Discovering Image Similarities.Tolerance Near Set Approach - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 231

common and a given equation have no exact solutions. An almost solution of an equation (or
a system of equations) is an object which, when substituted into the equation, transforms
it into a numerical 'almost identity', i.e., a relation between numbers which is true only
approximately (within a prescribed tolerance) (Sossinsky, 1986). Equality in the physical
world is meaningless, since it can never be verified either in practice or in theory. Hence, the
basic idea in a tolerance space view of images, for example, is to replace the indiscerniblity
relation in rough sets (Pawlak, 1982) with a tolerance relation in partitioning images into
homologous regions where there is a high likelihood of overlaps, i.e., non-empty intersections
between image tolerance classes. The use of image tolerance spaces in this work is directly to
recent work on tolerance spaces (see, e.g., (Bartol, Miro, Pioro, and Rossello, 2004; Gerasin,
Shlyakhov, and Yakovlev, 2008; Schroeder and Wright, 1992a; Shreider, 1970; Skowron and
Stepaniuk, 1996; Zheng, Hu, and Shi, 2005)). The contribution of this chapter is twofold,
namely, a proposed tolerance near set-based approach to solving the image correspondence
problem and a comparison of the results obtained using several tolerance space-based image
resemblance measures.
This chapter has the following organization. A brief introduction to tolerance near sets
is given in Sect. 12.2. This is followed in Sect. 12.3 by a presentation of four nearness
measures. A report on the results of experiments with pairs of images using the Meghdadi
image nearness toolset is given in Sect. 12.4.

TABLE 12.1 Nomenclature

Symbol Interpretation

O, X Set of perceptual objects, X, Y⊆O,

F ,B Sets of probe functions,B⊆ F ,

ε ε∈<(reals) such that ε≥0,

i th φ i (x) probe function representing feature of x,

, x∈O, description of x,
∼ = B,ε {(x, y)∈O×O :kφ B (x)−φ B (y)k 2 ≤ε}, tolerance relation
Preclass A⊂ ∼ = B,ε ⇐⇒∀x, y∈A, x ∼ = B,ε

φ B (x) (φ 1 (x), . . . , φ l (x)), φ i ∈ F

y, i.e.kφ B (x)−φ B (y)k 2 ≤ε,

maximal preclass in cover of X defined by ∼ = B, ,

X
i

C

T B,

X T B

Y ={(x, y)∈X×Y :kφ B (x)−φ B (y)k 2 ≤ε},

T ∼

i T ∼

X Y

j

X Y

j

= B,ε C = B,ε C

={(x, y)∈ C

i ×

C :kφ B (x)−φ B (y)k 2 ≤ε},

X ./ B,ε Y X resembles Y .

12.2 Tolerance Near Sets

This section introduces the basic notions underlying a tolerance near set approach to de-
tecting image resemblances.

12.2.1 Probe Function

Definition 1 Probe Function (nea). A probe function is a real-valued function repre-

senting a feature of a physical object.