sequence that are composed of the picture elements 
     
         a(k,h), k=1 . . . I, h=1 . . . J                           (1.1a) 
    
     and 
     
         b(i,j), i=1 . . . I, j=1 . . . J                           (1.1b) 
    
     whereby i and k are the row indices and j and h are the column indices. 
     The goal of the overall method is the determination of a motion vector field X(i,j), 
     
         i=1 . . . I, j=1 . . . J.                                  (1.1c) 
    
     The frame B is now subdivided into blocks 
     
         B(m,n), with m=1 . . . M, n=1 . . . N                      (1.2) 
    
     each of which contains 16×16 picture elements b(i,j), as shown in FIG. 2. 
     A preliminary motion vector 
     
         X16 (m,n)=[x16(m,n), y16(m,n)]                             (1.3) 
    
     serving as auxiliary is now identified for every block X16(m,n). This motion vector is composed of two components, namely, the horizontal displacement x16(m,n) and the vertical displacement y16(m,n). 
     On the basis of this motion vector X16(m,n) every picture element b(i,j) in the block B(m,n) has a picture element a(k,h) from frame A allocated to it, namely on the basis of the linking 
     
         k=i+x16(m,n) and                                           (1.4a) 
    
     
         h=j+y16(m,n),                                              (1.4b) 
    
     i.e. the coordinates [k,h] of the picture element a(k,h) that is allocated to the picture element b(i,j) result from addition of the displacement or motion vector X16(m,n) to the coordinates [i,j]. 
     For the calculation of X16(m,n) the &#34;displaced frame difference&#34; d(i,j,r,s) defines: 
     
         d(i,j,r,s)=b(i,j)-a(i+r,j+s),                              (1.5) 
    
     i.e. the difference compared to the corresponding picture element from frame A displaced by a motion vector [r,s], is defined for every picture element (pixel) b(i,j) in the block B(m,n) and for every motion vector [r,s]. 
     The absolute values of the differences d(i,j,r,s) within the block M(m,n) are then summed up, as a result whereof the L1-norm (absolute value norm) D16(m,n,r,s) of the &#34;Displaced Frame Difference&#34; for the block B(m,n) and the motion vector [r,s] is produced: 
     
         D16(m,n,r,s)=SUM abs (d(i,j,r,s)).                         (1.6) 
    
     
         i,j in B(m,n) 
    
     As a first measure for smoothing the motion vector fields, a &#34;penalty term&#34; (penalty) P16 (r,s) is now also added to this sum D(m,n,r,s), this evaluating or weighting the length of the motion vector [r,s] 
     
         D16&#39;(m,n,r,s)=D16(m,n,r,s)+P16(r,s)                        (1.7a) 
    
     whereby 
     
         P16(r,s)=256×βB×(abs(r)+abs(s))           (1.7b) 
    
     applies. 
     The &#34;penalty term&#34; is thus composed of the L1-norm of the motion vector, multiplied by a control parameter β and by the number of picture elements in a block, namely, 256. The parameter β can be used to determine how greatly the length of the motion vector enters into the objective function D16&#39;(m,n,r,s). (A typical value of β that has proven itself in simulation experiments is β=1.0). 
     The minimum of D16&#39;(m,n,r,s) is then identified by variation of [r,s] in a pre-established value range S, whereby the sought motion vector X16(m,n)=[x16(m,n),y16(m,n)] derives at: 
     
         D16&#39;(m,n,x16(m,n),y16(m,n))=min D16&#39;(m,n,r,s)              (1.8) 
    
     
         r,s, in S 
    
     A quadratic value range is usually selected for S, for example the set of all motion vectors [r,s] for which the maximum absolute value of the two components r and s does not exceed an upper limit. 
     What is achieved by the addition of the &#34;penalty term&#34; that represents a modified smoothness measure to D(m,n,r,s) is that short motion vectors are prioritized in uniform image regions, or at straight edges, where the motion cannot be unambiguously determined from the local picture content (no clear minimum of D(m,n,r,s)). The probability of the appearance of &#34;mavericks&#34; in the motion vector field is, thus, already reduced. 
     For β=1.0, for example, a motion vector [r,s]=[0,1] must yield a &#34;displaced frame difference&#34; d(i,j,r,s) that is lower on average by at least 1.0 so that it is privileged over the zero vector [r,s]=[0,0]. The analogous case applies to larger motion vectors. 
     Since a full search is usually too time-consuming, the value range S is best initially limited to a plurality of samples [r,s] in a fixed grid (for example, grid constant of 4) and further searching is subsequently carried out in the environment of the optimum (&#34;three step search&#34;, c.f., for example, H. G. Musmann, P. Pirsch, H. J. Grallert: &#34;Advances in Picture Coding&#34;, Proc. IEEE 73 (1985) 4, Pages 523-548. In this case, the hit reliability of the method can be increased in that the search for the optimum motion vector is partly carried out in low-pass-filtered image frames. A subsampling of the image frame can be combined with this in order to reduce the complexity of the apparatus required. 
     This first step of the method for the determination of the motion vector thus represents a known block matching method that, however, has been modified by the introduction of the &#34;penalty term&#34; P16 (r,s) according to equation 1.7a, b for the purpose of smoothing the motion vector fields. 
     Iterative Improvement of the Motion Vector Field By Relaxation In The Case Of Block Size 16 (ITER 16) 
     After the determination of the preliminary motion vectors X16(m,n) for all blocks B(m,n), as set forth above, an iterative improvement of this motion vector field (relaxation) is carried out. To that end, a new &#34;penalty term&#34; or, respectively, a new smoothness measure P16&#39;(m,n,r,s) is defined with whose assistance the deviation of the motion vectors X16(m,n) from their respectively four neighboring motion vectors X16(m-1,n), X16(m+1,n), X16(m,n-1) and X16(m,n+1) is measured, as shown in FIG. 3. 
     The smoothness measure P16&#39;(m,n,r,s) is defined by: ##EQU1## in which r and s are the components of a motion vector [r,s] that is to be introduced as new motion vector X16(m,n) for the block B(m,n). 
     The smoothness measure P16&#39;(m,n,r,s) thus represents the sum of the absolute value norms (L1-norms) of the four difference vectors between X16(m,n) and its neighbors, multiplied by a control parameter &#34;α&#34; and the side length of a block, namely, 16. The degree of the smoothness of the motion vector field in the relaxation step can be monitored or controlled with the control parameter &#34;α&#34;. The L1-norm was selected since edges in the motion vector field, which occur at subject edges, are preserved with it, in contrast to the &#34;quadratic norm&#34; (L2-norm) that privileges continuous transitions. 
     A new performance function D16&#34;(m,n,r,s), namely: 
     
         D16&#34;(m,n,r,s)=D(m,n,r,s)+P16&#39;(m,n,r,s)                     (2.2) 
    
     is formed from the smoothness measure P16&#39;(m,n,r,s) and from the &#34;displaced frame function&#34; D16(m,n,r,s) (Equation 1.6). 
     A smoothing of the motion vector field is now carried out in that the motion vector field determined according to equation 1.8 (without proximity relationships) first forms the basis and, proceeding from this, a new, optimum motion vector X16(m,n) is determined step-by-step for one block after the other, namely, by means of 
     
         X16(m,n)=[x16(m,n), y16(m,n)]                              (2.3a) 
    
     so that 
     
         D16&#34;(m,n,x16(m,n), y16(m,n))=min D16&#34;(m,n,r,s) 
    
     
         r,s in S(m,n)                                              (2.3b) 
    
     In order to limit the search complexity (and as a further measure for smoothing the vectors), the value range S(m,n) for every block is thereby adaptively designed, namely, such that the search is all the more inclusive the more different the motion vectors X16(m,n)=[x16(m,n), y16(m,n)] and their respectively neighboring motion vectors are: 
     
         S(m,n)=[rmin . . . rmax, smin . . . smax]                  (2.4a) 
    
     with ##EQU2## Only those motion vectors [r,s] for which: 
     
         rmin≦r≦rmax and smin≦s≦smax. 
    
     are valid and investigated. 
     In regions having a constant motion vector that also crosses over the block boundaries, rmin=x16(m,n)=rmax and smin=y16(m,n)=smax, apply, so that the value range shrinks to one point and no search expenditure arises. Improvement is sought only given the presence of discontinuities and other divergences and &#34;mavericks&#34; in the motion vector field. 
     After an improvement of the vector X16(m,n) has been sought once for all blocks B(m,n), the operation must be repeated again for all those blocks B(m,n) for which at least one of the four neighboring motion vectors X16(m-1,n), etc., changed in the preceding pass. An iterative process results therefrom, and this is continued until none of the motion vectors X16(m,n) can be improved anymore, namely, while keeping the four neighbors constant. 
     As a rule, about 5 through 10 iterations suffice, namely, dependent on the degree of motion in the picture, whereby it must be taken into consideration that all blocks have to be actually checked only in the first iteration and only those blocks then have to be checked subsequently in whose proximity changes continued to occur in the most recent pass. 
     The fact that the smoothness measure P16&#39;(m,n,r,s) (Equation 2.1) is based on the L1-Norm of neighboring vectors and not, for instance, on the L2-Norm, leads to the fact that edges are preserved in this smoothing process, similar to median filtering. In fact, of course, the median value of a set of numbers is that value which minimizes the sum of the absolute values of the differences, i.e. the sum of the L1-Norms. The smoothing set forth above on the basis of minimizing D16&#34;(m,n,r,s) can therefore also be interpreted as a generalized median filtering of the motion vectors that takes the &#34;displaced frame differences&#34; into consideration. 
     A local optimum of the overall performance function Z has now been achieved, this deriving by summation of D16&#34;(m,n,r,s) over all blocks B (m,n) in the frame, i.e., what is valid for Z is: 
     
         Z=Z1+αZ2,                                            (2.5) 
    
     whereby 
     
         Z1=SUM d(i,j,x(i,j), y(i,j))                               (2.6) 
    
     
         i,j in the frame 
    
     with d(i,j,x(i,j),y(i,j)) according to equation (1.5), and ##EQU3## 
     The values x(i,j) or, respectively, y(i,j), etc., are thereby the components of the motion vectors X(i,j), that derive from the motion vector field X16(m,n) in that all picture elements b(i,j) in the Block B (m,n) are assigned the same motion vector X16(m,n): 
     
         X(i,j)=[x(i,j),y(i,j)]=X16(m,n) when b(i,j) in B(m,n).     (2.8) 
    
     Subsequently, the motion vector field is further optimized--i.e. the performance function Z is further minimized--in that the size of the blocks in which the motion vector field is assumed to be uniform is halved. 
     Block Division of 16×16 to 8×8 picture elements (L=L/2) 
     A new motion vector field [X8(p,q)] for the 8×8 pixel blocks B8(p,q) is produced from the established motion vector field [X16(m,n)] for the 76×76 pixel blocks, this new field being composed of the motion vectors 
     
         X8(p,q)=[x8(p,q), y8(p,q)]                                 (3.1) 
    
     To that end, all blocks of 16×16 pixels are each subdivided into four blocks of eight by eight pixels and each of the sub-blocks is first assigned the same motion vector, namely, that of the block of 16. This new motion vector field serves as given for the next relaxation step. 
     Iterative Improvement of the Vector Field By Relaxation In The Case of Block Size 8 (ITER 8) 
     This method step exactly corresponds to the relaxation step for block size 16 as set forth above, but with the modification that &#34;16&#34; in the equations for the objective function is to be replaced by &#34;8&#34;. An objective function D8&#34;(p,q,r,s) corresponding to D16&#34;(m,n,r,s) is thus minimized according to equation 2.2, this containing a smoothness measure P8&#39;(p,q,r,s) as in equation 2.1. The same value as in the case of block size 16 can be employed here for α (see Equation 2.1). 
     Block Subdivision From 8×8 to 4×4 Picture Elements (L=L/2) 
     As in the block subdivision from 16×16 down to 8×8 picture elements, the motion vectors that were determined for blocks having the size 8×8 are now distributed onto four respective blocks having 4×4 picture elements. 
     Iterative Improvement Of the Vector Field Given Block Size 4 by Relaxation (ITER 4) 
     This method step exactly corresponds to the relaxation steps for block sizes 16 and 8. 
     The method steps of &#34;block subdivision&#34; and &#34;relaxation&#34; can be continued down to the block size of 1×1 picture element; however, a resolution of the motion vector field having a motion vector of respectively 4×4 picture elements is adequate for many applications. 
     General Block Circuit Diagram 
     FIG. 8 shows a functional arrangement with which the method for determining motion vectors set forth above can be implemented in principle. 
     Proceeding on the basis of two input frames A and B, the ultimate motion vector field is determined in a succession of variations of the motion vector field, taking the respective values of the performance function Z into consideration. 
     Determination of Motion Vectors For the Insertion of Intermediate Images (Image Interpolation) 
     The above-described method can also be utilized for the determination of motion vector fields for image interpolation. To this end, the values that measure the &#34;displaced frame differences&#34;, namely D16(m,n,r,s) (Equation 1.6) and the corresponding values for the blocks having the sizes 8*8 and 4*4 merely have to be somewhat modified. 
     When, for example, exactly one intermediate image is to be inserted, for example, between the given frames A and B by motion-adaptive interpolation such that moving subjects in the interpolated image have moved by exactly one-half of the displacement from frame A to frame B, then the new quantity D16i(m,n,r,s), derives instead of D16(m,n,r,s) in equation 1.6, whereby i stands for interpolation: 
     
         D16i(m,n,r,s)=SUM abs (d&#39;(i,j,r,s))                        (7.1) 
    
     
         i,j in B(m,n) 
    
     with 
     
         d&#39;(i,j,r,s)=b(i-r/2,j-s/2)-a(i+r/2,j+s/2)                  (7.2) 
    
     The motion vector [r,s], is thus now not completely applied to the frame A, but is only half applied to frame A and half to frame B, with inverted operational sign, so that, overall, frame A and frame B are again mutually shifted relative to one another by the full motion vector [r,s]. 
     The analogous case also applies to the interpolation by higher factors than 2, i.e. in case two or more frames are to be inserted between the given frames A and B. In general, the displacement [t*r, t*s] is applied to frame A and the displacement [(t-1)*r,(t-1)*s] is applied to frame B, namely, with 0&lt;t&lt;1. 
     In case the displacement does not lead to whole-numbered picture element coordinates, rounding is required. 
     Division of the Determination of Motion Vectors Onto Transmitter (Coder) and Receiver (Decoder) For Moving Picture Coding 
     In a moving picture coding method, the determination of the motion vectors is used for two purposes: 
     (1) Motion-compensating prediction at transmitter and receiver (2) Motion-adaptive interpolation of missing frames at the receiver. 
     A determination of motion vectors is thereby necessary at the transmission side in order to determine the motion vectors for the motion-compensating prediction. Since these motion vectors must be transmitted, the motion vector field cannot be arbitrarily refined herein. These motion vectors, however, can also be utilized for motion-compensating interpolation at the receiver in addition to being utilized for prediction provided that a finer motion vector field is previously acquired from the transmitted motion vector field and from the transmitted frames present at the receiver side. 
     The above-described, multi-stage method (See FIG. 1) can therefore be used in such fashion for application in image sequence coding that the method steps &#34;initialization with block size 16*16&#34; as well as &#34;iterations with block size 16&#34; are carried out at the transmitter side and the remaining method steps (&#34;block sub-division&#34; and &#34;iterations&#34; for blocks of 8 and blocks of 4) are carried out at the receiver, c.f. FIG. 9. 
     It has been shown in simulation experiments that the motion vector field acquired at the transmitter side for motion-compensating prediction is in fact suitable as a prescription for a refining at the receiver side for the purpose of interpolation, whereby the &#34;displaced frame difference&#34; D16(m,n,r,s) from Equation 1.6 is employed at the transmitter side and the function D16i(m,n,r,s) (Equations 7.1, 7.2) or, respectively, its corresponding forms for smaller blocks, adapted to the interpolation, are employed at the receiver side. 
     The execution of the method for determining motion vectors that was set forth above is illustrated by the flow charts shown in FIGS. 4-7. 
     FIG. 4 represents the overall programming flow chart of the method wherein it is shown that an initialization (INIT) of the vector field first occurs with the maximum block size L=L max  and that, following thereupon, an iterative improvement (ITER) occurs for all block sizes from L max  through L min  --whereby the block side length L is respectively halved. The execution of the initialization (INIT) is shown in FIG. 5. The motion vector that minimizes the objective function with modified smoothness measure is determined here for all blocks B(m,n). 
     The iterative improvement (ITER) follows the programming flow chart according to FIG. 6. The logging field FINISH (m,n) exists here, this indicating for every block B(m,n)--having the respective block size L--whether the block is still to be processed--i.e. FINISH(m,n)=0--or whether it is already situated in a local optimum (i.e., minimum) of the objective function--i.e. FINISH(m,n)=1. First, the FINISH field is set to 0 for all blocks. In the following loop, the minimum of the objective function is sought in a defined value range (OPTI), as shown in FIG. 7, being respectively sought for all blocks that do not yet have the &#34;FINISH&#34; equal to non-zero. FINISH(m,n)=1 is set for all of these blocks. In case the motion vector of the block has changed in the minimum search, the FINISH field for the neighboring blocks is set to 0 so that these are processed again. When all blocks have the value FINISH(m,n)=1, the iteration sequence has been ended. 
     FIG. 7 shows the execution of the optimization step (OPTI) that is carried out in ITER. 
     In summary, it is to be pointed out that the subdivision of the blocks is respectively preferably executed by halving the edge lengths of the blocks. Upon insertion of an intermediate image between the two frames, two picture elements, namely, one from the first frame and one from the second frame, are allocated to every picture element of this intermediate image. The differences in the luminance values of mutually corresponding picture elements of the two successive frames within a block of picture elements are evaluated or weighted by means of the sum of the absolute values of the differences of the luminance values and are used as addends in the performance function that is to be minimized, whereby these addends form a first component of the objective function. The differences of the luminance values of mutually corresponding picture elements of the two successive frames within a block of picture elements can also be evaluated or weighted by means of the sum of the squares of the differences of the luminance values and can be used as addends in the objective function to be minimized, whereby these addends form a first component of the performance function. 
     The method of the invention also provides that the differences between neighboring motion vectors are expressed by the absolute value norms of these differences, whereby the sum of these absolute value norms forms a second component of the objective function and is used as smoothness measure. At least one of the two components is multiplied by a weighting factor and the corresponding products form the objective function by summation, whereby the objective function preferably has the form Z=Z1+αZ2, whereby Z1 is the first component, Z2 is the second component and α is the weighting factor. 
     In the preferred exemplary embodiment, only those four neighboring motion vectors are used as neighbors of every motion vector whose coordinates are horizontally and vertically situated in proximity to the coordinates of the appertaining motion vector, i.e. that their coordinates differ from the coordinates of the appertaining motion vector by (0,1), (0,-1), (1,0) or (-1,0). 
     In the initialization of the motion vector field with large blocks--preferably 16×16 picture elements (pixels)--, namely, as long as a motion vector was not calculated at least once for every block, the smoothness measure is modified to the effect that the absolute value norms of the motion vectors to be optimized--multiplied by a weighting factor--are used in the objective function by means of the smoothness measure instead of the differences between neighboring motion vectors. 
     In every stage of the block subdivision, i.e. at the beginning given the maximum block size and, following thereupon, after every block sub-division that is preferably carried out down to a block size of 4×4 picture elements (pixels), every individual motion vector is optimized by variation in an appertaining value range in succession, until a smaller value of the objective function can no longer be found for any motion vector in this way. 
     The value range of the motion vector within which the individual motion vectors are varied, in order to minimize the performance function, is made dependent on which values the motion vectors have in the motion vector field that has already been calculated, so that the value range for the optimization of the motion vector field is small when neighboring motion vectors are the same or similar, and is only larger when neighboring motion vectors exhibit great differences compared to one another. 
     What is effected by a logging system, is that only those motion vectors are again optimized, in view of a possible minimization of the value of the performance function, whose neighboring motion vectors have changed since the most recent optimization of the appertaining motion vector, so that the smoothness measure may also have varied, whereby the logging system contains a logging field comprising a memory location per block for storing control information for the optimization sequence. 
     In the optimization of a specific motion vector, every motion vector of the appertaining value range need not necessarily be taken into consideration; rather, only a sub-set of the motion vectors need be taken into consideration according to a predetermined pattern. 
     In addition to the difference between neighboring motion vector fields to be calculated, the difference between the motion vectors of the current motion vector field and those of the motion vector field calculated immediately before from a preceding pair of successive frames, can be determined and used in the smoothness measure, namely, for the purpose of smoothing the motion vector field in the direction of the time axis, i.e. for matching successive motion vector fields. 
     The method of the invention can also be applied in instances wherein motion vector fields for the purpose of inserting intermediate images are calculated from respectively two successive frame transmitted with data compression from a coder via a channel to a decoder, namely, such that corresponding motion vector fields that had already been previously transmitted are used for the initialization of the method. 
     A complete program listing 42 pages in length (p1 . . . p42) may be found in the appendix hereto, which is incorporated herewith by reference, this reproducing the entire method execution of the invention for a preferred programming example in the FORTRAN programming language. 
     It will be apparent that various modifications and/or additions may be made in the apparatus of the invention without departing from the essential feature of novelty involved, which are intended to be defined and secured by the appended claims. ##SPC1##