Grid-free Monte Carlo methods are capable of solving Poisson equations on highly complex domains. However, existing methods operate solely in the primal domain and can converge slowly due to high variance. Inspired by gradient-domain rendering, we introduce a gradient-domain framework for Poisson problems. Specifically, we devise a new Monte Carlo estimator that directly targets differences of the solution between spatially varying query locations. Further, we adopt state-of-the-art reconstruction techniques originated in gradient-domain rendering to allow efficient reconstruction of the solutions without incurring additional bias. We demonstrate the effectiveness of our technique by comparing solutions obtained using our method and several state-of-the-art baselines.

Physically based differentiable rendering has become an increasingly important topic in recent years. A common pipeline computes local color derivatives of light paths or pixels with respect to arbitrary scene parameters, and enables optimizing or recovering the scene parameters through iterative gradient descent by minimizing the difference between rendered and target images. However, existing approaches cannot robustly handle complex illumination effects including reflections, refractions, caustics, shadows, and highlights, especially when the initial and target locations of such illumination effects are not close to each other in the image space.

Traditional differentiable rendering approaches are usually hard to converge in inverse rendering optimizations, especially when initial and target object locations are not so close. Inspired by Lagrangian fluid simulation, we present a novel differentiable rendering method to address this problem. We associate each screen-space pixel with the visible 3D geometric point covered by the center of the pixel and compute derivatives on geometric points rather than on pixels. We refer to the associated geometric points as point proxies of pixels. For each point proxy, we compute its 5D RGBXY derivatives which measures how its 3D RGB color and 2D projected screen-space position change with respect to scene parameters. Furthermore, in order to capture global and long-range object motions, we utilize optimal transport based pixel matching to design a more sophisticated loss function. We have conducted experiments to evaluate the effectiveness of our proposed method on various inverse rendering applications and have demonstrated superior convergence behavior compared to state-of-the-art baselines.