Anisotropic Image Sharpening via Well-Posed Sobolev Gradient Flows

We study well-posed perturbations of formally ill-posed diffusion equations which are used in image processing, such as the Perona–Malik equation. Our perturbation technique is to consider the diffusion equations as $L^2$ gradient flows on integral functionals and then modify the inner product from $L^2$ to a Sobolev inner product. We show that the functional differential equations obtained in this way are well-posed in both the forward and backward directions. We then show how to design a well-posed image sharpening algorithm via Sobolev gradient ascent on a Perona–Malik type functional. We provide full numerical implementation details and show experimental results on natural images which suggest that this method outperforms previous work by the authors as well as other sharpening algorithms such as the shock filter.

[1]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[2]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[3]  Kewei Zhang,et al.  Young measure solutions of the two-dimensional Perona-Malik equation in image processing , 2006 .

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  Antonin Chambolle,et al.  Image Decomposition into a Bounded Variation Component and an Oscillating Component , 2005, Journal of Mathematical Imaging and Vision.

[6]  K. Höllig.  Existence of infinitely many solutions for a forward backward heat equation , 1983 .

[7]  Massimo Gobbino,et al.  Entire Solutions of the One-Dimensional Perona-Malik Equation , 2007 .

[8]  Catherine Kublik,et al.  Coarsening in High Order, Discrete, Ill-Posed Diffusion Equations , 2010 .

[9]  Yehoshua Y. Zeevi,et al.  Forward-and-backward diffusion processes for adaptive image enhancement and denoising , 2002, IEEE Trans. Image Process..

[10]  Joachim Weickert,et al.  Coherence-Enhancing Diffusion Filtering , 1999, International Journal of Computer Vision.

[11]  Walter B. Richardson,et al.  High-order Sobolev preconditioning , 2005 .

[12]  Satyanad Kichenassamy,et al.  The Perona-Malik Paradox , 1997, SIAM J. Appl. Math..

[13]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[14]  Joachim Weickert,et al.  Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering , 2009, SSVM.

[15]  Nonlinear functional analysis and its applications, part I: Fixed-point theorems , 1991 .

[16]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[17]  Anthony J. Yezzi,et al.  Sobolev Active Contours , 2005, International Journal of Computer Vision.

[18]  John B. Greer,et al.  Traveling Wave Solutions of Fourth Order PDEs for Image Processing , 2004, SIAM J. Math. Anal..

[19]  Jean-Philippe Pons,et al.  Generalized Gradients: Priors on Minimization Flows , 2007, International Journal of Computer Vision.

[20]  L. Lieu,et al.  Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces , 2008 .

[21]  Massimo Gobbino,et al.  Gradient estimates for the Perona–Malik equation , 2007 .

[22]  Anthony J. Yezzi,et al.  Image Sharpening via Sobolev Gradient Flows , 2010, SIAM J. Imaging Sci..

[23]  M. Morini,et al.  Mumford–Shah Functional as Γ-Limit of Discrete Perona–Malik Energies , 2003 .

[24]  John William Neuberger,et al.  Sobolev gradients and differential equations , 1997 .

[25]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[26]  Massimo Gobbino,et al.  A class of local classical solutions for the one-dimensional Perona-Malik equation , 2009 .

[27]  Anthony J. Yezzi,et al.  Tracking With Sobolev Active Contours , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[28]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  Selim Esedoglu,et al.  Stability Properties of the Perona-Malik Scheme , 2006, SIAM J. Numer. Anal..

[30]  S. Esedoglu.  An analysis of the Perona‐Malik scheme , 2001 .

[31]  Mostafa Kaveh,et al.  Fourth-order partial differential equations for noise removal , 2000, IEEE Trans. Image Process..

[32]  G. Bellettini,et al.  A REGULARIZED PERONA-MALIK FUNCTIONAL: SOME ASPECTS OF THE GRADIENT DYNAMICS , 2005 .

[33]  Kewei Zhang,et al.  Young measure solutions and instability of the one-dimensional Perona–Malik equation , 2005 .

[34]  Andrea Braides.  Gamma-Convergence for Beginners , 2002 .

[35]  Sophia Demoulini,et al.  Young measure solutions for a nonlinear parabolic equation of forward-backward type , 1996 .

[36]  W. Richardson.  Sobolev gradient preconditioning for image‐processing PDEs , 2006 .

[37]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[38]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[39]  Padrón Victor,et al.  SOBOLEV REGULARIZATION OF A NONLINEAR ILL-POSED PARABOLIC PROBLEM AS A MODEL FOR AGGREGATING POPULATIONS , 1998 .

[40]  Guillermo Sapiro,et al.  New Possibilities with Sobolev Active Contours , 2007, International Journal of Computer Vision.

[41]  Kewei Zhang,et al.  Existence of infinitely many solutions for the one-dimensional Perona-Malik model , 2006 .

[42]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[43]  Stanley Osher,et al.  Shocks and other nonlinear filtering applied to image processing , 1991, Optics & Photonics.

[44]  L. Álvarez,et al.  Signal and image restoration using shock filters and anisotropic diffusion , 1994 .

[45]  F. Browder.  Nonlinear Functional Analysis and Its Applications, Part 1 , 1986 .

[46]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[47]  Jingxue Yin,et al.  Young measure solutions of a class of forward–backward diffusion equations , 2003 .

[48]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[49]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .