# Introduction

dge detection is one of the fundamental issues of digital image which is a method of segmentation. Edge detection significantly reduces the amount of data and filters out useless information, while preserving the important structural properties in an image. since edge detection is in the forefront of image processing for object detection, it is crucial to have good understanding of edge detection algorithms. Therefore, precise edge detection is required for numerous image analysis, evaluation and recognition techniques. In the past, a lot of research has been done in the area of image segmentation in various applications using edge detection.

The underlying idea of most edge detection techniques is by the computation of a local first or second derivative operator, followed by some regularization technique to reduce the effects of noise. Earlier edge detection methods, such as Sobel, Prewitt and Roberts' operator used local gradient method to detect edges along a specified direction.

The lack of noise control resulted in their poor performance on blurred or noisy images.

Canny [1] proposed a method to counter noise problems, wherein the image is convolved with the first order derivatives of Gaussian filter for smoothing in the local gradient direction followed by edge detection by thresholding. Marr and Hildreth [2] proposed an algorithm that finds edges at the zero-crossings of the Author ? : Department of ECE, GGSIPU(New Delhi), India. E-mails : gurpreet.preeti.82virk@gmail.com , varunraj.iitd@gmail.com image Laplacian. Non-linear filtering techniques for edge detection also saw much advancement through the SUSAN method [3], which works by associating a small area of neighboring pixels with similar brightness to each center pixel.

More recently, techniques have been proposed that characterize edge detection as a fuzzy reasoning problem.

Fuzzy logic by the local approach has been used in [4] for morphological edge extraction method.

Ho et al. [5] used both global and local image information for fuzzy categorization and classification based on edges.

In this paper, we have proposed a fuzzy-Canny based approach to edge detection that uses both global and local image information. Firstly, we used a modified Gaussian membership function to represent each pixel in the fuzzy domain. After which, a global contrast intensification operator is used to enhance the image by adjusting its parameters. In this process, pixels having more edginess will be enhanced while that with the lesser will be decreased. The optimization of the entropy function by gradient descent function produces new optimized parameters of contrast enhancement. The second phase involves the edge detection process with local image information by a local fuzzy mask, similar to the one suggested in [4,5]. Then simple thresholding method based on experimental observations and in the last step Canny edge detection is performed to link the edges obtained and results for very low contrast and noisy images are discussed in the paper.


# II.


# Global Contrast Intensification a) Fuzzy image representation

To represent an image in fuzzy domain from spatial domain, a gray tone image R of dimension M x N and L levels can be considered as an array of fuzzy singleton sets: R = {(µ mn , x mn ) where m= 1,?.M; n=1,?.N;}

Here each pixel has some intensity value x mn and its grade its membership grade µ mn (0? µ mn ?1) relative to some brightness level in the range 
µ mn = G(x mn ) = ?? [? (???????? ??????? ) 2 2ð??"ð??"? 2 ](2)
Here G (x mn ) is a Gaussian function, and x max , x mn are the minimum and (m, n) th gray values respectively. A fuzzy histogram is used to obtain the frequency of occurrence of membership functions of gray levels in the fuzzy image. Thus, R = U {µ(x), p(x)} = {µ mn /x mn } ; m=1,?.,M; n=1,2,?..,N

Where µ(x) is the membership of pixel with intensity value of x, and p(x) is the number of occurrences of the intensity value x, in the image R. The
distribution of p(x) is normalized such that ? ??(??) ???1 ??=0 = 1(4)
Membership function which is based on histogram by which pixels of spatial domain can be represented in fuzzy domain by histograrn fuzzification function as
µ(k) = ?? [?(???????? ???) 2 /2ð??"ð??"? 2 ]
(5)

k is in the range [0, L-1] and the fuzzifier parameter, f h can be determined as
f h = ? (???????? ???) 4 ??(??) ???1 ??=0 2 ? (???????? ???) 2 ??(??) ???1 ??=0(6)
p(k) stands for the frequency of occurrence of k in histogram R. In the fuzzy plane, a contrast -enhanced image is low perception (dark), µ ? [0, 0.5] or high perception (bright) µ ? [ 0.5, 1] values. The pixels near µ=0.5 which do not belong to any of the two classes describes the fuzzy boundary and hence they may contain edges.


# c) Contrast intensification function

We first enhance the image using non linear new contrast intensification function as image degradation is non linear in nature. NINT [µ(k)] having 3 tunable parameters, that are intensification operator t, fuzzifier f h and the crossover point x c defined as
µ'(k)= NINT[µ(k)] = 1/(1+exp[-t(µ(k)-x c )]) (7)
Here t controls the shape of the sigmoid function and the initial value of x c is taken as 0.5. And other two parameters are adjusted through µ(k) while t will be fixed instead to control the level of contrast enhancement in the image.


# d) Parameter x C and f h entropy optimization

To access the image quality different type of measures are reported which are difficult to be quantified. In the fuzzy based approach, entropy of the fuzzy set is a functional to measure the degree of fuzziness of a fuzzy set, giving the value of indefiniteness of an image. Entropy E can be defined in terms of Shannon's function S e 

Where
S e (µ'(k)) = -µ'(k) ln µ'(k) -(1-µ'(k)) ln(1-µ'(k)) and {0?µ'?1}(9)
Entropy optimization method with pre-set initial values of x c and f h . The derivatives of E w.r.to x c and f h are
???? ?????? = ???? ???? ? (??) ???? ? (??) ?????? = 1 ln 2 ? [?? 2 (??(??) ? ????)ð??"ð??"(?? ? )]??(??) ???1 ??=0 (10) ???? ??ð??"ð??" ? = ???? ???? ? (??) ??µ ? (??) ??µ(??) ???? (??) ??ð??"ð??" ? = 1 ???? 2 ? ? ?? 2 ?? (??)(?? (??)????? )(???????? ???) 2 ð??"ð??"??? ? ? ð??"ð??"? 2 ? ??(??) ???1 ??=0(11)
Where g(???) is defined by
g(µ')= µ'(k)(1-µ'(k)) = ?? ???(?? (??)????? [1+?? ???(?? (??)????? ) ]^2(12)
Gradient descent technique is used for the recursive learning of the parameters x c and f h
x c,new = x c,old -?? x ???? ?????? (13)f h,new = f h,old -?? x ???? ??ð??"ð??" ?(14)
Here Ñ?" x and Ñ?" f are learning factors or learning rates for parameters x c and f h. . If these two diverge and converge too quickly, the value of Ñ?" x and Ñ?" f have to be altered respectively in order that the convergence of these values is ensured. We note that the optimization of x c moves in both decreasing positive and negative search directions. The nearest optimization point of the both is taken as x c,new.


# III.

Local Edge Detection b) Entropy optimization of parameters ? and At the local window, optimization is also required to fine-tune parameters ? and ?, as the final edge output depends very much on the values of these two parameters. Taking into consideration that the edge mask is applied locally and does not involve the entire image, the entropy function is taken as
E(? (m, n))= ?(m, n)1n ? (m, n)+(1-? (m, n))1n-? (m, n))] (17)
Where the global membership value, ?(k) is now replaced by the local edge pixel ? (m, n).The derivatives of E with respect to ? and ? are obtained as: 
? new = ? old ?? ? ???? ???? (20) ? new = ? old ?? ? ???? ???? (21)
Where ? and ? are learning factors for both parameters ? and ? respectively. Similarly, if ? and ?? diverge or converge too quickly, the value of ? and ? have to be altered respectively to ensure stability.

Since the optimization formulae might be burdensome, we may not use all points (m, n) on the image. We proposed using only the maximum and minimum intensity points or a selection of points to represent different intensity ranges. Some Thus, the AND operation is taken to avoid such situations, so that the membership is within [0, 1], that is
? (m, n)= min[? (m, n)]? 1; ??????? ? (??, ??) > 1; ?????? ? (m, n)= max[? (m, n)]? 0; ??????? ? (??, ??) < 0; ??????? ? (??, ??) < 0 ; d) Edge image thresholding
After the edge image is produced through the edge detector, simple thresholding is required to binaries it according to a certain threshold level. An optimum threshold level ?? is determined through experiments to be in the range of 0.7 to 0.9, where ? (m, n)= ? 1 ?? ? 0.7 ? 0.9 0 ?? ? 0.7 ? 0.9

IV.


# Fuzzy-Canny Edge Detection

Now after fuzzifing the image we can simply apply canny operator to the resultant image. the simple algorithm for Canny edge detection is given below: V. The Fuzzy-Canny detector algorithm is implemented on low contrast noisy image and prior to the application of this algorithm, no pre-processing was done on the image. As the algorithm has two phasesfuzzy based detection and then implementing Canny edge detector, we present the results of implementation on these images separately.


# Results and Disscussions

For noisy low contrast images Fussy is a good approach because it involves the enhancement of an image before filtering the edges in which Canny failed to provide acceptable results. This can be explained by the fig. 5.2 and 5.3 given above. The resultant image can be further improved by using Canny operator which will help in linking the edges and enhancing the edge pixels as given by fig. 5.4.


# VI.


# Conclusion

The fuzzy -Canny edge detector presented in this paper uses both global (histogram of gray levels) and local(membership function in a window) information and finally edge linking which is one of the step of Canny immensely suitable for applications such as face recognition and fingerprint identification, as it does not distort the shape and is able to retain the important edges and continuous edges unlike the Canny and fuzzy-Canny edge detector. Choice of some of the parameters t, ?? and ?? is crucial for the success of this algorithm. 
![[0, L-1] b) Fuzzy membership function based on histogram Membership function is used for expressing the fuzzy property. Modified Gaussian membership function E a simple transformation functions containing only one fuzzifier is given by](image-2.png "")
![Local edge detector mask Redefining contrast intensification function, NINT(.) in terms of (m,n)th pixel µ'(k)= NINT[µ mn ] = 1/(1+exp[-t(µ mn -x c )]) Approach for Enhanced Edge Detection Algorithm by Entropy Optimization A fuzzy parameter-based new Gaussian-type edge detector is proposed as?(m, n)=?? ? ? ? [µ(?? +??,?? +?? )]?? ?? ?? 2(ð??"ð??"? )??(16)where i,j ? [ -(w -1) 2,(w -1) 2], and w x w is the size of the edge detector mask.?'(m, n) is the membership value of central pixel of the mask at location (m, n) and ?(m, n) is the output edge pixel replacing the previous central pixel. The fuzzifier f h is earlier optimized using equation. Parameters ?? and ? are adjustable and are preselected by experiments. As the mask is a generalized Gaussian function, different values of ?? and ? would yield different functions, i.e. selecting ? = ? = 1 would produce an exponential mask, while ? = ? = 2 would yield a normal Gaussian mask. The operation performed by the mask is a nonlinear mapping process and the output pixel value ?(m,n) ? [-?,?], though in general, the value of ? (m, n) lies in [0,1].](image-3.png "a)Fuzzy")
![used in the learning of the parameters xc and fh recursively by the gradient descent technique:](image-4.png "")
2![conditions and assumptions are needed to monitor the convergence of these values and prevent optimization process from yielding local minima or maxima. The following are the selection criteria and feasible range of values for ? and ? ? new ? 1 ?????? ? new ? old +0.new ? old /2 and ? new ? ? old If the value of ?? and ? converges outside the above range of values, optimization can be discarded, and the old values, ? old and ? old, are used. c) Removal of strong edges and noise However, when strong edge and impulse noise are encountered, ? (m, n) will have either very large values of ? (m, n) > 1; or very small values of ? (m, n) < 0](image-5.png "2 ?")
1![Minimum number of false negatives and false positives. 2. Good localization, report edge location at correct position. 3. Single response to single edge. Solving an optimization problem using variational calculus and the criteria specified above he arrived at an optimal edge enhancing filter, the derivative of a Gaussian Global Journal of Researches in Engineering Volume XII Issue v v v v II Version I Approach for Enhanced Edge Detection Algorithm by Entropy Optimization G (x, y) = ?? ??? 2 +?? 2 /2?? 2 Steps of Canny edge detection algorithm: 1. Convolve the image with the derivative of a Gaussian. 2. Apply non maxima suppression to the gradient magnitude image. 3. Use two thresholds T 1 > T 2 : ? a) Class = {edge if magnitude > T1 Candidate if magnitude > T2} b) Hysteresis: Any candidate which is a neighbor, in the gradient direction, of an edge is reclassified as an edge.](image-6.png "1 .Fuzzy")
5153![Fig 5.1 : Original low contrast noisy image Fig 5.2 : Detection of edges using Canny edge detection](image-7.png "Fig 5 . 1 :Fig 5 . 3 :")
![. The local information is fuzzified using a modified Gaussian membership function. Using the contrast intensification operator, the image is enhanced to the required level of visual quality by the entropy optimization of parameters fh and xc. Then, the local edge detection operator is applied on the enhanced image using parameters ?? and ? , which are again obtained from entropy optimization. Simple edge thresholding is applied and finally canny edge detection is performed. Results show that this edge detector is Global Journal of Researches in Engineering Volume Journals Inc. (US) J Fuzzy Approach for Enhanced Edge Detection Algorithm by Entropy Optimization 5.1](image-8.png "")
			© 2012 Global Journals Inc. (US)
		
		
			

				


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