Poisson Surface Reconstruction
📜 Abstract
We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.
✨ Summary
The paper introduces a method for reconstructing a watertight surface from an oriented point set by treating the samples as observations of the gradient of an implicit indicator function. The indicator function is smoothed so that its gradient can be represented as an integral of surface normals. The method approximates this gradient field using the input points and then solves the Poisson equation (\Delta \tilde{\chi}=\nabla\cdot\mathbf{V}) to obtain the scalar implicit function whose isosurface defines the reconstructed model.
The computational formulation uses an adaptive octree and a hierarchy of compactly supported basis functions. This produces a sparse, symmetric, well-conditioned system and concentrates computational effort near the reconstructed surface. Trilinear interpolation provides sub-node positional accuracy, while density estimation adjusts both the magnitude and spatial scale of point contributions for non-uniform sampling. The final surface is extracted using an adaptively modified marching-cubes procedure, with the isovalue selected from the implicit-function values at the input samples.
Experiments on noisy scans show that the method produces smooth, detailed reconstructions while avoiding many spurious sheets associated with methods that constrain the implicit function only near sample points. It compares favorably with several reconstruction techniques, particularly for sharp features and non-uniformly sampled data. The principal limitation identified by the authors is that the method does not use acquisition information such as line of sight; consequently, it may incorrectly connect regions where samples are absent. The reported implementation scales approximately quadratically with grid resolution and was demonstrated on very large scan datasets.
The method had substantial subsequent impact. Its core formulation became available as the actively maintained PoissonRecon implementation, which describes the codebase as originating from this work and later extending it to more general adaptive finite-element systems. (github.com) It was incorporated into widely used geometry-processing libraries, including Open3D and CGAL, where it remains an API-level surface-reconstruction method for oriented point clouds. (open3d.org) Follow-up research extended the approach to parallel computation, point-interpolation constraints through screened Poisson reconstruction, envelope constraints, and distributed processing. (hhoppe.com) Later work continues to describe Poisson surface reconstruction as a widely used baseline and builds probabilistic and application-specific variants on its formulation. (proceedings.mlr.press)