A Class of Priors for Color Image Restoration Parameterized by Lie Groups Acting on Pixel Values

In a recent paper [T. Batard, G. Haro, and C. Ballester, SIAM J. Imag. Sci., 14 (2021), pp. 1816-1847], a new prior for image restoration was introduced. It relies first on the observation that an image and a degraded version of it can share some visual content and then on the conjecture that an image restoration model can benefit from the use of an image prior encoding this invariance property. This prior considers the restored image as a parallel section of a connection (also called covariant derivative), this latter being a critical point of an energy associated to the Lie group R+* x SO(2) acting on image pixel values. In this paper, we propose a twofold generalization of this result. First, we consider other Lie groups acting on image pixels, yielding new optimal connections. Then, we derive a family of alpha-connections from the optimal connections. The corresponding parallel sections describe new invariance properties which we use as priors encoded as penalty terms in variational models for image restoration. Experiments conducted on color image deblurring show that the proposed generalization of the work of Batard, Haro, and Ballester outperforms the original approach.