Regularity for solutions of the total variation denoising problem. (English) Zbl 1228.94005

The authors study the local regularity properties of a local minimizer of the functional, \[ \int_\Omega|Du|+ {\lambda\over 2} \int_\Omega|u(x)- f(x)|^2\,dx, \] where \(\Omega\) is an open set in \(\mathbb{R}^N\), \(\lambda> 0\), and \(f: \omega\to\mathbb{R}\) is locally Hölder continuous. The purpose of this paper is to prove that \(u\) is also locally Hölder continuous (with the same exponent). In addition is to prove a local Hölder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Hölder continuous. The authors also prove a global estimate on the modulus of continuity of the solution in convex domain of \(\mathbb{R}^N\) and some extensions of this result for the total variation minimization flow.


94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
49N60 Regularity of solutions in optimal control
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