The Community for Technology Leaders
RSS Icon
Issue No.12 - December (2008 vol.30)
pp: 2175-2187
A general scale space algorithm is presented for denoising signals and images with spatially varying dominant scales. The process is formulated as a partial differential equation with spatially varying time. The proposed adaptivity is semi-local and is in conjunction with the classical gradient-based diffusion coefficient, designed to preserve edges. The new algorithm aims at maximizing a local SNR measure of the denoised image. It is based on a generalization of a global stopping time criterion presented recently by the author and colleagues. Most notably, the method works well also for partially textured images and outperforms any selection of a global stopping time. Given an estimate of the noise variance, the procedure is automatic and can be applied well to most natural images.
Smoothing, Parabolic equations, Partial Differential Equations
Guy Gilboa, "Nonlinear Scale Space with Spatially Varying Stopping Time", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 12, pp. 2175-2187, December 2008, doi:10.1109/TPAMI.2008.23
[1] L. Alvarez, F. Guichard, P.L. Lions, and J.-M. Morel, “Axioms and Fundamental Equations of Image Processing,” Archive for Rational Mechanics and Analysis, vol. 123, no. 3, pp. 199-257, 1993.
[2] G. Aubert and P. Kornprobst, “Mathematical Problems in Image Processing,” Applied Math. Sciences, vol. 147, 2002.
[3] J.F. Aujol, G. Aubert, L. Blanc-Féraud, and A. Chambolle, “Image Decomposition into a Bounded Variation Component and an Oscillating Component,” J. Math. Imaging and Vision, vol. 22, no. 1, Jan. 2005.
[4] J.F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-Texture Image Decomposition—Modeling, Algorithms, and Parameter Selection,” Int'l J. Computer Vision, vol. 67, no. 1, pp. 111-136, 2006.
[5] D. Barash, “A Fundamental Relationship between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Diffusion Equation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 6, pp. 844-847, June 2002.
[6] M.J. Black, G. Sapiro, D. Marimont, and D. Heeger, “Robust Anisotropic Diffusion,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 421-432, 1998.
[7] M.J. Black and G. Sapiro, “Edges as Outliers: Anisotropic Smoothing Using Local Image Statistics,” Proc. Second Int'l Scale-Space Theories in Computer Vision, pp. 259-270, 1999.
[8] M. Burger, G. Gilboa, S. Osher, and J. Xu, “Nonlinear Inverse Scale Space Methods,” Comm. Math. Sciences, vol. 4, no. 1, pp. 179-212, 2006.
[9] A. Chambolle and B.J. Lucier, “Interpreting Translation-Invariant Wavelet Shrinkage as a New Imagesmoothing Scale Space,” IEEE Trans. Image Processing, vol. 10, no. 7, pp. 993-1000, 2001.
[10] T.F. Chan and J. Shen, Image Processing and Analysis. SIAM, 2005.
[11] H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, 1996.
[12] I.A. Frigaard, G. Ngwa, and O. Scherzer, “On Effective Stopping Time Selection for Visco-Plastic Nonlinear Diffusion Filters Used in Image Denoising,” SIAM J. Applied Math., vol. 63, no. 6, pp.1911-1934, 2003.
[13] G. Gilboa, N. Sochen, and Y.Y. Zeevi, “Image Enhancement and Denoising by Complex Diffusion Processes,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 8, pp. 1020-1036, Aug. 2004.
[14] G. Gilboa, N. Sochen, and Y.Y. Zeevi, “Estimation of Optimal PDE-Based Denoising in the SNR Sense,” IEEE Trans. Image Processing, vol. 15, no. 8, pp. 2269-2280, 2006.
[15] G. Gilboa, N. Sochen, and Y.Y. Zeevi, “Variational Denoising of Partly-Textured Images by Spatially Varying Constraints,” IEEE Trans. Image Processing, vol. 15, no. 8, pp. 2280-2289, 2006.
[16] B. Hamza and A. Krim, “Image Denoising: A Nonlinear Robust Statistical Approach,” IEEE Trans. Signal Processing, vol. 49, pp.3045-3054, 2001.
[17] T. Iijima, “Basic Theory of Pattern Observation,” papers of Technical Group on Automata and Automatic Control, IECE, Japan (in Japanese), 1959.
[18] P.T. Jackway and M. Deriche, “Scale-Space Properties of the Multiscale Morphological Dilation-Erosion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 1, Jan. 1996.
[19] C. Kervrann, “An Adaptive Window Approach for Image Smoothing and Structures Preserving,” Proc. Eighth European Conf. Computer Vision, pp. 132-144, 2004.
[20] Kodak, Kodak Image Collection, , 2002.
[21] J.J. Koenderink, “The Structure of Images,” Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[22] Y.G. Leclerc, “Constructing Simple Stable Descriptions for Image Partitioning,” Int'l J. Computer Vision, vol. 3, no. 1, pp. 73-102, 1989.
[23] T. Lindeberg, “Feature Detection with Automatic Scale Selection,” Int'l J. Computer Vision, vol. 30, no. 2, pp. 79-116, 1998.
[24] F. Meyer and P. Maragos, “Nonlinear Scale-Space Representation with Morphological Levelings,” J. Visual Comm. and Image Representation, vol. 11, no. 2, pp. 245-265, 2000.
[25] Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, Mar. 2001.
[26] V.A. Morozov, “On the Solution of Functional Equations by the Method of Regularization,” Soviet Math. Dokl., vol. 7, pp. 414-417, 1966.
[27] P. Mrázek and M. Navara, “Selection of Optimal Stopping Time for Nonlinear Diffusion Filtering,” Int'l J. Computer Vision, vol. 52, nos. 2-3, pp. 189-203, 2003.
[28] S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An Iterative Regularization Method for Total Variation Based Image Restoration,” SIAM J. Multiscale Modeling and Simulation, vol. 4, pp. 460-489, 2005.
[29] Geometric Level Set Methods in Imaging, Vision, and Graphics, S.Osher and N. Paragios, eds. Springer, 2003.
[30] G. Papandreou and P. Maragos, “A Cross-Validatory Statistical Approach to Scale Selection for Image Denoising by Nonlinear Diffusion,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 625-630, 2005.
[31] P. Perona and J. Malik, “Scale-Space and Edge Detection Using Anisotropic Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629-639, July 1990.
[32] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection. John Wiley & Sons, 1986.
[33] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear Total Variation Based Noise Removal Algorithms,” Physica D, vol. 60, pp. 259-268, 1992.
[34] G. Sapiro, Geometric Partial Differential Equations and Image Processing. Cambridge Univ. Press, 2001.
[35] G. Sapiro and A. Tannenbaum, “Affine Invariant Scale-Space,” Int'l J. Computer Vision, vol. 11, no. 1, pp. 25-44, 1993.
[36] O. Scherzer and J. Weickert, “Relation between Regularization and Diffusion Filtering,” J. Math. Imaging and Vision, vol. 12, pp. 43-63, 2000.
[37] J. Shi and J. Malik, “Normalized Cuts and Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, Aug. 2000.
[38] N. Sochen, R. Kimmel, and R. Malladi, “A General Framework for Low Level Vision,” IEEE Trans. Image Processing, vol. 7, pp. 310-318, 1998.
[39] G. Steidl, J. Weickert, T. Brox, P. Mrázek, and M. Welk, “On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs,” SIAM J. Numerical Analysis, vol. 42, no. 2, pp. 686-713, 2004.
[40] E. Tadmor, S. Nezzar, and L. Vese, “A Multiscale Image Representation Using Hierarchical (BV, L2) Decompositions,” SIAM J. Multiscale Modeling and Simulation, vol. 2, no. 4, pp. 554-579, 2004.
[41] L. Vese and S. Osher, “Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing,” J.Scientific Computing, vol. 19, pp. 553-572, 2003.
[42] J. Weickert, “A Review of Nonlinear Diffusion Filtering,” Proc. First Int'l Conf. Scale-Space Theory in Computer Vision, pp. 3-28, 1997.
[43] J. Weickert, Anisotropic Diffusion in Image Processing. Teubner, 1998.
[44] J. Weickert, “Coherence-Enhancing Diffusion of Colour Images,” Image and Vision Computing, vol. 17, pp. 201-212, 1999.
[45] J. Weickert, S. Ishikawa, and A. Imiya, “On the History of Gaussian Scale-Space Axiomatics,” Gaussian Scale-Space Theory, J.Sporring, M. Nielsen, L.M. Florack, and P. Johansen, eds., pp.45-59, Kluwer Academic, 1997.
[46] A.P. Witkin, “Scale-Space Filtering,” Proc. Eighth Int'l Joint Conf. Artificial Intelligence, pp. 1019-1023, 1983.
18 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool