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Adaptively weighted discrete Laplacian for inverse rendering

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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.

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We use publicly available data (see the information on tested 3D models in the acknowledgments), and our source code will be available at


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We appreciate the reviewers for their helpful comments and thank the following providers of the tested 3D models: Stanford 3D repository (Armadillo, Bunny, Lucy), Oliver Laric (Dark-finger-reef-crab, Elbo-crab), Smithsonian National Museum of Natural History (Cranium), yeg3d (Deer), and Natural History Museum Vienna (Smilodon).


This work was supported in part by Ministry of Culture, Sports and Tourism and Korea Creative Content Agency (No. R2021080001) and by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2022-0-00566).

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An, H., Lee, W. & Moon, B. Adaptively weighted discrete Laplacian for inverse rendering. Vis Comput 39, 3211–3220 (2023).

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