The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory

Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by $\text{TV}_J$, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that $\text{TV}_J$ can be derived from the generalized Jacobians from geometric measure theory. Thus, within the context of this theory, $\text{TV}_J$ is the most natural form of a vectorial total variation. As an important feature, we derive how $\text{TV}_J$ can be written as the support functional of a convex set in $\mathcal{L}^2$. This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist of an extensive experimental section, where we compare the performance of a number of provable convergent algorithms for inverse problems with our proposed regularizer. In particular, we show in experiments for denoising, deblurring, superresolution, and inpainting that its use leads to a significantly better restoration of color images, both visually and quantitatively. Source code for all algorithms employed in the experiments is provided online.