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**Enclosed Laplacian Operator of Nonlinear Anisotropic Diffusion to preserve singularities and delete isolated points in image smoothing.**
*(English)*
Zbl 1213.94019

Summary: Existing Nonlinear Anisotropic Diffusion (NAD) methods in image smoothing cannot obtain satisfied results near singularities and isolated points because of the discretization errors. In this paper, we propose a new scheme, named Enclosed Laplacian Operator of Nonlinear Anisotropic Diffusion (ELONAD), which allows us to provide a unified framework for points in flat regions, edge points and corners, even can delete isolated points and spurs. ELONAD extends two diffusion directions of classical NAD to eight or more enclosed directions. Thus it not only performs NAD according to modules of enclosed directions which can reduce the influence of traction errors greatly, but also distinguishes isolated points and small spurs from corners which must be preserved. Smoothing results for test patterns and real images using different discretization schemes are also given to test and verify our discussions.

### MSC:

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

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\textit{Z. Liao} et al., Math. Probl. Eng. 2011, Article ID 749456, 15 p. (2011; Zbl 1213.94019)

### References:

[1] | H. K. Khalil, Nonlinear Systems, Prentice Hall, New York, NY, USA, 2001. |

[2] | M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002 |

[3] | M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space-A further study,” Applied Mathematical Modelling, vol. 31, no. 3, pp. 625-631, 2007. · Zbl 1197.94006 |

[4] | M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368-1372, 2010. |

[5] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002 |

[6] | C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, 31 pages, 2010. · Zbl 1189.92015 |

[7] | E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 2010. · Zbl 1191.35219 |

[8] | J. L. Casti, Nonlinear System Theory, vol. 175 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1985. · Zbl 0555.93028 |

[9] | M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, John Wiley & Sons, New York, NY, USA, 1980. · Zbl 0501.93002 |

[10] | C. L. Nikias and A. P. Petropuop, Higher-Order Spectra Analysis: A Non-linear Signal Processing Framework, Prentice-Hall, Englewood Cliffs, NJ, USA, 1993. · Zbl 0813.62083 |

[11] | M. Li, M. Scalia, and C. Toma, “Nonlinear time series: computations and applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 101523, 5 pages, 2010. · Zbl 1202.37117 |

[12] | Z. Liao, S. Hu, and W. Chen, “Determining neighborhoods of image pixels automatically for adaptive image denoising using nonlinear time series analysis,” Mathematical Problems in Engineering, vol. 2010, Article ID 914564, 14 pages, 2010. · Zbl 1189.94022 |

[13] | S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation for vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages, 2011. · Zbl 1202.94026 |

[14] | J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Applied Mathematical Modelling, vol. 35, no. 5, pp. 2516-2528, 2011. · Zbl 1217.94024 |

[15] | S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,” IEEE Sensors Journal, vol. 11, no. 2, pp. 389-390, 2011. |

[16] | S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167-176, 2008. · Zbl 05516617 |

[17] | C. Chaux, L. Duval, A. Benazza-Benyahia, and J.-C. Pesquet, “A nonlinear Stein-based estimator for multichannel image denoising,” IEEE Transactions on Signal Processing, vol. 56, no. 8, part 2, pp. 3855-3870, 2008. · Zbl 1390.94047 |

[18] | P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629-639, 1990. · Zbl 05111848 |

[19] | L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion. II,” SIAM Journal on Numerical Analysis, vol. 29, no. 3, pp. 845-866, 1992. · Zbl 0766.65117 |

[20] | L. Alvarez and L. Mazorra, “Signal and image restoration using shock filters and anisotropic diffusion,” SIAM Journal on Numerical Analysis, vol. 31, no. 2, pp. 590-605, 1994. · Zbl 0804.65130 |

[21] | J. Weickert, “Coherence-enhancing diffusion filtering,” International Journal of Computer Vision, vol. 31, no. 2, pp. 111-127, 1999. |

[22] | N. Sochen, G. Gilboa, and Y. Y. Zeevi, “Color image enhancement by a forward and backwardadaptive Beltrami flow,” in Proceedings of the International Workshop on Algebraic Frames for the Perception-Action Cycle (AFPAC ’00), G. Sommer and Y. Y. Zeevi, Eds., vol. 1888 of Lecture Notes in Computer Science, pp. 319-328, Springer, 2000. · Zbl 0973.68243 |

[23] | F. Zhang, Y. M. Yoo, L. M. Koh, and Y. Kim, “Nonlinear diffusion in laplacian pyramid domain for ultrasonic speckle reduction,” IEEE Transactions on Medical Imaging, vol. 26, no. 2, pp. 200-211, 2007. |

[24] | M. Ceccarelli, V. De Simone, and A. Murli, “Well-posed anisotropic diffusion for image denoising,” IEE Proceedings: Vision, Image and Signal Processing, vol. 149, no. 4, pp. 244-252, 2002. |

[25] | V. B. Surya Prasath and A. Singh, “Well-posed inhomogeneous nonlinear diffusion scheme for digital image denoising,” Journal of Applied Mathematics, vol. 2010, Article ID 763847, 14 pages, 2010. · Zbl 1189.94024 |

[26] | J. Ling and A. C. Bovik, “Smoothing low-SNR molecular images via anisotropic median-diffusion,” IEEE Transactions on Medical Imaging, vol. 21, no. 4, pp. 377-384, 2002. |

[27] | J. Yu, Y. Yang, and A. Campo, “Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain,” Mathematical Problems in Engineering, vol. 2010, Article ID 421657, 24 pages, 2010. · Zbl 1203.80026 |

[28] | G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Complex diffusion processes for image filtering,” in Proceedings of the 3rd International Conference on Scale-Space and Morphology in Computer Vision, vol. 2106 of Lecture Notes in Computer Science, pp. 299-307, Springer, 2001. · Zbl 0991.68590 |

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