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Image decomposition and restoration using total variation minimization and the H -1 norm. (English) Zbl 1051.49026
Summary: In this paper, we propose a new model for image restoration and image decomposition into cartoon and texture, based on the total variation minimization of L. I. Rudin, S. Osher and E. Fatemi [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] and on oscillatory functions, which follows results of Y. Meyer [“Oscillating pattern in image processing and nonlinear evolution equations” (2001; Zbl 0987.35003)]. This paper also continues the ideas introduced by the authors in a previous work on image decomposition models into cartoon and texture [L. A. Vese and S. Osher, J. Sci. Comput. 19, No. 1–3, 553–572 (2003; Zbl 1034.49039)]. Indeed, by an alternative formulation, an initial image f is decomposed here into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component is modeled by an oscillatory function, bounded in the norm dual to |·| H 0 1 . After some transformation, the resulting PDE is of fourth order, involving the Laplacian of the curvature of level lines. Finally, image decomposition, denoising, and deblurring numerical results are shown.

MSC:
49N90Applications of optimal control and differential games
35J35Higher order elliptic equations, variational problems
49Q20Variational problems in a geometric measure-theoretic setting
49J45Optimal control problems involving semicontinuity and convergence; relaxation
68U10Image processing (computing aspects)
94A08Image processing (compression, reconstruction, etc.)