49843608-Complete-thesis-Report-merged

# Figure 210 the intensity transformation of 283

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Figure 2.10 The Intensity Transformation of image [19].

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24 2.8.3 Histogram modification A histogram is a one-dimensional probability density function (PDF) of gray levels present in an image. The histogram describes the global gray level distribution of pixels within an image. Histogram-based enhancement techniques are devised to enhance an image by modifying its histogram. The histogram modification problem can be formulated as follows: Given: 1) PDF of the input image (r) 2) PDF of the desired image (s) Compute: A transformation function T, to modify gray values of the input image to achieve the desired PDF [20]. 2.9 Frequency Domain Filtering Fundamentals Filtering in the frequency domain consists of modifying the Fourier transform of an image and then computing the inverse transform to obtain the processed result. Thus, given a digital image, f(x, y) of size, M X N, the basic filtering equation in which we are interested has the form G(x, y) = Where is the Inverse Distributive Frequency Transformation, F( u , v) is the Distributive Frequency Transformation of the input image f( x, y), H( u, v) is a filter function and g( x, y ) is the filtered image [21]. 2.9.1 Gaussian Low pass Filters Gaussian low pass filters (GLPFs) of one dimension were introduced as an aid in exploring some important relationships between the spatial and frequency domains. The form of these filters in two dimensions is given by H (u, v) = .
25 Where D(u, v) is the distance from the center of the frequency rectangle. Here we do not use a multiplying constant [3]. 2.10 The Fuzzy set theory Fuzzy set theory is the extension of conventional (crisp) set theory. It handles the concept of partial truth (truth values between 1 (completely true) and 0 (completely false)). It was introduced by Prof. Lotfi A. Zadeh in 1965 as a mean to model the vagueness and ambiguity in complex systems [3]. Definition Fuzzy set: A fuzzy set is a pair ( A , m ) where A is a set and m: A > [0, 1]. For each, x A m ( x ) is called the grade of membership of x in ( A , m ). For a finite set A = { x 1 ,..., x n }, the fuzzy set ( A , m ) is often denoted by { m ( x 1 ) / x 1 ,..., m ( x n ) / x n }.Let x A Then x is called not included in the fuzzy set ( A , m ) if m ( x ) = 0, x is called fully included if m ( x ) = 1, and x is called fuzzy member if 0 < m ( x ) < 1.The set { x A | m(x) >0}is called the support of ( A , m ) and the set { x A | m(x) =1}is called its kernel. The idea of fuzzy sets is simple and natural. For instance, we want to define a set of gray levels that share the property dark. In classical set theory, we have to determine a threshold, say the gray level 100. All gray levels between 0 and 100 are element of this set; the others do not belong to the set (right image in Fig.). But the darkness is a matter of degree. So, a fuzzy set can model this property much better. To define this set, we also need two thresholds, say gray levels 50 and 150. All gray levels that are less than 50 are the full member of the set, all gray levels that are greater than 150 are not the member of the set. The gray levels between 50 and 150, however, have a partial membership in the set (left image in the below Fig.).

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26 Figure 2.11 Representation of "dark gray-levels" with a fuzzy and crisp set.
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