Color Bregman TV

In this paper we present a novel iterative procedure for multichannel image and data reconstruction using Bregman distances. The motivation for our approach is that in many applications multiple channels share a common subgradient with respect to a suitable regularization. This implies desirable properties such as a common edge set (and a common direction of the normals to the level lines) in the case of the total variation (TV). Therefore, we propose to determine each iterate by regularizing each channel with a weighted linear combination of Bregman distances to all other image channels from the previous iteration. In this sense we generalize the Bregman iteration proposed by Osher et al. in [Multiscale Model. Simul., 4 (2005), pp. 460-489] to multichannel images. We prove the convergence of the proposed scheme, analyze stationary points, and present numerical experiments on color image denoising, which show the superior behavior of our approach in comparison to TV, TV with Bregman iterations on each channel separately, and vectorial TV. Further numerical experiments include image deblurring and image inpainting. Additionally, we propose using the infimal convolution of Bregman distances to different channels from the previous iteration to obtain the independence of the sign and hence the independence of the direction of the edge. While this work focuses on TV regularization, the proposed scheme can potentially improve any variational multichannel reconstruction method with a one-homogeneous regularization.