  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
TVLp 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.
