Image Processing Reference

In-Depth Information

7.3

Perceptual Image Processing

Near set theory can be easily applied to images; for example, define a RGB image as

f ={p
1
, p
2
,..., p
T
}, where p
i
= (c,r,R,G,B)
T
, c∈[1,M], r∈[1,N], R,G,B∈[0, 255],

and M,N respectively denote the width and height of the image, and M×N = T. Further,

define a square subimage as f
i
⊂f with the following conditions:

f

∩f

...∩f
s
=∅,

1

2

f

∪f

...∪f
s
= f,

(7.4)

1

2

where s is the number of subimages in f. The approach taken in the NEAR system is to

restrict all subimages to be square except when doing so violates Eq. 7.4. For example, the

images in the Berkeley Segmentation Dataset (Martin, Fowlkes, Tal, and Malik, 2001) often

have the dimension 321×481. Consequently, a square subimage size of 25 will produce 6240

square subimages, 96 subimages of size 1×5, 64 subimages of size 5×1 and 1 subimage

consisting of a single pixel. Next, O can be defined as the set of all subimages, i.e., O =

{f
1
,...,f
s
}, and

is a set of functions that operate on images (see Section 7.4 for examples

of probe functions used in the NEAR system or (Marti, Freixenet, Batlle, and Casals, 2001)

for other examples). Once the setBhas been selected, the elementary sets are simply

created by grouping all objects with the same description, and the quotient set is made up

of all the elementary sets. Finally, a simple example of these concepts is given in Fig. 7.4

where the left image contains an octagon with a radius of 100 pixels located at the centre

of the 400×400 image, and the right image contains the elementary sets obtained using

B={φ
avg
(f
s
)}and a subimage size of 10×10.

F

(7.4a)

(7.4b)

FIGURE 7.4: Example of near set theory in the context of image processing: (a) Original

image, and (b) elementary sets obtained from (a) using φ

(f
s
).

avg

Observe that three elementary sets are obtained in Fig. 7.4b, namely, the blue back-

ground, the orange octagon interior, and the green squares along the diagonals. The green

squares are created by subimages that contain both black and white pixels (in the original

image) and are located only on the diagonals due to the subimage size and shape, and

the position and radius of the hexagon. All other subimages are uniformly white or black.

Thus, we are presented with perceptual information in the form of three equivalence classes

when restricted to only being able to describe the original image with the probe function

B={φ
avg
(f
s
)}and a subimage size of 10×10. This example clearly demonstrates that

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