  For nonconvex and nonsmooth restoration models, the lower bound theory
reveals their good edge recovery ability, and related analysis can help to
design convergent algorithms. Existing such discussions are focused on
isotropic regularization models, or only the lower bound theory of anisotropic
model with a quadratic fidelity. In this paper, we consider a general image
recovery model with a non-Lipschitz anisotropic composite regularization term
and an $\ell_q$ norm ($1\leq q<+\infty$) data fidelity term. We establish the
lower bound theory for the anisotropic model with an $\ell_1$ fidelity, which
applies to impulsive noise removal problems. For the general case with $1\leq
q<+\infty$, a support inclusion analysis is provided. To solve this
non-Lipschitz composite minimization model, we are then motivated to introduce
a support shrinking strategy in the iterative algorithm and relax the support
constraint to a thresholding support constraint, which is more computationally
practical. The objective function at each iteration is also linearized to
construct a strongly convex subproblem. To make the algorithm more
implementable, we compute an approximation solution to this subproblem at each
iteration, but not exactly solve it. The global convergence result of the
proposed inexact iterative thresholding and support shrinking algorithm with
proximal linearization is established. The experiments on image restoration and
two stage image segmentation demonstrate the effectiveness of the proposed
algorithm.
