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Well-posed inhomogeneous nonlinear diffusion scheme for digital image denoising. (English) Zbl 1189.94024
Summary: We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. By incorporating edges found at multiple scales via an adaptive edge detector-based indicator function, the proposed scheme removes noise while respecting salient boundaries. We create a smooth transition region around probable edges found and reduce the diffusion rate near it by a gradient-based diffusion coefficient. In contrast to the previous anisotropic diffusion schemes, we prove the well-posedness of our scheme in the space of bounded variation. The proposed scheme is general in the sense that it can be used with any of the existing diffusion equations. Numerical simulations on noisy images show the advantages of our scheme when compared to other related schemes.

94A08Image processing (compression, reconstruction, etc.)
94A13Detection theory
68U10Image processing (computing aspects)
Full Text: DOI EuDML
[1] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 2006. · Zbl 1110.35001
[2] 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. · doi:10.1109/34.56205
[3] Y.-L. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Transactions on Image Processing, vol. 5, no. 11, pp. 1539-1553, 1996.
[4] F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 182-193, 1992. · Zbl 0746.65091 · doi:10.1137/0729012
[5] D. Strong, Adaptive total variation minimizing image restoration, Ph.D. thesis, UCLA Mathematics Department, Los Angeles, Calif, USA, 1997, ftp://ftp.math.ucla.edu/pub/camreport/cam97-38.ps.gz.
[6] L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1-4, pp. 259-268, 1992. · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[7] A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, vol. 76, no. 2, pp. 167-188, 1997. · Zbl 0874.68299 · doi:10.1007/s002110050258
[8] 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. · doi:10.1049/ip-vis:20020421
[9] W. Kusnezow, W. Horn, and R. P. Würtz, “Fast image processing with constraints by solving linear PDEs,” Electronic Letters in Computer Vision and Image Analysis, vol. 6, no. 2, pp. 22-35, 2007.
[10] J. Yu, Y. Wang, and Y. Shen, “Noise reduction and edge detection via kernel anisotropic diffusion,” Pattern Recognition Letters, vol. 29, no. 10, pp. 1496-1503, 2008. · doi:10.1016/j.patrec.2008.03.002
[11] T. Barbu, V. Barbu, V. Biga, and D. Coca, “A PDE variational approach to image denoising and restoration,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1351-1361, 2009. · Zbl 1169.35341 · doi:10.1016/j.nonrwa.2008.01.017
[12] A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Measurement Science and Technology, vol. 18, no. 1, pp. 87-95, 2007. · doi:10.1088/0957-0233/18/1/011
[13] M. Basu, “Gaussian-based edge-detection methods-a survey,” IEEE Transactions on Systems, Man and Cybernetics Part C, vol. 32, no. 3, pp. 252-260, 2002. · doi:10.1109/TSMCC.2002.804448
[14] R. C. Gonzelez and R. E. Woods, Digital Image Processing, Pearson, Upper Saddle River, NJ, USA, 2002.
[15] J. F. Canny, “A computational approach to edge detection,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 6, pp. 679-698, 1986.
[16] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1984. · Zbl 0545.49018
[17] V. B. S. Prasath and A. Singh, “Edge detectors based anisotropic diffusion for enhancement of digital images,” in Proceedings of the 6th Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP ’08), pp. 33-38, IEEE Computer Society Press, Bhubaneswar, India, December 2008. · doi:10.1109/ICVGIP.2008.68
[18] F. Demengel and R. Temam, “Convex functions of a measure and applications,” Indiana University Mathematics Journal, vol. 33, no. 5, pp. 673-709, 1984. · Zbl 0581.46036 · doi:10.1512/iumj.1984.33.33036
[19] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1992. · Zbl 0804.28001
[20] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, no. 5, North-Holland, Amsterdam, The Netherlands, 1973. · Zbl 0252.47055