Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.

[1]  Tai Sing Lee,et al.  Texture Segmentation by Minimizing Vector-Valued Energy Functionals: The Coupled-Membrane Model , 1992, ECCV.

[2]  François Malgouyres,et al.  Combining total variation and wavelet packet approaches for image deblurring , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[3]  Guillermo Sapiro,et al.  Color Snakes , 1997, Comput. Vis. Image Underst..

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

[5]  Song-Chun Zhu.  Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling , 1998 .

[6]  T. S. Lee,et al.  A Bayesian framework for understanding texture segmentation in the primary visual cortex , 1995, Vision Research.

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

[8]  Frédérique Oru.  Rôle des oscillations dans quelques problèmes d'analyse non-linéaire , 1998 .

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

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

[11]  L. Vese,et al.  A Variational Method in Image Recovery , 1997 .

[12]  J. Morel,et al.  A multiscale algorithm for image segmentation by variational method , 1994 .

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

[14]  L. Vese.  A Study in the BV Space of a Denoising—Deblurring Variational Problem , 2001 .

[15]  F. Malgouyres,et al.  Mathematical analysis of a model which combines total variation and wavelet for image restoration 1 , 2002 .

[16]  Jesús Ildefonso Díaz Díaz,et al.  Some qualitative properties for the total variation flow , 2002 .

[17]  Manuel González,et al.  Affine Invariant Texture Segmentation and Shape from Texture by Variational Methods , 1998, Journal of Mathematical Imaging and Vision.

[18]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

[19]  W. Eric L. Grimson,et al.  Adaptive Segmentation of MRI Data , 1995, CVRMed.

[20]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[21]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

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

[23]  V. Caselles,et al.  Minimizing total variation flow , 2000, Differential and Integral Equations.

[24]  Song-Chun Zhu,et al.  Minimax Entropy Principle and Its Application to Texture Modeling , 1997, Neural Computation.

[25]  Tony F. Chan,et al.  A Level-Set and Gabor-based Active Contour Algorithm for Segmenting Textured Images , 2002 .

[26]  Pietro Perona,et al.  Boundary Detection in Piecewise Homogeneous Textured Images , 1992, ECCV.

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

[28]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[29]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .