Image processing with partial differential equations

In many applications computers analyse images or image sequences which are often contaminated by noise, and their quality can be poor (e.g. in medical imaging). We discuss how nonlinear partial differential equations (PDEs) can be used to automatically produce an image of much higher quality, enhance its sharpness, filter out the noise, extract shapes, etc. The models are based on the well-known Perona-Malik image selective smoothing equation and on geometrical equations of mean curvature flow type. Since the images are given on discrete grids, PDEs are discretized by variational techniques, namely by the semi-implicit finite element, finite volume and complementary volume methods in order to get fast and stable solutions. Convergence of the schemes to variational solutions of these strongly nonlinear problems and the extension of the methods to adaptive scheme strategies improving computational efficiency are discussed. Computational results with artificial and real 2D, 3D images and image sequences are presented.

[1]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

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

[3]  Alessandro Sarti,et al.  Nonlinear Multiscale Analysis of 3D Echocardiographic Sequences , 1999, IEEE Trans. Medical Imaging.

[4]  Joachim Weickert,et al.  Scale-Space Theories in Computer Vision , 1999, Lecture Notes in Computer Science.

[5]  Stephen Wolfram,et al.  The Mathematica book (3rd ed.) , 1996 .

[6]  Fiorella Sgallari,et al.  Variational Numerical Methods for Solving Nonlinear Diffusion Equations Arising in Image Processing , 2002, J. Vis. Commun. Image Represent..

[7]  Kunibert G. Siebert,et al.  ALBERT---Software for scientific computations and applications. , 2001 .

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

[9]  Gerhard Dziuk,et al.  CONVERGENCE OF A SEMI-DISCRETE SCHEME FOR THE CURVE SHORTENING FLOW , 1994 .

[10]  Luis Alvarez,et al.  Formalization and computational aspects of image analysis , 1994, Acta Numerica.

[11]  Alessandro Sarti,et al.  Numerical solution of parabolic equations related to level set formulation of mean curvature flow , 1998 .

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

[13]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

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

[15]  D. Evans.  Preconditioning Methods: Theory and Applications , 1983 .

[16]  Joachim Weickert,et al.  Coherence-enhancing diffusion of colour images , 1999, Image Vis. Comput..

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

[18]  Ricardo H. Nochetto,et al.  SHARP ERROR ANALYSIS FOR CURVATURE DEPENDENT EVOLVING FRONTS , 1993 .

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

[20]  Karol Mikula,et al.  Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing , 2001, Numerische Mathematik.

[21]  Martin Rumpf,et al.  DIFFUSION MODELS AND THEIR ACCELERATED SOLUTION IN IMAGE AN D SURFACE PROCESSING , 2000 .

[22]  Karol Mikula,et al.  Evolution of Convex Plane Curves Describing Anisotropic Motions of Phase Interfaces , 1996, SIAM J. Sci. Comput..

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

[24]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[25]  F. Guichard.  Axiomatisation des analyses multi-échelles d'images et de films , 1994 .

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

[27]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[28]  Karol Mikula,et al.  Solution of nonlinearly curvature driven evolution of plane curves , 1999 .

[29]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[30]  J. Kacur,et al.  Slow and fast diffusion effects in image processing , 2001 .

[31]  G. Dziuk,et al.  An algorithm for evolutionary surfaces , 1990 .

[32]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[33]  Thomas S. Huang,et al.  Image processing , 1971 .

[34]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[35]  Eberhard Bänsch,et al.  Adaptivity in 3D image processing , 2001 .

[36]  K. Mikula,et al.  A coarsening finite element strategy in image selective smoothing , 1997 .

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

[38]  Jozef Kačur,et al.  Method of rothe in evolution equations , 1986 .

[39]  Gene H. Golub,et al.  Matrix computations , 1983 .

[40]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[41]  Ron Kimmel,et al.  Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images , 2000, International Journal of Computer Vision.

[42]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[43]  Karol Mikula,et al.  Solution of nonlinear curvature driven evolution of plane convex curves , 1997 .

[44]  J. Kacur,et al.  Solution of nonlinear diffusion appearing in image smoothing and edge detection , 1995 .

[45]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.

[46]  Fiorella Sgallari,et al.  Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution , 2003, Numerische Mathematik.

[47]  Jean-Michel Morel,et al.  Introduction To The Special Issue On Partial Differential Equations And Geometry-driven Diffusion In Image Processing And Analysis , 1998, IEEE Trans. Image Process..

[48]  Karol Mikula,et al.  Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy , 2001, SIAM J. Appl. Math..

[49]  J. Koenderink.  The structure of images , 2004, Biological Cybernetics.

[50]  Masato Kimura,et al.  Numerical analysis of moving boundary problems using the boundary tracking method , 1997 .

[51]  Martin Rumpf,et al.  An Adaptive Finite Element Method for Large Scale Image Processing , 1999, J. Vis. Commun. Image Represent..

[52]  Baba C. Vemuri,et al.  Shape Modeling with Front Propagation: A Level Set Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[53]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

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

[55]  木村 正人,et al.  Numerical analysis for moving boundary problems using the boundary tracking method , 1996 .

[56]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[57]  P. Schönemann.  On artificial intelligence , 1985, Behavioral and Brain Sciences.

[58]  Eberhard Bänsch,et al.  Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..

[59]  Karol Mikula,et al.  An Adaptive Finite Volume Scheme for Solving Nonlinear Diffusion Equations in Image Processing , 2002, J. Vis. Commun. Image Represent..

[60]  N. Walkington.  Algorithms for Computing Motion by Mean Curvature , 1996 .

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

[62]  M. Beneš,et al.  Mathematical and computational aspects of solidification of pure substances. , 2001 .