Collaborative Total Variation

Even after two decades, the total variation (TV) remains one of the most popular regularizations for image processing problems and has sparked a tremendous amount of research, particularly on moving from scalar to vector-valued functions. In this paper, we consider the gradient of a color image as a three-dimensional matrix or tensor with dimensions corresponding to the spatial extent, the intensity differences between neighboring pixels, and the spectral channels. The smoothness of this tensor is then measured by taking different norms along the different dimensions. Depending on the types of these norms, one obtains very different properties of the regularization, leading to novel models for color images. We call this class of regularizations collaborative total variation (CTV). On the theoretical side, we characterize the dual norm, the subdifferential, and the proximal mapping of the proposed regularizers. We further prove, with the help of the generalized concept of singular vectors, that an $\ell^{\infty}$ channel coupling makes the most prior assumptions and has the greatest potential to reduce color artifacts. Our practical contributions consist of an extensive experimental section, where we compare the performance of a large number of collaborative TV methods for inverse problems such as denoising, deblurring, and inpainting.