Adaptively weighted discrete Laplacian for inverse rendering

Reconstructing a triangular mesh from images by a differentiable rendering framework often exploits discrete Laplacians on the mesh, e.g., the cotangent Laplacian, so that a stochastic gradient descent-based optimization in the framework can become stable by a regularization term formed with the Laplacians. However, the stability stemming from using such a regularizer often comes at the cost of over-smoothing a resulting mesh, especially when the Laplacian of the mesh is not properly approximated, e.g., too-noisy or overly-smoothed Laplacian of the mesh. This paper presents a new discrete Laplacian built upon a kernel-weighted Laplacian. We control the kernel weights using a local bandwidth parameter so that the geometry optimization in a differentiable rendering framework can be improved by avoiding blurring high-frequency details of a surface. We demonstrate that our discrete Laplacian with a local adaptivity can improve the quality of reconstructed meshes and convergence speed of the geometry optimization by plugging our discrete Laplacian into recent differentiable rendering frameworks.