This Article 
 Bibliographic References 
 Add to: 
Nonlinear Scale Space with Spatially Varying Stopping Time
December 2008 (vol. 30 no. 12)
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.

[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.

Index Terms:
Smoothing, Parabolic equations, Partial Differential Equations
Guy Gilboa, "Nonlinear Scale Space with Spatially Varying Stopping Time," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 12, pp. 2175-2187, Dec. 2008, doi:10.1109/TPAMI.2008.23
Usage of this product signifies your acceptance of the Terms of Use.