A NOVEL EDGE DETECTION ALGORITHM FOR ...

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Keywords: Edge Detection, Image Thresholding, Fisher Information, Histogram, Information ... The FI measure quantity is related to the Shannon entropy [10].
Appl. Comput. Math., V.14, N.3, 2015, pp.xx-xx

A NOVEL EDGE DETECTION ALGORITHM FOR IMAGE BASED ON NON-PARAMETRIC FISHER INFORMATION MEASURE GAMIL ABDEL-AZIM1 , S. ABDEL-KHALEK2 AND A. S. F. OBADA2 Abstract. In this paper, we proposed a novel edge detection algorithm based on the nonparametric Fisher information (FI) measure. It does not depend on the gradient or Gaussian smoothing. It takes advantage of the local thresholding to find edges. The algorithm firstly created a binary image by choosing a local threshold value using the non-parametric FI measure. Secondly, the usual masks used to detect the edges. The efficiency of the proposed approach is proved by using examples from the real-world. The performance evaluation of the proposed technique based on peak signal to noise ratio (PSNR) is presented. Experimental results show that the effect of the proposed method is comparable to the classic methods, such as Canny, and it is better than Sobel, Prewitt, and Robert methods. Keywords: Edge Detection, Image Thresholding, Fisher Information, Histogram, Information Theory. AMS Subject Classification: 68, 68T10, 68Q30, 68Q68.

1. Introduction Edge detection is an important domain in the image processing. Edges characterize object boundaries that are therefore useful for registration, segmentation and identification of objects in a scene. Effective detection edge reduces a large amount of data, but still retains most of the principal feature of the image. Edge detection refers to sharp discontinuities localization process in an image. These discontinuities are from different characteristics of the scene such as discontinuities in depth, discontinuities in surface orientation, and changes in the material properties and the changes in scene illumination. Image segmentation is the central part of technologies in the field of computer vision. So, a large plenty of literature aimed at image segmentation have been published. Thresholding segmentation algorithm is one of the most classical segmentation algorithms, and the most important step in this algorithm is obtaining the optimal threshold [13]. The obtained method for the optimal threshold in [23] depends on the histogram of the original image. Moreover, the optimal threshold achieved when peaks of the histogram vary significantly in size, and the Valley is relatively wide. The gradient method is introduced in [13] to obtain the optimal threshold. Another method proposed in [14]; it selects multi-level threshold using the method of mean shift mode seeking. Otsu introduced his method in [20]; this method maximizes the probability of the resultant classes in gray levels to get the optimal threshold, and this method is widely discussed and researched in recent years. The average amount of information that contained in the image measured by the entropy namely, entropies of quasi-segments measure how much information contained in these quasi-segments. Therefore, the entropy of quasi-segments reflects segmentation quality. 1

College of Computer & Informatics, Canal Suez University, Egypt e-mail: [email protected] 2 Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt e-mail: [email protected] Manuscript received xx . 1

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Shannon entropy has introduced in optimal threshold selection, and its effectiveness has proved [15, 21]. Many other methods were also put forward to obtain the optimal threshold using other entropies, such as cross-entropy [20], Reny entropy [6], and two-dimensional entropies [22] and others. All of these methods have their advantages, as well as disadvantages. The Otsu method was found to be infective while the difference of the gray scales between object and background is small [5]. The FI measure quantity is related to the Shannon entropy [10]. This quantity has two principals’ roles in the information theory. First, it is a core principle of the statistical field called parameter estimation. Secondly it is a measurement state of disorder of a system or phenomenon. Like FI, Shannon’s entropy is a function of an underlying probability density (mass) function p(x) and is an ”information.” Historically 25 years (1922 versus 1948), FI predates the Shannon entropy by approximately, for the relations that connect the two information concepts see [4, 25]. Shannon’s entropy can be, but is not always, the thermodynamic, Boltzmann entropy [10, 12]. For more information see[1, 7]. Recently Abdel Azim and Abo-Eleneen are proposed an Algorithm for Image thresholding using non-Parametric Fisher Information [1]. Also, the authors are proposed image thresholding using FI-based approach and extend of [17] Kittler and Illingworth’s MET method [3]. Moreover, in [2] they are developed method to segment a standard MRI brain images and a glioma MRI brain images. The FI based thresholding considers an image histogram to be a probability distribution and then selects an optimal threshold, which gives the maximum FI. The method based on a criterion that employs the combination of the FI measure and the intensity contrast. In this article, we suggested an edge detection algorithm based on the non-parametric FI measure. The rest of the paper organized as follows: In Section 2, Non-parametric FI measure and FI description for a gray scale image reviewed. FI Measure Thresholding Algorithm (FIMTHA), edge detection function and Performance measures presented in Section 3. Experimental Results and Discussion showed in section 4 and the conclusions presented in Section 5 1.1. Non parametric Fisher information measure. Given Y is a discrete random variable describing a system (or phenomena). Let Y takes on a finite or accountable infinite number N of values that characterized by the probability density function fj , j ∈ N where fi is the N ∑ probability of yj and yj ∈ (a, b) ⊆ ℜ, is assumed to be normalized to unity so that fj = 1. j=1

In this case, Y = {y1 , y2 , ....., yN } can be specified by a probability vector, f = {f1 , f2 , ..., fN }, the non-parametric FI measure [10, 9, 8] and the Shannon entropy [16] of the random variable Y are given by the following I(Y ) =

∑ {f (yj+1 ) − f (yj )}2 f (yj )

j

and H(Y ) = −



f (yj ) log f (yj ).

,

(1)

(2)

j

Compared with Shannon entropy, the FI is very sensitive to the density difference of the adjacent points of the variable. Indeed, when the density is subjected to a rearrangement points, although the shape of the density can significantly change the value of the entropy power remains constant as a function of the equation (2), but the local slope values vary considerably and thus the sum of the equation (1) Which defines the FI will also change substantially [10, 8]. We refer the reader to the books by Frieden [10, 1] to see FI measure with different applications. 1.2. Fisher information description for a gray scale image. Given I a gray-scale image with L gray levels [0, 1, ....L − 1]. The number of pixels with gray level j is denoted by nj and

G.ABDEL-AZIM, et al.: A NOVEL EDGE DETECTION ALGORITHM FOR IMAGE ...

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the total number of pixels is denoted by N = n0 + n1 + ... + nL−1 the probability of gray level appearing in the image is defined as the following: fj =

nj , N

fj ≥ 0,

L−1 ∑

fj = 1.

(3)

j=0

Suppose that the pixels in the image divided into two classes A and B. One usually corresponds to the object class and the second is the background class or vice versa. The probabilities of the two categories (classes) are given by the following. fA = fB =

f1 f2 ft , , ..., , w1 w1 w1 fL−1 ft+1 ft+2 , , ..., , w2 w2 w3

(4)

where w1 (t) =

t ∑

fj , w2 (t) = 1 − w1 (t).

(5)

j=0

The priori nonparametric FI[1, 3] for each distribution is defined as: IA (t) =

IB (t) =

t 1 ∑ (f (xi+1 ) − f (xi ))2 , w1 f (xi )

1 w2

i=0 L ∑

i=t+1

(f (xi+1 ) − f (xi ))2 . f (xi )

(6)

(7)

The FI I(t) is dependent on the threshold value t for the foreground and the background. We define the FI measure within the two classes as the following. I(t) = w1 IA (t) + (1 − w1 )IB (t).

(8)

We maximize the FI within the two categories (classes- the object and the background). When is maximized, the luminance level t is considered to be the optimum threshold value. This criterion can achieve with a small computational effort. topt = arg max {w1 IA (t) + (1 − w1 )IB (t)} .

(9)

2. Fisher measure thresholding algorithm (FIMTHA) We presented the newly proposed edge detection technique based on Fisher Measure Thresholding Algorithm (FIMTHA). The proposed algorithm is efficient and simple thresholding method. This approach defines a new criterion that based on the FI corresponding to two classes, then determines the optimal threshold by maximization the criterion. The proposed algorithm for image segmentation described by following steps: 1. Let max = 0 be the optimal threshold, and let max I be the maximum value of the objective function. 2. For t = 1 to Maximum of gray intensities 3. Compute the function objective value that corresponds to the gray level t If I(t) > max, Then max = Topt = t End Take Topt as the optimal threshold for segmenting the image.

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The relationship between a pixel and its neighborhoods in an image determine the edge information. If a pixel’s gray level value is similar to those around it, there is probably not an edge at that point. A pixel may present an edge point if it has neighbors with widely varying gray levels. Many Edge Detection Methods implemented with convolution mask and based on discrete approximations to differential operators. The rate of the change in the image brightness function is measured by differential operations. Some operators return orientation information. Other only return information about the existence of an edge at each point. In order to edge detection, firstly classification all pixels that satisfy the criterion of homogeneousness, and detection all pixels on the borders between different homogeneous areas. In the proposed algorithm, firstly a binary image is created by choosing a suitable threshold value using non-parametric Fisher information (FI) measure. Also, secondly, the usual masks used to detect the edges. A spatial filter mask may defined as a matrix w of size n × m [18, 19] for this research paper a smallest meaningful size of the mask is 3 × 3 is used and defined as following. 2.1. Edge Detection Function. Input: A grayscale image A of size M × N and t*. Output: The edge detection image g of A. Begin Step 1: Create a binary image using non-parametric Fisher information (FI) measure thresholding (t∗ ): Step 2: Create an M × N output image, g: For all x and y, Set g(x, y) = 0. Step 3: Checking for edge pixels: Create a mask w, with dimensions 3 × 3: Step 4: For 1 ≤ x ≤ M and 1 ≤ y ≤ N : Find g an output image by set g = f . Step 5: for all 2 ≤ y ≤ N − 1 and 2 ≤ x ≤ M − 1, Checking for edge pixels: i. sum = 0; ii. For all −1 ≤ k ≤ 1 and −1 ≤ j ≤ 1: If ( f (x, y) = f (x + j, y + k) ) Then sum = sum + 1. iii. If (sum