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Zbl 1051.49026
Osher, Stanley; Solé, Andrés; Vese, Luminita
Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm. (English)
[J] Multiscale Model. Simul. 1, No. 3, 349-370 (2003). ISSN 1540-3459; ISSN 1540-3467/e

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.

MSC 2000:: *49N90 Applications of optimal control and differential games
35J35 Higher order elliptic equations, variational problems
49Q20 Variational problems in geometric measure-theoretic setting
49J45 Optimal control problems inv. semicontinuity and convergence
68U10 Image processing
94A08 Image processing

Keywords: total variation; image decomposition; cartoon; texture; restoration; functional minimization; functions of bounded variation

Citations: Zbl 0780.49028; Zbl 0987.35003; Zbl 1034.49039

Cited in: Zbl 1217.65172 Zbl 1177.94035 Zbl 1171.94002 Zbl 1148.35329 Zbl 1134.35315 Zbl 1079.68104

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