Image Processing Reference

In-Depth Information

where sup and inf refer to the supremum and infimum, and d(x, y) is the distance metric

(in this case it is the l
2
norm). The Hausdorff distance is given graphically in Fig. 12.1a.

In this case, the distance from the x∈X to every element of set Y is determined, and

the shortest distance is selected as the infimum. This process is repeated for every x∈X

and the largest distance (supremum) is selected as the Hausdorff distance of the set X to

the set Y . Finally, this process is repeated for the set Y because the two distances will not

necessarily be the same. For example, the Hausdorff distance of the set X to the set Y would

be the distance from the lowercase letters in Fig. 12.1a, but the result would be smaller for

the distance of the set Y to the set X.

The method of applying the Hausdorff distance

X

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x

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Z = X
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(12.1b) Image Correspondence Prob-

lem

(12.1a) Hausdorff distance

FIGURE 12.1: Graphical Interpretations

to the image correspondence problem is the same as that described in Section 12.3.3. To

reiterate, consider Fig. 12.1b where each rectangle represents a set of subimages (obtained

by partitioning the original images X and Y ) and the coloured areas represent some of the

obtained tolerance classes. The tolerance relation covers both images, but not all the classes

are shown in the interest of clarity. Note, the tolerance classes are created based on the

feature values of the subimages, and consequently, do not need to be situated geographically

near each other
(as shown in Fig. 12.1b).
In the case of the tNM measure, the idea is that

similar images should produce tolerance classes with similar cardinalities. Consequently, we

are comparing the cardinalities of the portion of a tolerance class belonging to set X with

the portion of the tolerance class belonging to set Y (represented in
Fig. 12.1b
as sets with

the same colour). In contrast, the Hausdorff distance measures the distance between sets in

some metric space. As a result, we measure distance in the feature space between the portion

of a tolerance class belonging to set X with the portion of a tolerance class belonging to set

Y (again, represented as a sets with the same colour). Here, the idea is that images that are

similar should have tolerance classes are close to each other (in the Hausdorff sense) in the

feature space. As a result, low Hausdorff distances are desirable. Finally, it is important to

note, one may be inclined to intuitively think the Hausdorff distance should always be zero.

However, this is not the case because we are dealing with tolerance where the feature values

are similar, but not the same.

12.4

Illustration: Image Nearness Measures with Microfossil

Images

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