La Ineficacia Probatoria de la Prueba Ilícita y sus Efectos

Figure 9 The adjustment operations cannot change inclusion relationships.

Thus, they cannot successfully match the image on the left to the image on the right, although they are homeomorphic. Intuitively, these 'images have different structure.

adjustment operations. The operations are specified in terms of changes to the combinatorial cell configurations. The correspondence whose existence is guar-anteed, however, relates the underlying, infinite-resolution spaces represented by these complexes. Thus, when I say that the matcher preserves topological struc-ture, I mean that in the usual mathematical sense, not in some sense peculiar to digitized spaces. It is typical in computer vision algorithms to use approxima-tions to mathematical concepts, e.g. smoothness or differentiability. Although there may be noise in the boundaries that are input to the matcher the transfor-Nations performed by the adjustment phase of the matcher are mathematically exact.

Requiring two images to have the same topological structure, using the model of boundaries developed in Chapters 2 and 11, 'is a very weak condition on the images. It does not, for example, constrain the order of regions to be the same,, as shown in Figure 10. The four image adjustment operations cannot be used to relate any pair of images that have the same topological structure, as we saw in Section 2 However, they can scamble patterns of 2D regions 'in ways that are not desirable in image matching. The image matcher applies the operations only in limited ways, so as to make only small adjustments to the 'images.

x

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x 0

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Figure 10. These two images have the same topological structure.

Boundary adjustment is applied to an image in two phases. The input to adjustment is a pair of images, one of which is to be modified so as to match the other (target image) as well as possible. The first phase, thickening, 'identifies all cells whose labels are not the same in the two images and moves as many of these cells as possible into the boundaries. The second phase, thinning, then moves as many cells as possible out of the boundaries. A cell is moved out of the boundaries only 'if it can be re-assigned the label of the corresponding cell in the target 'image. As Figure 11 'ustrates, this process of thickening boundaries and then thinning them has the effect of moving boundary locations. The details of this process are described 'in Appendix B.

This pattern of applying adjustment operations restricts the ways in which

4 A

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Figure 11 A boundary location can be moved by thickening the boundary with cells from one sde and then moving these cells out the other side.

boundaries can be moved. Boundaries are only moved through regions in which labels conflict in the original 'images. Cells whose labels agree in the original images are not altered. This means that two regions can only be matched if they overlap in the original alignment. Furthermore a boundary can only be matched to one of the boundaries nearest to it in the original alignment and it cannot "hop over" any intervening boundaries.

Both the thinning and the thickening phase involve multiple passes through the 'image. Since the adjustment operations are local, they can be done at many image locations in parallel. However, each pass can only thicken or thin each boundary by one cell. Since most applications 'involve larger boundary motions, multiple passes are needed. In the current implementation, three passes are used in each phase, so each boundary can be moved approximately three cells in any direction.' This amount of motion seems sufficient for all of the applications I have considered, though t could be increased without great consequence.

LiMI'ting the number of adjustment passes restricts changes i4 region shape

3 Due to details of the algorithms, described in Appendix B, slightly more move-ment may be possible in some cases. The actual bound varies between 3 and 6 cells, depending on the details of image geometry.

to those that are plausible for the current application. More generally, the mini-mum number of operations required to transform one image into another can be used as a measure of how different two topologically equivalent representations are.4 This distance function measures roughly, the amount of work required to determine that the two representations are equivalent. The algorithms de-scribed in this thesis can only prove two representations topologically equivalent when this requires very little work, that is when the representations are also very similar in metric and cell structure. As Fgure 12 illustrates, 'it 'is difficult for people to determine whether two situations are topologically equivalent if their metric structure is very different. I doubt that the general problem of proving topological equivalence for cellular representations 'is omputationally tractable.

Figure 12. If the metric structure of two stuations is very different, it is difficult to determine whether they have the same topological structure.

After both phases of adjustment are finished, the adjusted image is compared to the target image. A cell is marked as matching if it has the same label in the adjusted and target images and as non-matching otherwise. Because boundaries in edge finder output are induced by label transitions, all boundary ismatches must involve label conflicts. Thus Iit is not necessary to flag boundary mis

The details of this dstance function depend, of course, on the details of the operations provided.

matches explicitly. Figure 13 shows match results for two images used in edge finder testing (see Chapter 9. These images represent the same scene, but have different samplings of random noise. The match results correctly 'identify which regions of the images have been corrupted by the noise.

Figure 13. Top: Noisy edge finder output for two images used 'in edge finder testing. These images reflect the same scene, but with different samplings of random noise. Bottom- the match between the two images before (left) and after (right)adjustment Matching cells are shown in white and non-matching cells in black.

Cells that match after adjustment are further classified into those whose label was changed during adjustment and those whose label was not altered. This is done by comparing the adjusted 'image to the original image from which it was derived. This information is used 'in the analysis phase to determine the amount of boundary motion. Thus, the output of adjustment is a three-way classification of cells 'Into matching, adjusted, and non-matching. I refer to this as the raw

match map.

The adjustment process described above does not treat the two images sym-metrically. When the 'images contain matching boundaries, the two outputs from the two directions differ primarily in that the final boundaries le to opposite s'des of the adjustment regions. However, f a boundary in one image does not correspond to any boundary in the other 'image, the two. outputs differ more sub-stantially. Consider two 'images, one blank and the other containing a dot, as 'in Figure 14. When the 'image containing the dot is adjusted, the mismatch can be reduced to a single point. When the other 'image is adjusted, however, the m-match covers the full area of the dot, because no adjustment is possible. In order to handle such cases properly, the matcher does adjustmentin both directions, in parallel. The two raw match maps are then reconciled by re-classifying a cell as non-matching 'in one image if it is non-matching in the other. In cases such as the missing dot, this combined match map contains a non-matching region covering the entire area of the dot.