An improvement of Rudin–Osher–Fatemi model

Abstract

In this article, we investigate some mathematical properties of a new algorithm proposed by Meyer [Y. Meyer, Oscillating patterns in image processing and in some nonlinear evolution equations, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, University Lectures Series, vol. 22, Amer. Math. Soc., Providence, RI, 2001] to improve the Rudin–Osher–Fatemi model (ROF) [L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259–268] in order to separate objects and textures contained in an image. He pointed out the crucial role played by a certain norm called the G-norm or “dual norm,” denoted ‖ ‖ ∗ , and the main drawback for the ROF model: any image is considered to have a textured component. We are then interested in minimizing the functional ‖ u ‖ BV + λ ‖ v ‖ ∗ . The main Theorem 6.1 is about invariance and stability properties of the new algorithm. It was first implemented by Osher and Vese [L. Vese, S.J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, UCLA C.A.M. Report 02-19, 2002]. In particular, we point out the role played by particular functions called extremal functions and characterize them.