Neural Laplacian Operator for 3D Point Clouds
SessionNeural Shapes
DescriptionThe Laplacian operator holds a crucial role in 3D geometry processing, yet it is still challenging to define it on point clouds.
Previous works mainly focused on constructing a local triangulation around each point to approximate the underlying manifold for defining the Laplacian operator, which may not be very robust or accurate.
In contrast, we simply use the $K$-nearest neighbors (KNN) graph constructed from the input point cloud and learn the Laplacian operator on the KNN graph with graph neural networks (GNNs).
However, the ground-truth Laplacian operator is defined on a manifold mesh with a different connectivity from the KNN graph and thus cannot be directly used for training.
To train the GNN, we propose a novel training scheme by imitating the behavior of the ground-truth Laplacian operator on a set of probe functions so that the learned Laplacian operator behaves similarly to the ground-truth Laplacian operator.
We train our network on a subset of ShapeNet and evaluate it across a variety of point clouds.
Compared with previous methods, our method reduces the error by \emph{an order of magnitude} and excels in handling sparse point clouds with thin structures or sharp features.
Our method also demonstrates a strong generalization ability to unseen shapes.
With our learned Laplacian operator, we further apply a series of Laplacian-based geometry processing algorithms directly to point clouds and achieve accurate results, enabling many exciting possibilities for geometry processing on point clouds.
\emph{We will release our code and trained models to ensure reproducibility.}
Previous works mainly focused on constructing a local triangulation around each point to approximate the underlying manifold for defining the Laplacian operator, which may not be very robust or accurate.
In contrast, we simply use the $K$-nearest neighbors (KNN) graph constructed from the input point cloud and learn the Laplacian operator on the KNN graph with graph neural networks (GNNs).
However, the ground-truth Laplacian operator is defined on a manifold mesh with a different connectivity from the KNN graph and thus cannot be directly used for training.
To train the GNN, we propose a novel training scheme by imitating the behavior of the ground-truth Laplacian operator on a set of probe functions so that the learned Laplacian operator behaves similarly to the ground-truth Laplacian operator.
We train our network on a subset of ShapeNet and evaluate it across a variety of point clouds.
Compared with previous methods, our method reduces the error by \emph{an order of magnitude} and excels in handling sparse point clouds with thin structures or sharp features.
Our method also demonstrates a strong generalization ability to unseen shapes.
With our learned Laplacian operator, we further apply a series of Laplacian-based geometry processing algorithms directly to point clouds and achieve accurate results, enabling many exciting possibilities for geometry processing on point clouds.
\emph{We will release our code and trained models to ensure reproducibility.}
Event Type
Technical Papers
TimeThursday, 5 December 20241:46pm - 1:58pm JST
LocationHall B5 (1), B Block, Level 5
