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Stochastic Poisson Surface Reconstruction with One Solve using Geometric Gaussian Processes
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Reformulates stochastic Poisson surface reconstruction so posterior surface samples need only one kernel linear solve (via SGD), using periodic Matérn geometric GP Fourier-series cross-covariances, enabling local, output-sensitive queries without volumetric Poisson solves.
Problem
Stochastic Poisson surface reconstruction (SPSR) requires a two-stage pipeline: GP interpolation of oriented point clouds followed by a global finite-element Poisson solve on a volumetric grid, with diagonal approximations of kernel inverses and grid-coupled length scales.
Approach
Treat the vector field and implicit surface as GPs on a torus using periodic Matérn kernels, so the Poisson solve reduces to a Fourier series; the cross-covariance between implicit surface and vector field is a truncated Fourier series, enabling joint posterior sampling and mean/covariance computation with a single kernel linear solve and local, grid-independent evaluation.
Key results
- Proposed method reproduces qualitative mean, variance and probability outputs of SPSR while avoiding volumetric FEM solves.
- Runtime scales with number of queried points rather than volumetric grid size; reported mean-computation times of 0.053, 0.54 and 9.1 seconds for decreasing length scales, with the method's runtime claimed to be roughly constant in length scale.
- SGD (dual variant) gives reasonable approximation after about 10 iterations and matches Cholesky solution within about 1000 iterations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the single-solve geometric-GP method instead of SPSR | Stochastic Poisson surface reconstruction · Surface reconstruction | qualitative reconstruction quality | No clear effect |
| Use the single-solve method for queries at selected points rather than a global volumetric solve | Stochastic Poisson surface reconstruction · Ray casting and collision detection | runtime scaling | Helped |
| Retain kernel correlations rather than use SPSR’s diagonal covariance approximation | Stochastic Poisson surface reconstruction · Posterior surface sampling | sample smoothness | Helped |
| Use an amortization grid that is too small (N = 5) (ablation) | Single-solve geometric-GP method · Surface reconstruction | reconstruction fidelity | Hurt |
| Increase the amortization grid density (ablation) | Single-solve geometric-GP method · Surface reconstruction | reconstruction fidelity | Helped |
| Use SGD for the kernel matrix solve instead of Cholesky factorization (ablation) | Single-solve geometric-GP method · Surface reconstruction | qualitative solution quality | No clear effect |
| Use 25^3 rather than 100^3 evaluation points for SPSR samples (ablation) | SPSR · Surface reconstruction samples | surface fidelity | Hurt |
| Decrease the kernel length scale (ablation) | SPSR · Surface reconstruction | runtime: dividing the length scale by 2 increases the runtime by roughly 2^d | Hurt |
| Decrease the kernel length scale (ablation) | Single-solve geometric-GP method · Surface reconstruction | runtime | No clear effect |
Surprising findings
- SPSR's diagonal approximation to the inverse kernel matrix yields irregular, over-smoothed samples, whereas the proposed method yields smooth sharp samples.
Limitations
- Formal collision likelihood computation may be too expensive for real-time applications.
- Fourier truncation with L=100^3 cross-covariance terms is costly, requiring amortization via interpolation on a grid.
- Requires periodic boundary conditions and product of stationary periodic kernels, differing from the original Neumann boundary setup.
Experimental setup
Datasets: Armadillo Bunny Falcon Scorpion Springer Tree Well Dragon
Metrics: kernel matrix MSE runtime qualitative comparison of posterior mean, variance and proba
Baselines: SPSR (Sellan et al. 2022) Poisson Surface Reconstruction Cholesky factorization
Builds on: Stochastic Poisson Surface Reconstruction Poisson Surface Reconstruction periodic Matérn kernels pathwise conditioning stochastic gradient descent for GPs
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