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Inverse total variation flow. (English) Zbl 1147.49026
Summary: 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 [{\it M. Burger, G. Gilboa, S. Osher}, and {\it J. Xu}, Commun. Math. Sci. 4, No. 1, 179--212 (2006; Zbl 1106.68117)] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with ${\text L}^p$-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 [{\it L. Ambrosio, N. Gigli}, and {\it G. G. Savaré}, Gradient flows in metric spaces and in the space of probability measures, Basel: Birkhäuser (2005; Zbl 1090.35002)]. 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.

MSC:
49N60Regularity of solutions in calculus of variations
35K90Abstract parabolic equations
47A52Ill-posed problems, regularization
68U10Image processing (computing aspects)
49J15Optimal control problems with ODE (existence)
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