From Tolerance Near Sets to Perceptual Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 177

Definition 9.3.4 Perceptual Tolerance Relation

Let O, F be a perceptual system and let ∈ R (set of all real numbers). For everyB⊆ F

the perceptual tolerance relation ∼ = B is defined as follows:
∼ = B, ={(x, y)∈O×O :kφ B (x)−φ B (y)k≤ε},

(9.2)

(x) ... φ l (x)] T where φ B (x) = [φ (x) φ is a feature-value vector obtained using all the probe

1 2

functions inBandk.kis L , instead

of ∼ = {φ} we write ∼ = φ . Further, for notational convenience, we will write ∼ = B instead of ∼ = B,ε
with the understanding that ε is inherent to the definition of the tolerance relation.

norm (L p norm in general). IfB={φ}for some φ∈ F

2

Let A⊂ ∼ = B . A is a preclass if∀x, y∈A, x ∼ = B y. A is a tolerance class if, and only if
A is a maximal preclass. Let x / ∼

= B ⊂ ∼ = B denote a tolerance class containing x. A covering
of X⊂O is the union of the tolerance classes in ∼ = B . Let p in Figure 9.3 denote a window
size. Figures 9.2 and 9.3 show examples of an image and its covering with different window
sizes.

(9.3a) Original Image (9.3b) Grayscale Image (9.3c) Covering, p=20

(9.3d) Covering, p=40 (9.3e) Covering, p=60 (9.3f) Covering, p=100

FIGURE 9.3: Coverings with different window sizes

9.4 Near Sets

It has been shown (Peters and Ramanna, 2008; Peters and Wasilewski, 2009; Peters, 2008a;
Henry and Peters, 2007) that near sets which are a generalization of rough sets (Polkowski,
2002) provide a good basis for classification of perceptual objects. Sets of perceptual objects
where two or more of the objects have matching descriptions are called near sets (Peters,
2007c). The basic idea in the near set approach to object recognition is to compare object
descriptions. Sample perceptual objects x, y∈O, x 6= y are near each other if, and only if
x and y have similar descriptions.