Comparative Study of Fractional Order Derivative Based Image Enhancement Techniques

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1 Comparative Study of Fractional Order Derivative Based Image Enhancement Techniques Pratee Kotha Department of Electronics and Communication Engineering, Jawaharlal Nehru Technological University - Kainada, Kainada, Andhra Pradesh, India pratei@gmail.com Abstract - In this paper, image enhancement based on fractional gradient is proposed. The Taylor s Series is used to obtain a generalized expression for this Fractional order derivative. The Image is differentiated in both x and y directions separately. Next, the Gradient is calculated which is nothing but an edge detection operation. For proving, that the proposed approach has better brightness and contrast it is compared to Grunwald Letniov fractional differential approach enhancement. The results obtained have shown good performance. KEYWORDS - FRACTIONAL ORDER, DIFFERNTIATOR, IMAGE ENHANCEMENT, EDGE DETECTION, DISCRETE FRACTIONAL DERIVATIVE, TAYLOR SERIES. I. INTRODUCTION Fractional Differentiation, a discipline in the field of mathematics mainly deals with the differentiation of arbitrary order, although it was proposed in 17th century, it is developed mainly in the 19th century [1]. Some of the popular Fractional Differentiation definitions are Grunwald-Letniov definition [2], Riemann-Liouville definition [3], Caputo definition [4], Taylor Series etc. The main difference between fractional differentiation and integer order differentiation mainly is the long-term memory associated with earlier one. The fractional differentiation started to play a major role in various research fields mainly in image and signal processing because of this long-term memory. Another main advantage of going with Fractional Differentiation when compared to integer order differentiation, is it s ability to deal with Fractal problem i.e., ability to deal with the damages of textural detail feature of image. In this paper we enhance the image using both Grunwald-Letniiv B.T.Krishna Department of Electronics and Communication Engineering, Jawaharlal Nehru Technological University - Kainada, Kainada, Andhra Pradesh, India tbattula@gmail.com method and Discrete Fractional derivative method to chec which procedure gives better enhancement. This paper is organised as follows. Conventional edge detection techniques are discussed in section 2. Fractional Derivative based techniques are studied in section 3. Results are presented in section 4 and conclusions are drawn in chapter 5. II. EDGE DETECTION An image can be interpreted as a twodimensional function, fx, y), where x, y are special coordinates and the amplitude of f at any pair of coordinatesx, y) is the intensity or gray level of the image at that point. The integer-order derivative operator is an important tool to detect the meaningful discontinuities in intensity values. First order derivative edge gradient is obtained by forming running difference of pixels along rows and columns of the image. The gradient of a function f x, y) at coordinates x, y) is defined as two dimensional column vector given as f = gradf) = g g = f x f x 1) The magnitude length) of vector f denoted as Mx, y), where Mx, y) = mag f) = g + g 2) Mx, y) is referred to as the gradient image. It has the same size as the original, created when x and y are Page 231

allowed to vary over all pixel locations in f. Some of the first order derivatives in image processing are gradient operator, prewitt operator, sobel operator.

2 allowed to vary over all pixel locations in f. Some of the first order derivatives in image processing are gradient operator, prewitt operator, sobel operator. Second order derivatives are employed when only edge magnitudes are of interest and without regard to their orientations. The second-order derivative in image processing is generally computed by using the Laplacian. The Laplacian of a image fx, y) is defined by = f x + f 3) y III. FRACTIONAL ORDER DERIVATIVES A. Grunwald-Letniov Fractoinal Derivarive Grunwald-letniov Fractional Derivative Definition is introduced in the original paper of Grunwald[5] and letniov[6] in 1867 an 1868 respectively. Let p>0, f C [ ] [a, b] and a < b. D ft) = lim 1 h 1) p ft Kh ) 4) With h = t a) N is called the Grunwald-Letniov Fractional derivative of order p of the function f. Let w ) = 1) p ) represent the coefficients of the polynomial 1-z) p. The coefficients can also be obtained recursively from w ) = 1, w ) = 1 p + 1 w ), = 1,2.. 5) Based on definition, the fractional order differentiation can easily be caculated from D ft) 1 h w ) ft h ) 6) We can see that the fractional differential mas can be estimated by W ) and W ) can be calculated. These fractional differential mass which are respectively on the directions of positive x-coordinate W ), positive y- coordinate W ), negative x-coordinate W ), negative y- coordinate W ),right downward diagonal W ), right upward diagonal W ), left upward diagonal W ), left downward diagonal W ) and the p th order derivative of Gx, y) is calculated by[9] ), ) = max ), ) Ω 7) WhereΩ {,,,,,,, }, and Satx) is the saturated function defined by 0 < 0 ) = [0, ] > 8) L is the gray level and ) is the enhanced image. B. Discrete Fractional Derivative The fractional calculus can be considered a branch of mathematical analysis, which deals with integro-differntial equations with inclusion of integration and derivation of any positive real order. Then, the calculus of derivatives is not straight forward as the calculus of integer order derivatives. It is quite complex but the reader can find concise descriptions of this calculus [7],[8].Since image processing is usually woring on quantized and discrete data, we discuss just the discrete implementation of fractional derivation. We have to deal with twodimensional image maps, which are two-dimensional arrays of pixels. The recorded image is a discrete signal, because it is the discrete position of pixel in the map. Moreover, the signal is quantized, since the colour tones are ranging from 0 to 255. To define the partial derivative suitable for calculations on the images, let us define the discrete fractional differentiation in the following way. Let us suppose, to have a signal st), where t can have only discrete values, t = 1,2,...,n.[10],[11] The fractional differentiation of this signal is given by ) = st) + v)st 1) + ) ) st 2) + ) ) ) st 3) 9) Where, v is a real number. If we have a bi-dimensional map sx, y),where x, y can have only discrete values, that is x=1,2,..,n x and y=1,2,...,n y, the partial derivatives are, ) = sx, y) + v)sx 1, y) + ) ) sx 2,y+ v v+1 v+26sx 3,y 10), ) = sx, y) + v)sx, 1y) + ) ) sx, y 2) + ) ) ) sx, 3y) 11) Page 232

3 We can also define the fractional gradient = + = + 12) Where and are unit vectors of the two space directions. Given the two components of the gradient, we easily evaluate the magnitude G v ) p = 0.5 = ) + 13) Here it is preferable to approximate the magnitude with = +. In the case when the fractional order parameter is v=1, it is gradient. IV. RESULTS The following results are shown when both Grunwald Letniiv and Discrete Fractional Derivative methods are applied to LENNA image at various p and values. P = 0.8 Original Fig.1 Enhanced LENNA image at different p values using Grunwald-Letniov Fractional Derivative Definition p = 0.3 Original Page 233

Fractional Order Calculus is playing a vital role in many of the applications of science and engineering. In this project wor, application of Fractional derivative to image enhancement is studied.

4 = 0.3 = 0.5 = 0.8 Fig.2.Enhanced LENNA image at different values using Taylor Series based Discrete Fractional Derivative Definition V. CONCLUSIONS Edge detection of an image is useful in many areas of engineering lie Medical Imaging, Astronomical Image Processing etc. Fractional Order Calculus is playing a vital role in many of the applications of science and engineering. In this project wor, application of Fractional derivative to image enhancement is studied. In the first part of the project wor, Image Texture Enhancement is carried out by using Grunwald Letniov Differential Filter.The classical texture enhancement technique suffers from certain drawbacs lie susceptibility to noise. Different enhanced images are obtained by implementing the improved Grunwald - Letniov Fractional Differential Filter and the textural features of image are more efficient than the classical enhancement techniques. The proposed methodology of this project wor is as follows. The generalized expression for The Fractional Order Derivative, using Discrete Fractional Derivative definition is taen into consideration. The image is differentiated in both x and y directions separately. Next, the Gradient is calculated which is nothing but an Edge Detection Operation. It has been observed that, the proposed approach produces better brightness and contrast compared to previous techniques. So, the proposed approach can be used for the Image Enhancement Applications in many areas. V. REFERENCES [1] I. Podlubny, Fractional Differential Equations. San Diego: Acadamic Press,1999. [2] Dervatives of noninteger,mathemat,mag., 683): ,1995. [3] B.Riemann,Versuch einer allgemeinen auffassung der integration and differentiation, Gesammelte Mathematics Were, , [4] M.caputo, Linear models of dissipation whose Q is almost frequency independent- II, Geophysical Journal of the Royal Astronomical Society, 135): ,1967. [5] A.K.Grunwald,Uber Begrenzte derivationen und deren anwendung,z.angew.math.phy., 12: ,1867. [6] A.V.Letniov, Theory of differentiation with an arbitrary index, mat.sb., 3:1-66, Page 234

5 [7] FractionalCalculus, [8] A.Beardon,FractionalCalculus [9] Dali chen,yang Quan Chen,Dingue Xue,Feng Pan, Adaptive Image Based on Fractional Differntial Mas,CCDC [10] Jia Hauding and Pu Yifei, Fractional Calculus method for enhancing digital image of ban slip,proceedings of the 2008 Congress on Image and Signal Processing, Vol.3,pp ,CSIP 2008,2008. [11] Y.Pu,W.X.Wang,J.L.Zhou,Y>Y.Y.Wand and H.D.Jia, Fractional Scientific approach to detecting textural features of digital image and it s fractional differential filter implementation, Sci.China Ser.F Inf.Sci.,519),2008,pp Page 235

Filtering and Edge Detection

Filtering and Edge Detection Local Neighborhoods Hard to tell anything from a single pixel Example: you see a reddish pixel. Is this the object s color? Illumination? Noise? The next step in order of complexity