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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