Perceptual Systems Approach to Measuring Image Resemblance - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 154

O /∼ B = {x /∼ B | x ∈ O}, (8.4)

x∈O x /∼ B =

O, (8.5)

∀x, y ∈ O ( x /∼ B ) ∩ ( y /∼ B )= ∅. (8.6)

Example 8.2

Figure 8.3a shows an image of size 256 × 256 pixels and their subimages. Let O, F

be a perceptual system where O denotes the set of 25 × 25 subimages.

A) The original image B) A subimage in the covering A) The original image B) A subimage in the covering

(b)

(a) (a) (b)

C) The equivalence class on a white background D) The equivalence class marked on image C) The equivalence class on a white background D) The equivalence class marked on image

(c) (d) (c) (d)

(8.3a) An equivalent class (8.3b) A tolerance class

FIGURE 8.3: An image and one of its equivalence (Left) and tolerance (Right)
classes

Let B = {φ 1 ( x ) }⊆ F is the set of only one probe function

φ 1 where φ 1 ( x )=

gray . The two marked subimages

are perceptually indiscernible with respect to their gray level values and hence they
are named an equivalence class . The equivalence class is shown both individually
(c) and on top of the image where the rest of image is blurred (d).

( x ) is the average gray scale value of subimage x

Perceptual indiscernibility relation can also be defined in a weak sense as follows

Perceptual Weak Indiscernibility Relation (Peters and

DEFINITION 8.3

Wasilewski, 2009; Peters, 2009)

Let O, F be a perceptual system. Let B = {φ 1 ,φ 2 , ..., φ l } and x, y ∈ O . A percep-

tual weak indiscernibility relation ∼ B is defined relative to B as follows