Image Processing Reference
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as good models of how one perceives, how one sees. Tolerance relations are also consid-
ered as a basis for studying similarities between visual perceptions (Zeeman, 1961; Peters,
In this chapter, first, the formal definitions of perceptual systems, equivalence and toler-
ance relations, and near set theory will be reviewed in sections 9.2, 9.3, and 9.4, respectively.
Then, the new nearness measure called tolerance cardinality distribution measure (TCD)
will be introduced in section 9.5. More details of the implementation of a tolerance-based
perceptual image analysis system will be reviewed in section 9.6. Finally, section 9.7 will
conclude the chapter.
Perceptual systems
A perceptual system is a real valued, total, deterministic, information system. Such a
system consists of a set of perceptual objects and set of probe functions representing object
features (Peters and Wasilewski, 2009), (Peters, 2007b,a; Peters and Ramanna, 2008). A
perceptual object (x∈O) is something presented to the senses or knowable by human
mind (Murray et al., 1933). For example, a pixel or a group of pixels in an image can
be perceived as a perceptual object. Features of an object such as color, entropy, texture,
contour, spatial orientation, etc. can be represented by probe functions. A probe function
can be thought of as a model for a sensor. A probe function φ(x) is a real-valued function
representing features of the physical object x. A set of prob functions
={φ 1 , φ 2 , ..., φ l }
can be defined to generate all the features for each object x, where φ i : O→<. However, all
the of the probe functions (features) in
are used all of the time. The setB⊆
the probe functions in use.
Example 9.1
Perceptual images & subimages An image in a set of images can be considered as a
perceptual object. Anther example for a perceptual object is a pixel or a group of pixels
inside an specific image. Also different regions in an image can be considered as perceptual
objects. For example, an image can be divided into different regions(subimages), and each
of these subimages can be considered as a perceptual object. Figure 9.1 shows an example
image. This image has been divided into square subimages(windows) with the same size.
The group of pixels belonging to each window(subimage) creates a perceptual object.
The set of probe functions or different features for the perceptual objects represented in
an image can include average gray level, color, entropy, edge, texture, and spatial orientation
of the pixels of that perceptual object in the image. For instance, in the example of figure
9.1, we would like to use only average gray level of the pixels in each subimage (window) as
a probe function. Hence, sets
will be,
={φ 1 , φ 2 , ..., φ l }
={avg gray level, color, entropy, edge, texture, spatial orientation}, e.g.,
={φ 1 }
={avg gray level of the pixels in each subimage (window)}, e.g.
Figure 9.1d shows feature values of the perceptual objects in Figure 9.1b. The average
gray scale value of all of the pixels in a perceptual object(a window box) has been replaced
with the value of the pixels in that specific region.
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