用于总缩放梯度变分模型的锥约束双线性分解

中文摘要 AI 辅助

总缩放梯度变分（TSGV）正则化项源于分段线性结构的稀疏建模，已被证明可在图像复原中保留边缘和角点。然而，其高度非凸且非线性的特性带来了严峻的计算挑战，现有方法常存在参数敏感性问题或缺乏收敛保证。为克服这一问题，我们提出一种定制的双线性分解方法，该方法可解耦TSGV正则化项中的非线性加权梯度。该方法会产生一个等价优化问题，其约束取决于所选缩放函数，可为锥约束或球约束；其中锥约束在表征边缘与角点保留特性方面起着核心作用。我们采用交替最小化方法（AMM）结合 majorization-minimization 策略求解该重构问题，确保能量单调下降且无需步长调整。此外，我们通过分析图像奇点附近的渐近行为，对这些约束的边缘保留特性给出了几何解释。我们在Kurdyka–Łojasiewicz框架内证明了所提方法收敛至临界点的全局收敛性。针对高斯去噪和非视距（NLOS）成像的大量数值实验表明，所提方法的峰值信噪比（PSNR）和结构相似性（SSIM）与代表性变分方法相当或更优，尤其在高噪声水平下表现突出，且能在密集和稀疏扫描下改善结构重建效果。

英文摘要

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.

发表机构

• Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
• School of Mathematical Sciences, Nankai University（南开大学数学科学学院）
• Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学与交叉科学研究院）

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