A Cone-Constrained Bilinear Decomposition for Total Scaled-Gradient Variation Models
Authors: Haibin Su, Chunlin Wu, Huibin Chang, Zhifang Liu
Submitted: 29 August 2026
Abstract
The total scaled-gradient variation (TSGV) regularizer, derived from the sparse modeling of piecewise-linear structures, has demonstrated strong potential for preserving edges and corners in image restoration. However, its inherent nonconvexity and nonlinearity present significant computational challenges. Existing optimization methods often suffer from parameter sensitivity or lack rigorous convergence guarantees, limiting their practical applicability.
To address these limitations, we propose a tailored bilinear decomposition that effectively decouples the nonlinear weighted gradient in the TSGV regularizer. This decomposition yields an equivalent optimization problem governed by cone or sphere constraints, depending on the chosen scaling function. Notably, the cone constraint plays a central role in characterizing edge- and corner-preserving behavior.
We solve this reformulated problem using an alternating minimization method (AMM) enhanced with a majorization–minimization strategy. This approach ensures a monotonic decrease in the energy function without the need for step-size tuning. In addition, we provide a geometric interpretation of the edge-preserving properties of these constraints by analyzing their asymptotic behavior near image singularities.
We establish the global convergence of the proposed method to a critical point within the Kurdyka–Łojasiewicz (KL) framework. Extensive numerical experiments on Gaussian denoising and non-line-of-sight (NLOS) imaging demonstrate that the proposed method achieves PSNR and SSIM values competitive with, or superior to, representative variational methods—particularly at high noise levels—and improves structural reconstruction under both dense and sparse scanning conditions.
Keywords
total scaled-gradient variation, image restoration, bilinear decomposition, cone constraint, alternating minimization, nonconvex optimization, Kurdyka–Łojasiewicz framework, NLOS imaging
via ArXiv CV