A Cone-Constrained Bilinear Decomposition for Total Scaled-Gradient Variation Models

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.

Non-line-of-sight imaging with arbitrary relay surface geometries via 3D Gaussian Transient Rendering

Imaging objects hidden outside the direct line of sight expands the effective field of view and is critical for applications such as autonomous driving and robotic perception. Despite impressive progress in time-of-flight (ToF)-based non-line-of-sight (NLOS) imaging, real-world deployment remains challenging because practical measurements are often collected over spatially limited, arbitrarily shaped relay regions-conditions that violate the planar-wall and dense-sampling assumptions made by most existing methods. To address these limitations, we propose a LOS-guided NLOS imaging pipeline that imposes no geometric assumptions on the relay surface and naturally supports both confocal and non-confocal configurations. Our method represents the hidden scene using 3D Gaussian primitives and couples them with an efficient, differentiable transient rendering model, enabling end-to-end optimization directly from measured transients. We validate our approach on real-world measurements from both a public dataset and a custom-built capture system. Across settings, our method achieves state-of-the-art reconstruction fidelity under spatially limited, sparsely sampled conditions, and significantly outperforms existing methods on complex, arbitrary relay surface geometries.