Building discriminative representations for 3D data has been an important task in computer graphics and computer vision research. Convolutional Neural Networks (CNNs) have shown to operate on 2D images with great success for a variety of tasks. Lifting convolution operators to 3D (3DCNNs) seems like a plausible and promising next step. Unfortunately, the computational complexity of 3D CNNs grows cubically with respect to voxel resolution. Moreover, since most 3D geometry representations are boundary based, occupied regions do not increase proportionately with the size of the discretization, resulting in wasted computation.

As a flexible representation, NeRF has been first adopted for 3D representation. With density-based volumetric rendering, it however suffers both intensive computational overhead and inaccurate mesh extraction. Using a signed distance field and Marching Tetrahedra, DMTet allows for precise mesh extraction and real-time rendering but is limited in handling large topological changes in meshes, leading to optimization challenges. Alternatively, 3D Gaussian Splatting (3DGS) is favored in both training and rendering efficiency while falling short in mesh extraction. In this work, we introduce a novel 3D representation, Tetrahedron Splatting (TeT-Splatting), that supports easy convergence during optimization, precise mesh extraction, and real-time rendering simultaneously.

Neural implicit function based on signed distance field (SDF) has achieved impressiveprogress inreconstructing 3Dmodels withhighfidelity. However,such approaches canonlyrepresent closed surfaces.

In this work we target a learnableoutputrepresentation that allows continuous, high resolution outputs of arbitrary shape. Recent works represent 3D surfaces implicitly with a Neural Network, thereby breaking previous barriers in resolution, and ability to represent diverse topologies. However, neural implicit representations arelimited to closed surfaces, which divide the space into inside and outside. Many real world objects such as walls of a scene scannedby a sensor,clothing,or a car with innerstructuresare not closed. Thisconstitutesa significant barrier,in termsof datapre-processing (objects need to be artificially closed creating artifacts), and the ability to output open surfaces. In this work, we proposeNeural Distance Fields (NDF), a neural network based model which predicts the unsigned distance field for arbitrary 3D shapes given sparse point clouds. NDF represent surfaces at high resolutions as prior implicit models, but do not require closed surface data, and significantly broaden the class of representable shapes in the output.

Geodesics play a critical role in many geometry processing applications.