Problem
Research questions and friction points this paper is trying to address.
Total Scaled-Gradient Variation
nonconvex
nonlinear
image restoration
convergence
A Cone-Constrained Bilinear Decomposition for Total Scaled-Gradient Variation Models
🤖 AI Summary
为解决TSGV正则化器的计算难题，提出了一种双线性分解方法，并通过交替最小化法求解，保证了收敛性和边缘保持性能。
📝 Abstract
The total scaled-gradient variation (TSGV) regularizer, derived from sparse modeling of piecewise-linear structures, has been shown to preserve edges and corners in image restoration. However, its highly nonconvex and nonlinear nature poses severe computational challenges, as existing methods often suffer from parameter sensitivity or lack convergence guarantees. To overcome this, we propose a tailored bilinear decomposition that decouples the nonlinear weighted gradient in the TSGV regularizer. This approach yields an equivalent optimization problem governed by cone or sphere constraints, depending on the chosen scaling function. In particular, the cone constraint plays a central role in characterizing edge- and corner-preserving behavior. We solve this reformulation using the alternating minimization method (AMM) equipped with a majorization--minimization strategy, ensuring a monotonic decrease in energy without step-size tuning. Furthermore, 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 framework. Extensive numerical experiments on Gaussian denoising and non-line-of-sight (NLOS) imaging show that the proposed method achieves PSNR and SSIM competitive with or superior to representative variational methods, especially at high noise levels, and improves the structural reconstruction under dense and sparse scanning.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Cone-Constrained Bilinear Decomposition
Total Scaled-Gradient Variation (TSGV)
Alternating Minimization Method (AMM)
Edge-Preserving
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