# Image Restoration by Combined Order Regularization with Optimal Spatial Adaptation

## TL;DR

This paper introduces a novel non-convex regularization method for image restoration that adaptively combines Hessian-Schatten norm and TV1 with spatially varying weights, improving performance in applications like MRI and microscopy.

## Contribution

It proposes a new adaptive non-convex regularization functional with a joint minimization approach and convergence proof, advancing multi-order image restoration techniques.

## Key findings

- Improved image recovery in MRI and microscopy tasks.
- Effective adaptive weighting enhances restoration quality.
- Convergence of the proposed optimization algorithm is proven.

## Abstract

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.