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

${|\xb7|}_{{H}_{0}^{1}}$. 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.