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An improvement of Rudin-Osher-Fatemi model. (English) Zbl 1113.49037

Summary: We investigate some mathematical properties of a new algorithm proposed by Y. Meyer [Oscillating patterns in image processing and in some nonlinear evolution equations. University Lectures Series. Vol. 22, Providence, RI: AMS (2001; Zbl 0987.35003)] to improve the Rudin-Osher-Fatemi model (ROF) [L. Rudin, S. Osher and E. Fatemi, Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] 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 \(\| \, \|_{\ast}\), 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 \(\| u \|_{BV}+\lambda \| v \|_{*}\). The main Theorem 6.1 is about invariance and stability properties of the new algorithm. It was first implemented by L. Vese and S. J. Osher [J. Sci. Comput. 19, No. 1–3, 553–572 (2003; Zbl 1034.49039)]. In particular, we point out the role played by particular functions called extremal functions and characterize them.

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49M30 Other numerical methods in calculus of variations (MSC2010)
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