Tuberías

Figure 17. Two stuations that do not have the same topological structure.

intuitively, continuous correspondences must match boundaries wth boundaries.

This model of matching in terms of continuous functions is a standard mathe-matical approach, but one that may seem unfamiliar to researchers in other fields.

In low-level vision, for example, the matching problem has typically been stated as a problem of matching discrete features, such as short sections of boundary.

In high-level reasoning, topological structure is typically approached via topolog-ical properties such as the presence or absence of holes. Because the continuous function approach is more general, 'it can lead to more powerful constraints on algorithm behavior, as we will see in later chapters. It also extends well to cases in which we may only be able to construct a continuous correspondence between subsets of the two situations and in which additional considerations may limit the choice of correspondences.

Consider the process of representing a camera 'image for computer vsion analysis. Because a computer can only store finite amounts of 'information, we cannot store the exact 'intensity value at each point in the image. Rather, only a finite number of intensity values are stored, each one representing an average over a small patch of the 'image. Each intensity value is represented wth only finitely many bts of precision. We can model this as a mapping between two cellular representations, as shown in Fgure 18. The real intensity function maps points in the image onto exact intensity values. The approximation maps cells 'in the image onto intensity cells. Such approximations are not peculiar to computer vision, but occur 'in any application that 'involves interpreting measurements of real situations.

I

I 10

9 8 7 6

image intensities

Figure 18 A digitized image.

It is important to realize that digitized functions are not maps between spaces of discrete values, but rather approximations to continuous functions. Suppose that we labelled the image with two discrete values, dark and light, as shown in Figure 1. Whenever a dark cell is adjacent to a light cell in the image, there

must be a boundary in the mage, because a continuous function on a connected region cannot jump between two discrete values. Adjacent cells in the image can, however bear different ntensity values without there being a boundary in the image, because intensities form a connected space.

I

0 light

& dark I

' I I

Figure 19. Labelling an image with discrete values.

Except in rare cases, such as functions with discrete values, topological analy-sis of the raw digitized values does not provide sufficient information for practical applications. Consider first the relationship between the dgitized function F and the continuous function f that 'it approximates. Each digitized value has two as-sociated neighborhoods.- a support neighborhood and an error neighborhood.

The support neighborhood at a point x contains all the points whose values (from the function f) were used to derive the dgitized value F(x). The support neighborhood for each cell must include at least all the points 'in the cell and often points from other cells. Types of support regions are discussed in Section 7.

The error neighborhood at a point x consists of the points 'in the range that might be represented by the digitized value F(x). Since the value F(x) 'is reported

-1

...-only to the nearest cell, the error neighborhood must clearly include all points in the cell F(x), including the boundaries it shares with adjacent cells. Error neighborhoods are typically somewhat larger than this, due to various sources of noise present in real measurements.

In designing algorithms that operate on digitized functions it is important to be aware of the error neighborhoods associated with the values of these func-tions. This is particularly important when comparing the values at two cells.

Following Poston 1971) I refer to two values as indistinguishable if their error neighborhoods overlap. Indistinguishable values could represent measurements of the same underlying value.

Using error neighborhoods, 'it is possible to deduce the presence of boundaries even when function values form one connected region. Two cells that are adja-cent, but not separated by a boundary, overlap along their common face, edge, or vertex. The underlying values for each common point must belong to the error neighborhoods of the digitized values for both cells. Thus, the values at the two cells must be indistinguishable. If a digitized function assigns distinguishable values to two adjacent cells, they must be separated by a boundary.

Algorithms using digitized functions may also be able to take advantage of constraints on the class of continuous functions under consideration. For exam-ple, it may be possible to assume that the underlying function satisfies certain bounds on slopes, second differences, or derivatives of various orders. Depend-ing on the application, these constraints may be formulated so as to respect the topological structure. For example, bounds on slopes might apply only to differ-ences taken along connected paths. If so, a topological boundary would license apparent violations of these constraints, just a 'it lcenses apparent violations of continuity.

Although constraints on slopes or differences may be formulated as constraints on the underlying function, they often 'imply similar constraints on digitized approximations to that function.7 For example, the smoothing and sampling procedures commonly used in computer vision do not increase the magnitude of finite differences. Thus if a difference of the sampled function exceeds a given bound, the underlying infinite-resolution function must also contain a dfference that exceeds the bound. Thus, the presence of boundaries can be inferred from apparent violations of the constraints, even when only digitized approximations to function values are available.