Collaborative Total Variation: A General Framework for Vectorial TV Models

TL;DR

This paper introduces a flexible framework called collaborative total variation (CTV) for vector-valued image regularization, analyzing its mathematical properties and demonstrating its effectiveness in various image restoration tasks.

Contribution

It develops a general class of vectorial TV models based on tensor norms, characterizes their mathematical properties, and shows their superiority in reducing color artifacts.

Findings

Channel coupling with $$ norm reduces color artifacts.

CTV models outperform traditional TV in denoising, deblurring, and inpainting.

Theoretical analysis supports the effectiveness of the proposed regularizers.

Abstract

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.