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**A new construction of a fractional derivative mask for image edge analysis based on Riemann-Liouville fractional derivative.**
*(English)*
Zbl 1419.34009

Summary: We present a new way of constructing a fractional-based convolution mask with an application to image edge analysis. The mask was constructed based on the Riemann-Liouville fractional derivative which is a special form of the Srivastava-Owa operator. This operator is generally known to be robust in solving a range of differential equations due to its inherent property of memory effect. However, its application in constructing a convolution mask can be devastating if not carefully constructed. In this paper, we show another effective way of constructing a fractional-based convolution mask that is able to find edges in detail quite significantly. The resulting mask can trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges. The experiments conducted on the mask were done using some selected well known synthetic and Medical images with realistic geometry. Using visual perception and performing both mean square error and peak signal-to-noise ratios analysis, the method demonstrated significant advantages over other known methods.

### MSC:

34A08 | Fractional ordinary differential equations |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

92C55 | Biomedical imaging and signal processing |

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\textit{P. Amoako-Yirenkyi} et al., Adv. Difference Equ. 2016, Paper No. 238, 23 p. (2016; Zbl 1419.34009)

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### References:

[1] | Ghimpeteanu, G, Batard, T, Bertalmio, M, Levine, S: A decomposition framework for image denoising algorithms. IEEE Trans. Image Process. 25, 388-399 (2016) |

[2] | Yan, R, Shao, L: Blind image blur estimation via deep learning. IEEE Trans. Image Process. 25, 1910-1921 (2016) |

[3] | Jalab, HA, Ibrahim, RW: Texture enhancement based on the Savitzky-Golay fractional differential operator. Math. Probl. Eng. 2013, Article ID 149289 (2013) · Zbl 1296.94031 |

[4] | Oram, JJ, McWilliams, JC, Stolzenbach, KD: Gradient-based edge detection and feature classification of sea-surface images of the Southern California Bight. Remote Sens. Environ. 112(5), 2397-2415 (2008) |

[5] | Shrivakshan, GT, Chandrasekar, C: A comparison of various edge detection techniques used in image processing. Int. J. Comput. Sci. Issues 9(1), 269-276 (2012) |

[6] | Yasri, I.; Hamid, NH; Yap, VV, An FPGA implementation of gradient based edge detection algorithm design (2009), New York |

[7] | El-Sayed, MA: A new algorithm based entropic threshold for edge detection in images. Int. J. Comput. Sci. Issues 8(1), 71-78 (2011) |

[8] | Guo, W, Lai, M-J: Box spline wavelet frames for image edge analysis. SIAM J. Imaging Sci. 6(3), 1553-1578 (2013) · Zbl 1278.62100 |

[9] | Yang, Z, Lang, F, Yu, X, Zhang, Y: The construction of fractional differential gradient operator. J. Comput. Inf. Syst. 7, 4328-4342 (2011) |

[10] | Oustaloup, A.; Mathieu, B.; Melchior, P., Edge detection using non integer derivation, Copenhagen, Denmark, 3-6 September |

[11] | Garg, V, Singh, K: An improved Grunwald-Letnikov fractional differential mask for image texture enhancement. Int. J. Adv. Comput. Sci. Appl. 3(3), 130-135 (2012) |

[12] | Gao, C, Zhou, J, Zhang, W: Edge detection based on the Newton interpolationâ€™s fractional differentiation. Int. Arab J. Inf. Technol. 11(3), 223-228 (2014) |

[13] | Pu, Y, Wang, W, Zhou, J, Wang, Y, Jia, H: Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation. Sci. China, Ser. F 51(9), 1319-1339 (2008) · Zbl 1147.68814 |

[14] | Jalab, HA; Ibrahim, RW, Texture feature extraction based on fractional mask convolution with Cesaro means for content-based image retrieval, 170-179 (2012) |

[15] | Dalir, M, Bashour, M: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021-1032 (2010) · Zbl 1195.26011 |

[16] | Jalab, HA, Ibrahim, RW: Denoising algorithm based on generalized fractional integral operator with two parameters. Discrete Dyn. Nat. Soc. 2012, Article ID 529849 (2012) · Zbl 1244.94007 |

[17] | McAndrew, A: An introduction to digital image processing with Matlab notes for SCM2511 image processing. School of Computer Science and Mathematics, Victoria University of Technology (2004) |

[18] | El-Zaart, A: A novel method for edge detection using 2 dimensional gamma distribution. J. Comput. Sci. 6(2), 199-204 (2010) |

[19] | Roushdy, M: Comparative study of edge detection algorithms applying on the grayscale noisy image using morphological filter. GVIP, special issue on edge detection, 51-59 (2007) |

[20] | Wang, M.; Yuan, S., A hybrid genetic algorithm based edge detection method for SAR image, 9-12 May |

[21] | Chan, TF, Vese, LA: Active contours without edges. IEEE Trans. Image Process. 10, 266-277 (2001) · Zbl 1039.68779 |

[22] | Mumford, D, Shah, J: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 557-685 (1989) · Zbl 0691.49036 |

[23] | He, W, Lai, MJ: Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities. Appl. Comput. Harmon. Anal. 6, 53-74 (1999) · Zbl 0960.42017 |

[24] | Lai, MJ: Construction of multivariate compactly supported prewavelets in L \(2L_2\) spaces and pre-Riesz basis in Sobolev spaces. J. Approx. Theory 142, 83-115 (2006) · Zbl 1105.42025 |

[25] | Aurich, V.; Weule, J., Nonlinear Gaussian filters performing edge preserving diffusion, Bielefeld, Germany, 13-15 September, Berlin |

[26] | Basu, M: A Gaussian derivative model for edge enhancement. Pattern Recognit. 27, 1451-1461 (1994) |

[27] | Deng, G.; Cahill, LW, An adaptive Gaussian filter for noise reduction and edge detection, 31 October-6 November, San Francisco |

[28] | Kang, C, Wang, W: A novel edge detection method based on the maximizing objective function. Pattern Recognit. 40, 609-618 (2007) · Zbl 1118.68178 |

[29] | Siuzdak, J: A single filter for edge detection. Pattern Recognit. 31, 1681-1686 (1998) |

[30] | Zhu, Q: Efficient evaluations of edge connectivity and width uniformity. Image Vis. Comput. 14, 21-34 (1996) |

[31] | Ambrosio, L, Tortorelli, VM: On the approximation of free discontinuity problems. Boll. Unione Mat. Ital., B 7(6), 105-123 (1992) · Zbl 0776.49029 |

[32] | Osher, S, Sethian, JA: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12-49 (1988) · Zbl 0659.65132 |

[33] | Lai, MJ, Stockler, J: Construction of multivariate compactly supported tight wavelet frames. Appl. Comput. Harmon. Anal. 21, 324-348 (2006) · Zbl 1106.42028 |

[34] | Qi, D.; Guo, F.; Yu, L., Medical image edge detection based on omnidirectional multi-scale structure element of mathematical morphology, 2281-2286 (2007), Washington |

[35] | Stefan, W, Renaut, RA, Gelb, A: Improved total variation-type regularization using higher-order edge detectors. SIAM J. Imaging Sci. 3, 232-251 (2010) · Zbl 1195.41007 |

[36] | Mahmoodi, S: Edge detection filter based on Mumford-Shah Green function. SIAM J. Imaging Sci. 5, 343-365 (2012) · Zbl 1250.65039 |

[37] | Zhang, L, Butler, A, Sun, C: Fractal dimension assessment of brain white matter structural complexity post stroke in relation to upper-extremity motor function. Brain Res. 1228, 229-240 (2008) |

[38] | Hristov, J: Transient heat diffusion with a non-singular fading memory Therm. Sci. 20(2), 757-762 (2016) |

[39] | Caputo, M, Fabrizio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2, 73-85 (2015) |

[40] | Atangana, A, Baleanu, D: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763-769 (2016) |

[41] | Ibrahim, RW: On generalized Srivastava-Owa fractional operators in the unit disk. Adv. Differ. Equ. 2011, 55 (2011) · Zbl 1273.35295 |

[42] | Zhang, Y, Pu, Y, Zhou, J: Construction of fractional differential masks based on Riemann-Liouville definition. J. Comput. Inf. Syst. 6(10), 3191-3199 (2010) |

[43] | Chen, X.; Fei, X., Improving edge-detection algorithm based on fractional differential approach, 1-6 (2012) |

[44] | Loverro, A: Fractional calculus: history, definitions and applications for the engineer. In: Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, pp. 1-28 (2004) |

[45] | Sheikh, HR, Wang, Z, Bovik, AC: LIVE Image Quality Assessment Database, Release 2. http://live.ece.utexas.edu/research/quality (2005). Accessed 25 Jan 2015 |

[46] | Ma, X.; Li, B.; Zhang, Y.; Yan, M., The Canny edge detection and its improvement, No. 7530, 50-58 (2012) |

[47] | Wang, Z, Bovik, AC, Sheikh, HR, Simoncelli, EP: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600-612 (2004) |

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