Multi-channel/modality image joint reconstruction has gained much research interest in recent years. In this paper, we propose to use a nonconvex and nonLipschitz joint regularizer in a general variational model for joint reconstruction under additive measurement noise. This framework has good ability in edge-preserving
by sharing common edge features of individual images. We study the lower bound
theory for the non-Lipschitz joint reconstruction model in two important cases with
Gaussian and impulsive measurement noise, respectively. In addition, we extend previous works to propose an inexact iterative support shrinking algorithm with proximal linearization for multi-channel image reconstruction (InISSAPL-MC) and prove
that the iterative sequence converges globally to a critical point of the original objective
function. In a special case of single channel image restoration, the convergence result
improves those in the literature. For numerical implementation, we adopt primal dual
method to the inner subproblem. Numerical experiments in color image restoration
and two-modality undersampled magnetic resonance imaging (MRI) reconstruction
show that the proposed non-Lipschitz joint reconstruction method achieves considerable improvements in terms of edge preservation for piecewise constant images compared to existing methods.
