A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

Martin Burger, Yury Korolev, Carola-Bibiane Schönlieb, Christiane Stollenwerk

Abstract: We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers $${{\,\mathrm{{{\,\mathrm{TVL}\,}}^p}\,}}$$ and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.