Color Image Segmentation by the Vector-Valued Allen–Cahn Phase-Field Model: A Multigrid Solution

We present an efficient numerical solution of a PDE-driven model for color image segmentation and give numerical examples of the results. The method combines the vector-valued Allen-Cahn phase field equation with initial data fitting terms with prescribed interface width and fidelity constants. Efficient numerical solution is achieved using a multigrid splitting of a finite element space, thereby producing an efficient and robust method for the segmentation of large images. We also present the use of adaptive mesh refinement to further speed up the segmentation process.

[1]  Harald Garcke,et al.  On anisotropic order parameter models for multi-phase system and their sharp interface limits , 1998 .

[2]  Zeyun Yu,et al.  Anisotropic vector diffusion in image smoothing , 2002, Proceedings. International Conference on Image Processing.

[3]  Yoon Mo Jung,et al.  Multiphase Image Segmentation via Modica-Mortola Phase Transition , 2007, SIAM J. Appl. Math..

[4]  Luminita A. Vese,et al.  Multiphase Object Detection and Image Segmentation , 2003 .

[5]  Jean-François Aujol,et al.  Color image decomposition and restoration , 2006, J. Vis. Commun. Image Represent..

[6]  Jinchao Xu.  The method of subspace corrections , 2001 .

[7]  S. Esedoglu,et al.  Threshold dynamics for the piecewise constant Mumford-Shah functional , 2006 .

[8]  Tony F. Chan,et al.  An Active Contour Model without Edges , 1999, Scale-Space.

[9]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[10]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .

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

[12]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[13]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities I , 1994 .

[14]  K. Mikula,et al.  Geometrical image segmentation by the Allen-Cahn equation , 2004 .

[15]  Ralf Kornhuber,et al.  On multigrid methods for vector-valued Allen-Cahn equations , 2003 .

[16]  S. Osher,et al.  Geometric Level Set Methods in Imaging, Vision, and Graphics , 2011, Springer New York.

[17]  Ralf Kornhuber,et al.  Nonlinear Multigrid Techniques , 2001 .

[18]  James F. Blowey,et al.  Theory and Numerics of Differential Equations , 2001 .

[19]  Tony F. Chan,et al.  Total Variation Denoising and Enhancement of Color Images Based on the CB and HSV Color Models , 2001, J. Vis. Commun. Image Represent..

[20]  L. Vese,et al.  A level set algorithm for minimizing the Mumford-Shah functional in image processing , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[21]  Jean Petitot.  An introduction to the Mumford–Shah segmentation model , 2003, Journal of Physiology-Paris.

[22]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[23]  Ralf Kornhuber,et al.  Robust Multigrid Methods for Vector-valued Allen–Cahn Equations with Logarithmic Free Energy , 2006 .

[24]  Tony F. Chan,et al.  Variational PDE models in image processing , 2002 .

[25]  Tony F. Chan,et al.  Image segmentation using level sets and the piecewise-constant Mumford-Shah model , 2000 .

[26]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[27]  N. Fusco.  An Overview of the Mumford-Shah Problem , 2003 .

[28]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[29]  Steven J. Ruuth.  Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature , 1998 .

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

[31]  Guillermo Sapiro,et al.  Color image enhancement via chromaticity diffusion , 2001, IEEE Trans. Image Process..

[32]  Nicolas Neuss.  V-cycle Convergence with Unsymmetric Smoothers and Application to an Anisotropic Model Problem , 1998 .

[33]  Jinchao Xu,et al.  Global and uniform convergence of subspace correction methods for some convex optimization problems , 2002, Math. Comput..