Overview
TetraSDF is presented as an analytic isosurface extraction framework designed for neural signed distance functions (SDFs). The core objective is to obtain an explicit surface that precisely aligns with the zero-level set of a neural SDF. This framework integrates a multi-resolution tetrahedral positional encoder with a ReLU Multi-Layer Perceptron (MLP) to represent the SDF.
Research Context
The extraction of an explicit surface corresponding to the zero-level set of a neural signed distance function (SDF) poses a persistent challenge. Existing sampling-based isosurfacing methodologies, exemplified by Marching Cubes, inherently introduce discretization error. Conversely, continuous piecewise affine (CPWA) analytic approaches, while aiming for precision, commonly necessitate the use of plain ReLU MLPs. This requirement practically restricts their capacity to accurately learn high-frequency SDFs.
Approach
TetraSDF addresses these challenges by enabling exact zero-level set extraction while preserving the expressiveness characteristic of grid-based encoders. The framework achieves this through the representation of the SDF using a ReLU MLP, which is composed with a multi-resolution tetrahedral positional encoder. A key component of this approach is the positional encoder's utilization of barycentric interpolation. This interpolation method is designed to maintain a global CPWA structure. The preservation of this structure facilitates the tracking of ReLU linear regions within a polyhedral complex induced by the encoder.
Furthermore, the TetraSDF framework incorporates a fixed analytic input preconditioner. This preconditioner is derived from the encoder's metric and is introduced with the specific aim of reducing directional bias. By mitigating directional bias, the preconditioner contributes to stabilizing the training process of the neural SDF.
Findings
Evaluations conducted across multiple benchmarks indicate that TetraSDF achieves or surpasses the SDF reconstruction accuracy of existing grid-based encoders. Concurrently, the framework accurately recovers the network's zero-level set, which is represented as a triangle mesh.
Why This Matters
This approach offers a method for accurately extracting explicit surfaces from neural SDFs without the discretization errors associated with sampling techniques, while also allowing for the representation of complex shapes.