  Total Variation (TV) and related extensions have been popular in image
restoration due to their robust performance and wide applicability. While the
original formulation is still relevant after two decades of extensive research,
its extensions that combine derivatives of first- and second-order are now
being explored for better performance, with examples being Combined Order TV
(COTV) and Total Generalized Variation (TGV). As an improvement over such
multi-order convex formulations, we propose a novel non-convex regularization
functional which adaptively combines Hessian-Schatten (HS) norm and first order
TV (TV1) functionals with spatially varying weight. This adaptive weight itself
is controlled by another regularization term; the total cost becomes the sum of
this adaptively weighted HS-TV1 term, the regularization term for the adaptive
weight, and the data-fitting term. The reconstruction is obtained by jointly
minimizing w.r.t. the required image and the adaptive weight. We construct a
block coordinate descent method for this minimization with proof of
convergence, which alternates between minimization w.r.t. the required image
and the adaptive weights. We derive exact computational formula for
minimization w.r.t. the adaptive weight, and construct an ADMM algorithm for
minimization w.r.t. to the required image. We compare the proposed method using
image recovery examples including MRI reconstruction and microscopy
deconvolution.
