**3. Type-2 fuzzy system**

282 Fuzzy Inference System – Theory and Applications

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

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

systems it will not be biased or affected very much by the lower level decisions).

convey this information while classifying a pattern or clustering a data set.

In an image *I* with dimension *M*x*N* and levels *L* (based on individual pixel as well as a

��� � ��������

*<sup>r</sup>* denotes the *i*th combination (sequence) of *r* pixels in *I; k* is the number of such

����� �

��� (1)

������ � ��� ��� �� � �� ��� �� (2)

��� . exp������� (3)

��� . exp����� (4)

�������������

*<sup>r</sup>*, as a whole, possesses

���������

�� � �� �� � �� � ��� �

�

�� � �� �� � �� � ������� �

�

*r)* denotes the degree to which the combination *- si*

���

���

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

**2.1 Grayness ambiguity measures** 

collection of pixels) are listed below.

����� � ������ � ������

*rth Order Fuzzy Entropy :* 

some image property *µ.* 

sequences; and *µ(si*

**2.2 Hybrid entropy** 

where *si*

with

respectively.

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 [Karnik'2001].

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

Fig. 2. Block Diagram of Type-2 FIS

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-

Type-2 Fuzzy Logic for Edge Detection of Gray Scale Images 285

Pre-processing: Before a computer vision method can be applied to image data in order to extract some specific piece of information, it is usually necessary to process the data in order

b. Noise reduction in order to assure that sensor noise does not introduce false

a. **Contrast adjustment:** The contrast of an image is the distribution of its dark and light pixels. A low-contrast image exhibits small differences between its light and dark pixel values. The histogram of a low-contrast image is narrow. Since the human eye is sensitive to contrast rather than absolute pixel intensities, a perceptually better image could be obtained by stretching the histogram of an image so that the full dynamic range of the image. After stripping away the color from an image (done by setting the saturation control to zero) the grayscale image that remains, represents the Luma component of the image. Luma is the portion of the image that controls the *lightness* of the image and is derived from a weighted ratio of the red, green, and blue channels of the image which corresponds to the eye's sensitivity to each color. The Luma component of images can be manipulated using the contrast controls in color image.

d. Scale space representation to enhance image structures at locally appropriate scales

to assure that it satisfies certain assumptions implied by the method. Examples are

a. Re-sampling in order to assure that the image coordinate system is correct.

c. Contrast enhancement to assure that relevant information can be detected.

Extreme adjustments to the image contrast will affect image saturation.

intensities of *I*. This increases the contrast of the output image.

b. **Intensity adjustment:** Image enhancement techniques are used to improve an image, where "improve" is sometimes defined objectively (i.e., increase the signal-to-noise ratio), and sometimes subjectively (i.e., making certain features easier to see by modifying the colors or intensities). Intensity adjustment is an image enhancement technique that maps the image intensity values to a new range. The low-contrast images have its intensity range in the centre of the histogram. Mapping the intensity values in grayscale image *I* to new values, such that 1% of data is saturated at low and high

c. **Histogram Equalization:** The purpose of a histogram is to graphically summarize the distribution of a uni-variate data set. In an image processing context, the histogram of an image normally refers to a histogram of the pixel intensity values. This histogram is a graph showing the number of pixels in an image at each different intensity value found in that image. For an 8-bit grayscale image there are 256 different possible intensities, and so the histogram will graphically display 256 numbers showing the distribution of pixels amongst those grayscale values. Histograms can also be taken of color images. Either individual histogram of red, green and blue channels can be taken, or a 3-D histogram can be produced with the three axes representing the red, blue and green channels. The brightness at each point representing the pixel count. The exact output from the operation depends upon the implementation. It may simply be a picture of the required histogram in a suitable image format, or it may be a data file of

The following are the generally applied preprocessing methods.

information.

a. Contrast adjustment b. Intensity adjustment c. Histogram equalization d. Morphological operation

dimensional plots of a Type-1 MFs. Another way to visualize Type-2 fuzzy sets is to plot their so-called Footprint Of Uncertainty (FOU). The Type-2 MFs, MF(x, w), sits atop a twodimensional x-w plane. It sits only on the permissible (sometimes called "admissible") values of x and w. This means that x is defined over a range of values (its domain)—say, X. In addition, w is defined over its range of values (its domain)—say, W.

From the Figure 2, the measured (crisp) inputs are first transformed into fuzzy sets in the fuzzifier block because it is fuzzy set, not the number, that activates the rules which are described in terms of fuzzy sets.

Three types of fuzzifiers are possible in an interval Type-2 FLS. When measurements are:


after fuzzification of measurements (inputs), the resulting input fuzzy sets are mapped into fuzzy output sets by the Inference block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are M rules, the fuzzy input sets to the Inference block will activate only a subset of those rules usually fewer than M rules. So, at the output of the Inference block, there will be one or more fired-rule fuzzy output sets.

The fired-rule output fuzzy sets have to be converted into a number by Output Processing block as shown in the Figure 2. Conversion of an interval Type-2 fuzzy set to a number (usually) requires two steps. In the first step, an interval Type-2 fuzzy set is reduced to an interval-valued Type-1 fuzzy set called type-reduction. There are many type-reduction methods available [Karnik'2001]. Karnik and Mendel have developed an algorithm, known as the KM Algorithm, used for type-reduction. It is very fast algorithm but iterative. The second step of output processing, after type-reduction, is defuzzification. Since a typereduced set of an interval Type-1 fuzzy set is a finite interval of numbers, the defuzzified value is just the average of the two end-points of this interval. If a type-reduced set of an interval Type-2 fuzzy set is a Type-1 fuzzy set, the defuzzified value can be obtain by any of the defuzzification method applied to Type-1 FL.
