Inverse Total Variation Flow

Abstract

In this paper we analyze iterative regularization with the Bregman distance of the total variation seminorm. Moreover, we prove existence of a solution of the corresponding flow equation as introduced in [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with ${\rm L}^p$-norm fit-to-data terms and Bregman distance regularization terms. For the associated flow equations well-posedness is derived using recent results on metric gradient flows from [L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005]. In contrast to previous work the results of this paper apply for the analysis of variational denoising methods with the Bregman distance under adequate noise assumptions. Aside from the theoretical results we introduce a level set technique based on Bregman distance regularization for denoising of surfaces and demonstrate the efficiency of this method. (A corrected version of this paper has been appended to the PDF file.)