Adaptively weighted discrete Laplacian for inverse rendering

Hyeonjang An , Wonjun Lee , Bochang Moon

Gwangju Institute of Science and Technology

The Visual Computer

Reconstruction results of the *Dark-finger-reef-crab* using an inverse rendering framework.
Reconstruction results of the *Dark-finger-reef-crab* using an inverse rendering framework, LSIG[8], which employs the discrete Laplacian of the mesh in their optimization, a pre-conditioned gradient descent. We vary the discrete Laplacian into the uniform Laplacian (LSIG-uni), the cotangent Laplacian (LSIG-cot)[3, 10], and our method. We plot Mean Absolute Errors (MAEs) over 1.2K iterations, and the three zoomed images are visualized results from each method after 1.2K iterations. As can be seen in the figure, our method boosts the convergence of LSIG compared to the other tested Laplacians while capturing fine structures of the mesh. Also, our improved reconstruction quality is achieved in a shorter reconstruction time (22 secs) than those of LSIG-uni and LSIG-cot (38 secs and 36 secs) due to our fewer number of vertices than the others.


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.