A total variation based regularizer promoting piecewise-Lipschitz reconstructions
This work addresses image reconstruction challenges, particularly in regularization methods, but is incremental as it builds upon existing total variation frameworks.
The authors tackled the problem of image reconstruction by introducing a new total variation-based regularizer that enforces a given Lipschitz constant, including spatial variation, and proved its regularizing properties. Their numerical experiments showed it achieves performance similar to total generalized variation while offering an intuitive free parameter interpretation and enabling spatially adaptive regularization.
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.