Image Processing Reference

In-Depth Information

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.

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