  Even after over 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 to move 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 extend, the differences to other pixels, and the spectral channels.
The smoothness of this tensor is then measured by taking different norms along
the different dimensions. Depending on the type 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
like denoising, deblurring and inpainting.
