*(English)*Zbl 0969.65081

The authors consider a new numerical approach for the solution of the total variation minimization problem for denoising and/or deblurring of an image $u$ introduced by *L. I. Rudin*, *S. Osher* and *E. Fatemi* [Phys. D 60, 259-268 (1992; Zbl 0780.49028)]. The Euler-Lagrange equation for this problem takes the form of a stationary nonlinear diffusion equation. In all the following considerations the Euler-Lagrange equation is considered as the steady state of the corresponding time dependent equation.

It is observed by the authors that the right-hand side of the time evolution model in the original form suffers the drawback of a parabolic term which becomes singular for small gradients, which makes it necessary to choose small time-steps in the numerical realization. It is therefore proposed to regularize the singularity by multiplying the right-hand side by the factor $\left|\nabla u\right|$. The evolution model then has the form of a level set equation, yielding evolution of the level sets of the given image driven by a morphological convection term and a mean-curvature diffusion term.

The authors propose to calculate the steady state of the evolution equation by an explicit numerical scheme using upwind discretization for the convective an dcentral differencing for the diffusive terms. Fast convergence due to comparatively large CFL numbers and a reduced staircasing effect are the characteristic features of the presented numerical experiments.

##### MSC:

65M06 | Finite difference methods (IVP of PDE) |

65K10 | Optimization techniques (numerical methods) |

49J20 | Optimal control problems with PDE (existence) |

49M25 | Discrete approximations in calculus of variations |

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

35K55 | Nonlinear parabolic equations |