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 is an element of 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.