The Correspondence Problem

The aperture problem is caused by the fact that we cannot fi nd the corre- sponding point at an edge in the following image of a sequence, because we have no means of distinguishing the diff erent points at an edge. In this sense, we can comprehend the aperture problem only as a special case of a more general problem, the correspondence problem. Generally

380                                                                                                                  14 Motion

 

a                                                                    b

Figure 14.6: Illustration of the correspondence problem: a deformable two- dimensional object; b regular grid.

 

a                                   b

       

Figure 14.7: Correspondence problem with indistinguishable particles: a mean particle distance is larger than the mean displacement vector; b the reverse case. Filled and hollow circles: particles in the fi rst and second image.

 

speaking, this is that we are unable to fi nd unambiguously correspond- ing points in two consecutive images of a sequence. In this section we discuss further examples of the correspondence problem.

Figure 14.6a shows a two-dimensional deformable object — like a blob of paint — which spreads gradually. It is immediately obvious that we cannot obtain any unambiguous determination of displacement vec- tors, even at the edge of the blob. In the inner part of the blob, we cannot make any estimate of the displacements because there are no features visible which we could track.

At fi rst we might assume that the correspondence problem will not occur with rigid objects that show a lot of gray value variations. The grid as an example of a periodic texture, shown in Fig. 14.6b, demonstrates that this is not the case. As long as we observe the displacement of the grid with a local operator, we cannot diff erentate displacements that diff er by multiples of the grid constant. Only when we observe the whole grid does the displacement become unambiguous.

Another variation of the correspondence problem occurs if the image includes many objects of the same shape. One typical case is when small particles are put into a fl ow fi eld in order to measure the velocity fi eld

14.2 Basics                                                                                  381

 

(Fig. 14.7). In such a case the particles are indistinguishable and we generally cannot tell which particles correspond to each other. We can fi nd a solution to this problem if we take the consecutive images at such short time intervals that the mean displacement vector is signifi cantly smaller than the mean particle distance. With this additional knowledge, we can search for the nearest neighbor of a particle in the next image. Such an approach, however, will never be free of errors, because the particle distance is statistically distributed.

These simple examples clearly demonstrate the basic problems of motion analysis. On a higher level of abstraction, we can state that the physical correspondence, i. e., the real correspondence of the real objects, may not be identical to the visual correspondence in the image. The prob- lem has two faces. First, we can fi nd a visual correspondence without the existence of a physical correspondence, as in case of objects or periodic object textures that are indistinguishable. Second, a physical correspon- dence does not generally imply a visual correspondence. This is the case if the objects show no distinctive marks or if we cannot recognize the visual correspondence because of illumination changes.

 

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