Edge Detection Based on the Newton Interpolation s Fractional Differentiation

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1 Te International Arab Journal of Information Tecnology, Vol. 11, No. 3, May Edge Detection Based on te Newton Interpolation s Fractional Differentiation Caobang Gao 1,, Jiliu Zou, 3, and Weiua Zang 3 1 College of Information Science and Tecnology, Cengdu Uniersity, Cina Key Laboratory of Pattern Recognition and Intelligent Information Processing, Cengdu Uniersity, Cina 3 Scool of Computer Science, Sicuan Uniersity, Cina Abstract: In tis paper, according to te deelopment of te fractional differentiation and its applications in te modern signal processing, we improe te numerical calculation of fractional differentiation by Newton interpolation equation, and propose a new mask, te Newton Interpolation s Fractional Differentiation (NIFD. Ten, we apply tis new mask to image edge detection and can obtain te better edge information image. In order to get continuous and tin edges, we syntesize a new gradient and adopt te non-maxima suppression metod. For a comparison, we consider te edge map yielded by te sobel operator and canny operator. By contrast, we discoer tat te edge image obtained by NIFD operator is better tan tose of sobel and canny operators, and specially for a noisy image, NIFD operator as te best anti-noise ability. Keywords: NIFD operator, edge detection, newton interpolation, fractional differentiation. Receied September 4, 011; accepted May, 01; publised online April 4, Introduction Fractional differentiation [10], also called non-integer differentiation, is not a new concept: it dates back to Caucy, Riemann, Liouille and Letniko in te 19 t century. Since ten, seeral teoretical pysicists and matematicians ae studied fractional differential equations, especially fractional-order linear differential equations. In comparison wit integer-order differentiation, te fractional differentiation of direct current or low-frequency signal is often nonzero. Fractional differential processing is not only nonlinearly keeping signal s low-frequency and direct current components, but also nonlinearly enancing te signal s ig-frequency and middle-frequency components. Based on te special caracteristics of fractional differentiation in te last two decades, fractional differentiation as played a ery important role in arious pysical sciences fields, suc as mecanics, electricity, cemistry, biology, economics, time and frequency domains system identification, notably control teory, and robotics. Recently, fractal teory is already used in fractal image processing [3, 4, 5, 6, 9, 1, 13]. In image processing, edge detection often makes use of integer-order differentiation operators, especially order 1 used by te gradient and order by te Laplacian [1,, 7, 8, 14]. In [4, 5, 6, 9, 11, 1], te principles of non-integer order differentiation operators in edge detection is introduced. Tis paper demonstrates wit details ow using an edge detector Newton Interpolation s Fractional Differentiation (NIFD can improe te criterions of tin detection and immunity to noise, wic can be interpreted in term of robustness to noise in general. Based on te existed teories of fractional calculus and teir specific applications in digital image processing, we improe te numerical calculation of fractional differentiation by Newton interpolation equation and propose a new operator (NIFD operator tat can improe te criterions of tin detection and immunity to noise.. Fractional Differentiation Reiew Differential operation is a basic type of matematical calculations, it is widely used in te fields of signal analysis and processing. Recently, many researces ae been reported about fractional calculus and its applications, and it as become more and more important in foundational researc and engineering application. From a teoretical point of iew, fractional differentiation extends te order of signal processing from integer-order to any order, wic is an extension of information processing metods and means [3, 4, 5, 6, 9, 1, 13]. Grümwald-Letniko definition of fractional calculus originates from te classical definition of integer-order differentiation for continuous function, wic is deduced by generalizing differential order from integer to fraction [10]. Assume tat R (R represents te real set, [] is its integral part, te signal F(t [a,t], a<t, a R, t R, as m(m Z, Z represents te integer set order continuous

2 4 Te International Arab Journal of Information Tecnology, Vol. 11, No. 3, May 014 differentiation. Wen >0, m is no less tan [], so order differentiation could be expressed in te form [10]: Γ D F(t = lim F(t m (m 1 ( m 1 n 1 m G ( 1 ( 1 a t 0 Γ Γ n= t a m = 0 d F Equation 1 is also written as, and te difference of dt fractional differentiation equation 1 is expressed as [4, 5, 6, 1]: = F(t ( F(t 1 F(t d F ( 1 dt ( 1( F(t 3 6 Γ(n F(t n Γ(n 1 Γ( Similarly, for te signal F(x, y, te order differentiation could be expressed: t a m Γ lim 0 m= 0 Γ Γ F ( 1 ( 1 = F(x m,y x (m 1 ( m 1 t a m Γ lim 0 m= 0 Γ Γ F ( 1 ( 1 = F( x,y m y (m 1 ( m 1 (3 Te differences of fractional partial differentiation respectiely are expressed as [4, 5, 6, 1]: F ( 1 = F(x,y ( 1( F(x 3,y 6 Γ(n F(x n,y Γ(n 1 Γ( F(x,y ( F(x 1,y x F ( 1 = F( x,y ( F( x,y 1 F( x,y y ( 1( F( x,y 3 6 Γ(n F( x,y n Γ(n 1 Γ( According to equations 5 and 6, among te n non-zero coefficients, only te coefficient of te first term is te constant 1, te oter n-1 non-zero coefficients are functions wit respect to te fractional order. We can proe te sum of n non-zero coefficients is non-zero [5, 1]. And it is one of te significant differences in te caracteristics between te fractional differentiation and te integer-order differentiation. 3. Teory of NIFD (1 ( (4 (5 (6 To make te fractional differential operator more precise, we can improe fractional differentiation algoritm. Next, te points F(t - m, m = 0, 1,,..., n in equation are iewed as nodes. To express any point between t m - and t-m, let ξ = t - m. Wen [-, ], ξ [t m -, t-m ]. Tus, for any tree nodes F(t - m -, F(t - m, F(t - m, using Newton interpolation equation, one as interpolation expression of te signal function F(t in equation : Were F ( ξ = F [ t m ] F [ t m,t m ] ( ξ ( t m F [ t m,t m,t m ] ( ξ ( t m ( ξ ( t m F[ t m ] = F ( t m (8 F( t m F( t m (9 F[ t m, t m ] = F[ t m, t m, t m ] F[ t m, t m ] F[ t m, t m] = F( t m F( t m F( t m F( t m = F( t m F( t m F( t m = Taking equations 8 to 10 into equation 7, we ae: F ( ξ = F ( t m F ( t m F ( t m ( ξ ( t m F ( t m F ( t m F ( t m ( ξ ( t m ( ξ ( t m Since ξ=t-m, equation 11 becomes: F ( t m F ( t m F ( ξ = F ( t m ( t m ( t m F ( t m F ( t m F ( t m ( t m ( t m ( t m ( t m = F ( t m 1 ( F ( t m F ( t m ( ( F ( t m F ( t m F ( t m 8 1 = 1 1 ( F ( t m 8 ( 1 1 F ( t m ( F ( t m = F ( t m 1 F ( t m ( 8 4 F t m (7 (10 (11 (1 Compare wit F(t in equations and 1 as introduced te signal alue F(ξ on non-node. And te non-node signal alue F(ξ is a linear combination of te nodes F(t - m -, F(t - m, and F(t m, wic implies tat F(ξ contains te more information in its neigborood. As we known, te processed object of computer or digital filter is te limit number, te biggest ariable of gray-leel of digital image signal is also limited, and te sortest canging distance of gray-leel is one pixel, tat is =1. Ten F(t is replaced by F(ξ and taking equation 1 into equation, we can obtain te approximation of

3 Edge Detection Based on te Newton Interpolation s Fractional Differentiation 5 equation as follows: n 1 m d F ( t ( 1 Γ ( dt m= 0 Γ ( m 1 Γ ( m F( t m 1 F(t m F( t m = F( t 1 1 F( t F( t F( t 1 F(t 1 F( t ( 1 F( t 3 1 F( t Γ ( 1 F( t Γ ( m 1 Γ ( m 1 F ( t m 1 1 F( t m F ( t m = F(t 1 1 F( t ( F ( t ( ( 1( F ( t ( 1 ( 1( ( 1( ( 3 F( t Γ ( n 1 Γ ( n Γ ( n Γ ( 4 Γ ( n 1 Γ ( 1 1 Γ ( n 1 F 8 4 Γ ( n Γ ( ( t n 3 = F( t 1 1 F(t F(t F(t F( t Γ ( n 1 Γ ( n Γ ( n Γ ( 4 Γ ( n 1 Γ ( Γ ( n 1 F(t n 8 4 Γ ( n Γ ( (13 Equation 13 is called te NIFD of F(t. Indeed, te expression can only get te approximated alue due it simplifies fractional differentiation to multiplication and add. Similarly, we coose te top n terms as te fractional differentiation approximation of F(t. Let: Tus, d F ( t dt a F ( t 1 a F ( t a F ( t a F ( t n n Similarly, for te signal F(x,y, from equation 13, te approximate backward differences of fractional partial differentiation respectiely on negatie x-coordinate and y-coordinate, are expressed as: 3 F = F( x 1,y 1F(x,y x F( x 1,y F( x,y Γ(n 1 F( x 3,y Γ(n Γ( Γ(n Γ(n 1 1 F( x n,y 4 Γ(n 1 Γ( 8 4 Γ(n Γ( 3 F = F( x,y 1 1F( x,y y F( x,y 1 F( x,y Γ(n 1 F( x,y Γ(n Γ( Γ(n Γ(n 1 1 F( x,y n 4 Γ(n 1 Γ( 8 4 Γ(n Γ( (15 (16 (17 To obtain new mask along eigt symmetric directions and make tem ae anti-rotation capability, eigt new masks wic are respectiely along te directions of negatie x-coordinate, negatie y-coordinate, positie x-coordinate, positie y-coordinate, left downward diagonal, rigt upward diagonal, left upward diagonal, and rigt downward diagonal are implemented, wic is noted as NIFD operators and te operator along eery direction is written as NIFD i, i = 0, 1,..., 7, as sown in Figure a 1 = a 0 = a 1 = a = a 3 = Γ ( n 1 Γ ( n a n = Γ ( n Γ ( 4 Γ ( n 1 Γ ( 1 1 Γ ( n Γ ( n Γ ( (14 Figure 1. NIFD mask.

4 6 Te International Arab Journal of Information Tecnology, Vol. 11, No. 3, May Experiment Results and Analysis According to Figure 1, we adopt 5 5 mask. For an image, first do Gauss filter, and ten calculate te fractional differentiation by NIFD mask, tus eigt edge information images are obtained by NIFD i, i=0,1,,7 along eigt directions respectiely. Some results are sown in Figure. a Original. gradf ( x, y = d f ( x, y i d f ( x, y j [ x = NIFD f ( x, y NIFD f ( x, y 3 ( NIFD 4 f ( x, y NIFD 5 f ( x, y NIFD6 f ( x, y NIFD7 f ( x, y i NIFD f ( x, y NIFD f ( x, y [ 0 1 ( NIFD4 f ( x, y NIFD6 f ( x, y NIFD5 f ( x, y NIFD7 f ( x, y j Te norm of te syntesized gradient is: y gradf ( x, y = d x d y (0 In order to get continuous and tin edges, next we need to track and connect te edges of image based on te norm image of te gradient. Te idea is te same as te non_maxima suppression metod []. For a comparison, we considered te edge map yielded by sobel and canny operators. Some results are sown in Figure 3. b Edge information by NIFD 0. c Edge information by NIFD 1. a Original. b NIFD = 0.1. d Edge information by NIFD. e Edge information by NIFD 3. c NIFD = 0.3. d NIFD = 0.5. f Edge information by NIFD 4. g Edge information by NIFD 5. e NIFD = 0.7. f NIFD = 0.9. Edge information by NIFD 6. i Edge information by NIFD 7. Figure. Te edge information image of 0.8 order by NIFD i, i = 0, 1,, 7. From te eigt edge information images obtained by NIFD i, i = 0, 1,, 7, we can project tem to two directions (i.e., linear combination: d f ( x, y = NIFD f ( x, y NIFD f ( x, y x 3 ( NIFD 4 f ( x, y NIFD 5 f ( x, y NIFD f ( x, y NIFD f ( x, y 6 7 d f( x, y = NIFD f( x,y NIFD f( x, y y 0 1 ( NIFD 4 f( x,y NIFD 6 f( x,y NIFD f( x,y NIFD f( x,y 5 7 Ten, we syntesize a new gradient for an image f(x,y: (18 (19 g Sobel. Canny. Figure 3. Comparison of edge images. Obsering carefully Figure 3, we see tat NIFD operator and canny operator ae stronger edge searc capability and more complete edge, but sobel operator can not. In addition, NIFD operator and canny operator almost completely maintain contour information of te original image, but only NIFD operator can weaken false negaties in te textured regions.

5 Edge Detection Based on te Newton Interpolation s Fractional Differentiation 7 It is known tat accurate detection of edges in noisy data sould comply wit two possibly conflicting requirements. Te edge detection process sould aoid false edges produced by noise and ensure tat actual edges are correctly detected. In order to alidate te performance of NIFD operator, we generate some noisy images by adding zero-mean Gaussian noise wit standard deiation 0.05, ten detect edge of noisy images by sobel, Canny and NIFD operators. A part of results are sown in Figure 4. a Original noisy image. b Sobel. c Canny. d NIFD. e Original noisy image. f Sobel. g Canny. NIFD. i Original noisy image. j Sobel. k Canny. l NIFD. Figure 4. Comparison of edge images obtained by different operators for noisy images. From Figure 4, we see tat te better performance of NIFD operator is apparent. Te edge cure obtained by te NIFD operator is always smooter tan tat of canny operator. For sobel operator, te edge map is noisy, and tis effect is ery annoying. And we can easily see tat te edge map gien by NIFD looks sarper tan tose yielded by sobel operator and canny operator. Te lower resolution yielded by te sobel operator is clearly perceiable. Te edge cures ae not been detected. Altoug, canny operator can extract edge cures, te result is more noisy and te edge cures contain always some glitc. Since canny operator tracks and connects te edges of image based on te norm image of gradient tat is obtained by filtering along two directions. Wile NIFD operator is based on te norm image of gradient tat is syntesized by filtering along eigt directions suc as equations 18 and 19, tus it as anti-rotation capability and anti-noise ability. Terefore, te NIFD operator offers te best result among tem. Te edge cures can satisfactorily been detected and be muc less noisy by NIFD operator, and te image edges look ery sarp. NIFD operator as stronger edge searc capability and more complete edge, and may oercome te sortcomings of sobel and canny operators. 5. Conclusions Fractional differentiation as played a ery important role in digital image processing fields respectiely and more and more researcers begin to study tem. Tis paper intends to deduce a new operator, NIFD operator, wic may be applied to edge detection. Experiments sow tat te NIFD operator as excellent edge detection capabilities and especially plays an important role in reducing te noise sensitiity. Acknowledgements Tis work is supported by te National Natural Science Foundation of Cina (No , te Support Project of Science & Tecnology Department of Sicuan Proince (No. 011JY0077, No. 010JQ003, No. 010SZ0004, and te Natural Science Foundation of Education Department of Sicuan Proince (No. 09ZB110. References [1] Balamurugan V. and Kannan S., Detection of Traffic Signal by Adaptie Approac and Sape Constraints, te International Arab Journal of Information Tecnology, ol. 8, no. 4, pp , 011. [] Canny J., A Computational Approac to Edge Detection, IEEE Transaction on Pattern Analysis and Macine Intelligence, ol. 8, no. 6, pp , 1986.

6 8 Te International Arab Journal of Information Tecnology, Vol. 11, No. 3, May 014 [3] Fiser Y., Fractal Image Compression: Teory and Application, Springer, New York, USA, [4] Gao C. and Zou J., Image Enancement Based on Quaternion Fractional Directional Differentiation, Acta Automatica Sinica, ol. 37, no., pp , 011. [5] Gao C., Zou J., Hu J., and Lang F., Edge Detection of Color Image Based on Quaternion Fractional Differential, IET Image Processing, ol. 5, no. 3, pp. 61-7, 011. [6] Gao C., Zou J., Zeng X., and Lang F., Image Enancement Based on Improed Fractional Differentiation, Journal of Computational Information Systems, ol. 7, no. 1, pp , 011. [7] Heat M., Sarkar S., Sanocki T., and Bowyer K., Comparison of Edge Detectors: A Metodology and Initial Study, in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, USA, pp , [8] Martin D., Fowlkes C., and Malik J., Learning to Detect Natural Image Boundaries Using Local Brigtness, Color, and Texture Cues, IEEE Transaction on Pattern Analysis and Macine Intelligence, ol. 6, no. 5, pp , 004. [9] Matieu B., Melcior P., Oustaloup A., and Ceyral C., Fractional Differentiation for Edge Detection, Signal Processing, ol. 83, no. 11, pp , 003. [10] Oldam K. and Spanier J., Fractional Calculus: Teory and Applications, Differentiation and Integration to Arbitrary Order, Academic Press, New York, [11] Oustaloup A., Matieu B., and Melcior P., Edge Detection Using Non Integer Deriation, in Proceedings of Presanted at te IEEE European Conference on Circuit Teory and Design, Copenagen, Denmark, [1] Pu Y., Zou J., and Yuan X., Fractional Differential Mask: A Fractional Differential Based Approac for Multi-Scale Texture Enancement, IEEE Transactions on Image Processing, ol. 19, no., pp , 010. [13] Sabatier J., Agrawal O., and Tenreiro J., Adances in Fractional Calculus: Teoretical Deelopments and Applications in Pysics and Engineering, Springer, New York, 007. [14] Tai Y., Jia J., and Tang C., Soft Color Segmentation and its Applications, IEEE Transaction on Pattern Analysis and Macine Intelligence, ol. 9, no. 9, pp , 007. Caobang Gao is a PD and associate professor wit te College of Computer Science and Tecnology, Cengdu Uniersity. His researc interests coer digital image processing, artificial intelligence, biology feature recognition, te teory of fractional calculus and its application in signal processing. He studies currently te teory and application of fractional directional differentiation. He publised or will publis more tan 30 papers wic inole fractional directional differentiation and quaternion fractional calculus. Jiliu Zou is currently a Professor wit te Scool of Computer Science, Sicuan Uniersity. His researc is mainly in te field of image processing, artificial intelligence, fractional differential application on te latest signal and image processing, and so on. He as publised more tan 100 papers, of wic more tan 80 papers are indexed by SCI, EI, or ISTP. Weiua Zang is currently a PD wit te Scool of Computer Science, Sicuan Uniersity. His researc interests coer digital image processing, artificial intelligence, biology feature recognition, te teory of fractional calculus and its application in signal processing.