Non-convex regularization based on shrinkage penalty function

TL;DR

This paper introduces a non-convex shrinkage penalty for Hessian Schatten norm regularization to improve image recovery by reducing smoothing and staircase effects, with a provably convergent algorithm.

Contribution

It proposes a novel non-convex regularization method with a provably convergent algorithm for sharper image recovery, improving upon existing convex approaches.

Findings

Recovered images are sharper than convex counterparts.

The proposed method effectively reduces staircase effects.

Convergence of the algorithm is theoretically guaranteed.

Abstract

Total Variation regularization (TV) is a seminal approach for image recovery. TV involves the norm of the image's gradient, aggregated over all pixel locations. Therefore, TV leads to piece-wise constant solutions, resulting in what is known as the "staircase effect." To mitigate this effect, the Hessian Schatten norm regularization (HSN) employs second-order derivatives, represented by the pth norm of eigenvalues in the image hessian, summed across all pixels. HSN demonstrates superior structure-preserving properties compared to TV. However, HSN solutions tend to be overly smoothed. To address this, we introduce a non-convex shrinkage penalty applied to the Hessian's eigenvalues, deviating from the convex lp norm. It is important to note that the shrinkage penalty is not defined directly in closed form, but specified indirectly through its proximal operation. This makes constructing a provably convergent algorithm difficult as the singular values are also defined through a non-linear operation. However, we were able to derive a provably convergent algorithm using proximal operations. We prove the convergence by establishing that the proposed regularization adheres to restricted proximal regularity. The images recovered by this regularization were sharper than the convex counterparts.