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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 {\it L. I. Rudin}, {\it S. Osher} and {\it E. Fatemi} [Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)] and on oscillatory functions, which follows results of {\it 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 [{\it L. A. Vese} and {\it 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 $\vert \cdot \vert_{H^1_0}$. 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.

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.)
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