Possible application of fractional order derivative to image edges detection. Oguoma Ikechukwu Chiwueze 1 and Alain Cloot 2.

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1 Life Science Journal 3;(4) Poible application of fractional order derivative to image edge detection Oguoma Iechuwu hiwueze and Alain loot. Department of Mathematic and Applied Mathematic Facult of Natural and Agricultural Science Univerit of the Free State Bloemfontein 93 South Africa. Abtract: Thi paper focue on the poible application of the concept of fractional calculu to image proceing. In particular we generalized the Prewitt operator uing the fractional order derivative to detect the edge in a given image. The comparion of reult obtained via the modified operator give more detail than the eiting operator that ue the ordinar derivative. The fractional derivative ued in thi wor i in the aputo ene; in addition the numerical evaluation of the fractional operator i done uing a finite difference cheme. [Oguoma Iechuwu hiwueze and loot Alain. Poible application of fractional order derivative to image edge detection. Life Sci J 3;(4):7-76]. (ISSN:97-835).. Keword: Edge detection; Fractional-Prewitt operator; aputo derivative; numerical reult.. Introduction The edge of an image i the mot baic feature of the image. Edge i baicall the mbol and reflection of dicretene of partial image []. It contain a wealth of internal information of the image. Therefore edge detection i one of the e reearch wor in image proceing. The current image edge detection method are mainl differential operator technique and high-pa filtration. Among thee method the mot primitive of the differential and gradient edge detection method are comple and the effect are not atifactor. The widel ued operator uch a Sobel Prewitt Robert and Laplacian are enitive to noie and their anti-noie performance are poor. The Log and ann edge detection operator which have been propoed ue Gauian function to mooth or do convolution to the original image but the computation are ver large. Prewitt operator i a dicrete differentiation operator computing an approimation of the gradient of the image intenit function at each point in the image. It i ued in image proceing particularl within edge detection algorithm. In thi paper we ue the fractional order derivative in the aputo ene applied on the Prewitt operator to perform the edge detection under the gradient method.. Bacground of edge detection The edge detection method ma be grouped into two categorie: The gradient method and Laplacian method []. The gradient method detect the edge b looing for the maimum and minimum in the firt derivative of the image. The Laplacian method earche for the zero croing in the econd derivative of the image to find edge. The e tep i to decompoe a large and comple image into mall image with independent feature. The primar obective for uing computer to do image proceing are: Firtl to create more uitable image for people to oberve identif and undertand. Secondl to mae ure that computer can automaticall recognize and undertand image [8]. Mathematicall the gradient of a two-variable function or image intenit function i at each image point a two-dimenional vector with the component given b the derivative in the horizontal and vertical direction... Prewitt Operator The Prewitt operator i one tpe of an edge model operator. The ernel can be applied eparatel to the input image to produce eparate meaurement of the gradient component in each orientation a G and G. The firt derivative in image proceing are implemented uing the magnitude of the gradient. For a function f f at coordinate i defined a the twodimenional column vector the gradient of The magnitude of thi vector i given b () 7

2 Life Science Journal 3;(4) () The component of the gradient vector itelf are linear operator but the magnitude of thi vector obvioul i not becaue of the quaring and quare root operation. Thee can then be combined together to find the abolute magnitude of the gradient at each point and the orientation of that gradient []. The gradient magnitude i given b: f G G G G (3) The gradient G and G for the Prewitt operator are calculated uing the dicrete approimation: G G f 7 f8 f9 f f f3 f f f f f f (4) Table below how the Prewitt operator coniting of a pair of 3 3 convolution ernel Table : A pair of 33 convolution ernel Figure : Prewitt operator edge detection operation Figure give a practical eample of the Prewitt operator edge enhancement operation. The reulting image appear a a directional outline of the obect in the original image. The contant bright region became blac and changing bright region became highlighted. Thi operator doe not place an emphai on piel that are cloer to the center of the ma [3]... Fractional order derivative There are man definition of fractional derivative but in thi ection we preent the fundamental definition of fractional order derivative that are motl ued which include aputo [4] and Riemann-Liouville [6 7] reult relative to the Gamma function and dicu briefl the advantage and diadvantage of thee fractional order definition. Riemann-Liouville gave the mot popular definition of fractional derivative of order a: f a D RL n n n d n f d n a (5) where i the Gamma function. aputo gave the econd popular definition ued a: a D f n n. n a f n d n (6) Advantage The aputo repreentation ha advantage over Riemann-Liouville repreentation. aputo mot well nown advantage i that it allow traditional initial and boundar condition to be included in the formulation of the problem [4]. Alo it fractional derivative or aputo derivative of a contant i zero wherea for the Riemann-Liouville the derivative of a contant i not zero. The Laplace tranform of the Riemann-Liouville derivative lead to boundar term containing the limit value of the Riemann- Liouville fractional derivative at the lower boundar of integration a and inpite of the fact that mathematicall uch problem can be olved there i no phical interpretation for uch tpe of condition. On the other hand the Laplace tranform of aputo derivative impoe boundar condition involving integer-order derivative at the lower boundar a which uuall are acceptable phical condition. With the Riemann-Liouville fractional derivative an arbitrar function need not to be continuou at the origin and it need not to be differentiable. 7

3 Life Science Journal 3;(4) Diadvantage Function that have no firt order derivative might have fractional derivative of all order le than one in the Riemann-Liouville ene. aputo derivative require higher condition of regularit with repect to the differentiabilit of a function. A aputo derivative i defined onl for function that are differentiable and in the claical ene. f f f d f f f 3 d where d. Thi numerical cheme i onl firt order accurate and we have the following reult: Propoition : Let f be a function in a b and. Then with D f D f E E O 3 A econd order accurate numerical cheme can be obtained uing the concept of pline: For... we need to calculate. d f t t. (8) We compute thee integral b approimating the econd order derivative b a linear pline t whoe node and not are choen at.... The pline t i of the form Dicretization of the fractional derivative [9] d f In thi ection we decribe how to dicretize the t t (9) fractional derivative in the aputo ene. Firtl we derive numerical approimation baed on the aputo with derivative definition (6): t in each d f t interval for D f t. given b t (7) t A uual wa of approimating the aputo derivative D f read: t t t f f f D f t otherwie. For t i of the form and t t t t otherwie. t t otherwie. Therefore an approimation for (7) i of the form t t a d f t t and after ome traightforward Mathematical manipulation we obtain t t d f a () 4 where a 73

4 Life Science Journal 3;(4) () () For the meh point... N the econd order derivative can be approimated b f / where i the central econd order differential operator f f f f. Additionall we alo need to now the value of the econd order derivative at the boundar point. If we have a phical boundar condition of the tpe d f d b (3) we can conider the given value. If thi value i not available at the econd order derivative can be approimated b U / where operator: f f 5 f 4 f f 3 Finall an approimation for D i the (4) can be written a D f a f a f 4. For which the following propoition hold: Propoition : Let D 3 a b and with f be a function in. Then D f f E E 3 Note that D that i D f D RL d f a a d D. a ' a f f f a f a RL. Jutification of the Algorithm In thi ection the reult obtained b appling aputo derivative numerical cheme to the Error function are preented numericall and analticall with different value of alpha. The analtical reult are in agreement with the numerical reult which indicated that the numerical code i efficient and accurate. The red dot in Figure to Figure 5 below how the numerical reult while the green line indicate the analtical reult. F() Figure : The analtical and numerical reult with alpha =.5 F() Figure 3: The analtical and numerical reult with alpha =.75 F() Figure 4: The analtical and numerical reult with alpha =.85 74

5 Life Science Journal 3;(4) F() Figure 5: The analtical and numerical reult with alpha =.5 3. Numerical Simulation In order to ae the poible effect of the order of the fractional derivative in detecting edge in an image or enhancing we mae ue of the fractional- Prewitt operator method. The matri (alo called in image proceing language ma ) obtained are repreented in -direction and -direction. The approimation of the addition of both - direction and -direction in thi wor i called the fractional Prewitt operator. The fractional-prewitt operator (ma) i ued to convolute with the original image. The aim of thi convolution i to detect the edge of an image. Now we mae ue of the aputo derivative for -direction -direction and the fractional- Prewitt operator (both direction) to acce the edge contained in the -ra picture below. Thi i done for different value of alpha ( and.95) a indicated in the Figure 6 to Figure. Figure 7: Derivative with alpha =.5 Figure 8: Derivative with alpha =.75 Figure 6: Original image Figure 9: Derivative with alpha =.85 75

6 Life Science Journal 3;(4) The purpoe of thi tud i to how that fractional order derivative can be ued a a tool for edge detection during image enhancement. Therefore uing aputo derivative to detect image edge a een above how that fractional order derivative can be ued a one of the tool in image proceing enhancement method. Figure : Derivative with alpha =.95 It i oberved from the above reult that appling aputo derivative to thi original image we are able to ee the effect of fractional order derivative on an image in all direction. We oberved from thee image that there are ignificant effect of different alpha value on the ame original image. The moothing or edge detecting power of alpha i noticeable a alpha increae for to. More image intenit detail and harp edge are detected and een on both direction with the Fractional Prewitt operator. 4. oncluion Given that the principal obective of enhancement i to proce an image o that the reult i more uitable than the original image for a pecific application and obcured detail are detected and enhanced or certain feature of interet in an image are highlighted or harpening of image feature uch a edge boundarie or contrat to mae an image more ueful for dipla and anali nowing that there i no particular wa to determine a perfect or ideal or good enhanced image but whenever an image loo good we a it ha been enhanced. Reference [] L.P. Han and W.B. Yin. An Effective Adaptive Filter Scale Adutment Edge Detection Method (hina Tinghua univerit 997). [] Gonzalez R.. and Wood R.E.; Digital Image Proceing; Second Edition; Prentice; Hall;. Univerit of Tenneee and MedData Interavtive. [3] A. Seif et.al.; A hardware architecture of Prewitt edge detection Sutainable Utilization and Development in Engineering and Technolog (STUDENT) IEEE onference Malaia pp Nov.. [4] M. aputo [967] Linear model of diipation whoe W i almot frequenc independent II Geophical Journal International. Atr. Soc. 3: [5] S. G. Samo A.A. Kilba and O.I. Maritchev [987]. Integral and Derivative of the Fractional Order and ome of their Application. Naua: Tehnia Min [in Ruian]. [6] I. Podlubn; Geometric and Phical interpretation of fractional integration and fractional differentiation Fract. alculu Appl. Anal. 5() [8] G. Wenhuo; Y. Lei; Z. Xiaoguang and L. Huizhong; An improved Sobel Edge detection; Digital media department. ommunication Univerit of hina; IIT ; Beiing hina [9] E. Soua Fractional Differentiation and it Application; MU Department of Mathematic Univerit of oimbra oimbra Portugal. [] D. Marr and E. Hildreth Theor of Edge Detection (London 98). /6/3 76