Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acar, R., and Vogel, C. R. (1994). Analysis of bounded variation penalty methods of ill-posed problems. Inverse Problem 10, 1217-1229.
Alvarez, L., Guichard, F., Lions, P. L., and Morel, J.-M. (1993). Axioms and fundamental equations of image-processing. Archive for Rational Mechanics and Analysis 123, 199-257.
Andreu, F., Ballester, C., Caselles, V., and Mazon, J. M. (2000). Minimizing total variation flow. CRAS I-Mathématique 331(11), 867-872.
Andreu, F., Caselles, V., Diaz, J. I., and Mazon, J. M. (2002). Some qualitative properties for the total variation flow. J. Func. Anal. 188(2), 516-547.
Aubert, G., and Vese, L. (1997). A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948-1979.
Ballester, C., and Gonzalez, M. (1998). Affine invariant texture segmentation and shape from texture by variational methods. J. Math. Imaging Vis. 9(2), 141-171.
Casadei, S., Mitter, S., and Perona, P. (1992). Boundary Detection in Piecewise Homogeneous Textured Images, Lecture Notes in Computer Science, Vol. 588, pp. 174-183.
Chan, T., and Vese, L. (1999). An active contour model without edges. In Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Vol. 1682, pp. 141-151.
Chan, T. F., and Vese, L. A. (2001). Active contours without edges. IEEE Transactions on Image Processing 10(2), 266-277.
Chambolle, A., and Lions, P. L. (1997). Image recovery via total variation minimization and related problems. Numer. Math. 76, 167-188.
Evans, L. C., and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions, CRC Press, London.
Koepfler, G., Lopez, C., and Morel, J.-M. (1994). A multiscale algorithm for image segmentation by variational method. SIAM J. Numer. Anal. 31(1), 282-299.
Lee, T. S. (1995). A Bayesian framework for understanding texture segmentation in the primary visual-cortex. Vision Research 35(18), 2643-2657.
Lee, T. S., Mumford, D., and Yuille, A. (1992). Texture Segmentation by Minimizing Vector-Valued Energy Functionals—the Coupled-Membrane Model, Lecture Notes in Computer Science, Vol. 588, pp. 165-173.
Malgouyres, F. (2002). Mathematical analysis of a model which combines total variation and wavelet for image restoration, Journal of Information Processes 2(1), 1-10.
Malgouyres, F. (2001). Combining total variation and wavelet packet approaches for image deblurring. Proceedings, IEEE Workshop on Variational and Level Set Methods in Computer Vision, 57-64.
Meyer, Y. (2002). Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, Vol. 22, Amer. Math. Soc.
Mumford, D., and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42(5), 577-685.
Oru, F. (1998). Le rôle des oscillations dans quelques problèmes d'analyse non-linéaire, Thèse, CMLA, ENS-Cachan, June 9, 1998.
Perona, P., and Malik, J. (1990). Scale-space and edge-detection using anisotropic diffusion. IEEE on PAMI 12(7), 629-639.
Rudin, L., and Osher, S. (1994). Total Variation Based Image Restoration with Free Local Constraints, Proc. IEEE ICIP, Vol. I, Austin (Texas) USA, pp. 31-35.
Rudin, L., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60, 259-268.
Sandberg, B., Chan, T., and Vese, L. (2002). A Level-Set and Gabor-Based Active Contour Algorithm for Segmenting Textured Images, UCLA CAM Report 02-39.
Sapiro, G. (1997). Color Snakes. Computer Vision and Image Understanding 68(2), 247-253.
Vese, L. (2001). A study in the BV space of a denoising-deblurring variational problem. Applied Mathematics and Optimization 44(2), 131-161.
Wells, W. M.,III, Grimson, W. E. L., Kikinis, R., and Jolesz, F. A. (1996). Adaptive Segmentation of MRI Data. IEEE Transactions on Medical Imaging, Vol. 15, No. 4.
Zhu, S. C., Wu, Y. N., and Mumford, D. (1998). Filters, random fields and maximum entropy (FRAME): Towards a unified theory for texture modeling. Intern. J. Comput. Vision 27(2), 107-126.
Zhu, S. C., Wu, Y. N., and Mumford, D. (1997). Minimax entropy principle and its application to texture modeling. Neural computation 9(8), 1627-1660.
Candés, E. J., and Guo, F. (2002). New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction. Signal Processing 82, 1519-1543.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vese, L.A., Osher, S.J. Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing. Journal of Scientific Computing 19, 553–572 (2003). https://doi.org/10.1023/A:1025384832106
Issue Date:
DOI: https://doi.org/10.1023/A:1025384832106