Behavioral analysis of anisotropic diffusion in image processing

In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.

[1]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[2]  V. Caselles,et al.  A geometric model for active contours in image processing , 1993 .

[3]  Guido Gerig,et al.  Vector-Valued Diffusion , 1994, Geometry-Driven Diffusion in Computer Vision.

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

[5]  Ross T. Whitaker,et al.  A multi-scale approach to nonuniform diffusion , 1993 .

[6]  J. Sethian.  AN ANALYSIS OF FLAME PROPAGATION , 1982 .

[7]  S. Angenent.  Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature , 1990 .

[8]  Benjamin B. Kimia,et al.  On the evolution of curves via a function of curvature , 1992 .

[9]  S. Angenent.  Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions , 1991 .

[10]  Lawrence C. Evans.  Estimates for smooth absolutely minimizing Lipschitz extensions. , 1993 .

[11]  Arch W. Naylor,et al.  Linear Operator Theory in Engineering and Science , 1971 .

[12]  W. Rudin.  Real and complex analysis , 1968 .

[13]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[14]  Peter Smith.  Convexity methods in variational calculus , 1985 .

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

[16]  M. Grayson.  The heat equation shrinks embedded plane curves to round points , 1987 .

[17]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[18]  Wenyuan Xu,et al.  Analysis and design of anisotropic diffusion for image processing , 1994, Proceedings of 1st International Conference on Image Processing.

[19]  M. Grayson.  Shortening embedded curves , 1989 .

[20]  J. Sethian.  Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws , 1990 .

[21]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[22]  K. Höllig,et al.  A Diffusion Equation with a Nonmonotone Constitutive Function , 1983 .

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

[24]  S. Zucker,et al.  Toward a computational theory of shape: an overview , 1990, eccv 1990.

[25]  Mark Nitzberg,et al.  Nonlinear Image Filtering with Edge and Corner Enhancement , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  J. Sethian.  Curvature and the evolution of fronts , 1985 .

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

[28]  P. Lions,et al.  TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS , 1992 .

[29]  Farzin Mokhtarian,et al.  A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[30]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[31]  F. Guichard,et al.  Axiomatisation et nouveaux opérateurs de la morphologie mathématique , 1992 .

[32]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[33]  M. Bertero,et al.  Ill-posed problems in early vision , 1988, Proc. IEEE.