**2. Uncertainties in a recognition system and relevance of fuzzy set theory**

A gray scale image possesses some ambiguity within the pixels due to the possible multivalued levels of brightness. This pattern uncertainty is due to inherent vagueness rather than randomness. The conventional approach to image analysis and recognition consists of segmenting (hard partitioning) the image space into meaningful regions, extracting its different features (e.g. edges, skeletons, centroid of an object), computing the various properties of and relationships among the regions, and interpreting and/or classifying the image. Since the regions in an image are not always clearly defined, uncertainty can arise at every phase of the job. Any decision taken at a particular level will have an impact on all higher level activities. In defining image regions, its features and relations in a recognition system (or vision system) should have sufficient provision for representing the uncertainties

Type-2 Fuzzy Logic for Edge Detection of Gray Scale Images 283

The basic geometric properties of and relationships among regions are generalized to fuzzy subsets. Such an extension, called fuzzy geometry [Rosefeld'84, Pal'90 & 99], includes the topological concept of connectedness, adjacency and surroundedness, convexity, area, perimeter, compactness, height, width, length, breadth, index of area coverage, major axis, minor axis, diameter, extent, elongatedness, adjacency and degree of adjacency. Some of these geometrical properties of a fuzzy digital image subset (characterized by piecewise constant membership function *μI (Imn)* or simply *μ*. These may be viewed as providing

The original fuzzy logic (FL), Type-1 FL, cannot handle (that is, model and minimize the effects of) uncertainties sounds paradoxical because the word fuzzy has the connotation of uncertainty. A user believes that Type-1 FL captures the uncertainties and vagueness. But, in reality Type-1 FL handles only the vagueness, not uncertainties, by using precise membership functions (MFs). When the Type-1 MFs have been chosen, all uncertainty disappears because Type-1 MFs are totally precise. Type-2 FL, on the other hand, handles uncertainties hidden in the information/data as well as vagueness by modeling these using Type-2 MFs. All set theoretic operations, such as union, intersection, and complement for Type-1 fuzzy sets, can be performed in the same for Type-2 fuzzy sets. Procedures for how to do this have been worked out and are especially simple for Type-2 fuzzy sets

First, let's recall that FL is all about IF-THEN rules (i.e., IF the sky is blue and the temperature is between 60 and 75° Fahrenheit, THEN it is a lovely day). The IF and THEN parts of a rule are called its antecedent and consequent, and they are modeled as fuzzy sets. Rules are described by the MFs of these fuzzy sets. In Type-1 FL, the antecedents and consequents are all described by the MFs of Type-1 fuzzy sets. In Type-2 FL, some or all of

> Output processing **De-fuzzifier**

> > **Type-Reducer**

CRISP OUTPUTS

TYPE REDUCED OUTPUTS (INTERVAL)

The Type-2 fuzzy sets are three-dimensional, so they can be visualized as three-dimensional plots. Unfortunately, it is not as easy to sketch such plots as it is to sketch the two-

the antecedents and consequents are described by the MFs of Type-2 fuzzy sets.

**Rules** 

**Inference** 

**2.3 Spatial ambiguity measures based on fuzzy geometry of image** 

measures of ambiguity in the geometry (spatial domain) of an image.

**3. Type-2 fuzzy system** 

[Karnik'2001].

CRISP INPUT

Fig. 2. Block Diagram of Type-2 FIS

**Fuzzifier** 

involved at every level.[Lindeberg'1998] The system should retains as much as possible the information content of the original input image for making a decision at the highest level. The final output image will then be associated with least uncertainty (and unlike conventional systems it will not be biased or affected very much by the lower level decisions).

Consider the problem of determining the boundary or shape of a class from its sampled points or prototypes. There are various approaches[Murfhy'88, Edelsbrunner'83, Tousant'80] described in the literature which attempt to provide an exact shape of the pattern class by determining the boundary such that it contains (passes through) some of the sample points. This need not be true. It is necessary to extend the boundaries to some extent to represent the possible uncovered portions by the sampled points. The extended portion should have lower possibility to be in the class than the portions explicitly highlighted by the sample points. The size of the extended regions should also decrease with the increase of the number of sample points. This leads one to define a multi-valued or fuzzy (with continuum grade of belonging) boundary of a pattern class [Mandal'92 & 97]. Similarly, the uncertainty in classification or clustering of image points or patterns may arise from the overlapping nature of the various classes or image properties. This overlapping may result from fuzziness or randomness. In the conventional classification technique, it is usually assumed that a pattern may belong to only one class, which is not necessarily true. A pattern may have degrees of membership in more than one class. It is, therefore, necessary to convey this information while classifying a pattern or clustering a data set.

#### **2.1 Grayness ambiguity measures**

In an image *I* with dimension *M*x*N* and levels *L* (based on individual pixel as well as a collection of pixels) are listed below.

*rth Order Fuzzy Entropy :* 

$$H^r(I) = (-1/k) \sum\_{l=1}^k \left[ \{ \mu(\mathbf{s}\_l^r) \log \mu(\mathbf{s}\_l^r) \} + \left\{ \{1 - \mu(\mathbf{s}\_l^r)\} \log \{1 - \mu(\mathbf{s}\_l^r) \} \right\} \right] \tag{1}$$

where *si <sup>r</sup>* denotes the *i*th combination (sequence) of *r* pixels in *I; k* is the number of such sequences; and *µ(si r)* denotes the degree to which the combination *- si <sup>r</sup>*, as a whole, possesses some image property *µ.* 

#### **2.2 Hybrid entropy**

$$H\_{\rm hy}(I) = -P\_{\rm w} \log E\_{\rm w} - P\_{\rm b} \log E\_{\rm b} \tag{2}$$

with

$$E\_{\mathcal{W}} = \{1/MN\} \Sigma\_{m=1}^{M} \Sigma\_{n=1}^{N} \mu\_{mn} \cdot \exp(1 - \mu\_{mn}) \tag{3}$$

$$E\_w = \left(1/MN\right)\Sigma\_{m=1}^M \Sigma\_{n=1}^N \left(1 - \mu\_{mn}\right). \exp(\mu\_{mn})\tag{4}$$

where *µmn* denotes the degree of "whiteness" of the (m,n)th pixel. *Pw* and *Pb* denote probability of occurrences of white (*µmn* =1) and black (*µmn*=0) pixels respectively; and *Ew* and *Eb* denote the average likeliness (possibility) of interpreting a pixel as white and black respectively.

#### **2.3 Spatial ambiguity measures based on fuzzy geometry of image**

The basic geometric properties of and relationships among regions are generalized to fuzzy subsets. Such an extension, called fuzzy geometry [Rosefeld'84, Pal'90 & 99], includes the topological concept of connectedness, adjacency and surroundedness, convexity, area, perimeter, compactness, height, width, length, breadth, index of area coverage, major axis, minor axis, diameter, extent, elongatedness, adjacency and degree of adjacency. Some of these geometrical properties of a fuzzy digital image subset (characterized by piecewise constant membership function *μI (Imn)* or simply *μ*. These may be viewed as providing measures of ambiguity in the geometry (spatial domain) of an image.
