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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:
 49N90 Applications of optimal control and differential games 35J35 Higher order elliptic equations, variational problems 49Q20 Variational problems in a geometric measure-theoretic setting 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 68U10 Image processing (computing aspects) 94A08 Image processing (compression, reconstruction, etc.)