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Fitting Flats to Flats
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Generalizes PCA-style least-squares fitting so a k-flat can be fit to a set of flats of varying dimension by summing their squared distance fields and taking the eigendecomposition of the summed matrix.
Problem
Fitting a flat of given dimension to a set of flats (possibly of different dimensions) in the least-squares sense; existing Riemannian center-of-mass approaches on the affine Grassmannian are complex, slow, and not equivariant to translations.
Approach
Represent each flat as a quadratic squared-distance field (Q, r, s); the least-squares fit is obtained by averaging the fields and projecting onto flats via eigendecomposition, giving PCA-like complexity, rigid-motion equivariance, and an L1 (Weiszfeld) extension.
Key results
- Mean-SDF and Median-SDF are substantially faster than the iterative affine Grassmannian optimization across ambient dimensions 4-20.
- In R^3 with 20 planes at noise sigma=0.5, Median-SDF achieves d_min 0.1192 versus 0.2174 for Graff.
- Under outliers, Median-SDF gives the best overall results, though Mean-SDF is not robust to outliers (e.g., d_min 15.06 vs 0.39 for Graff at 50% outliers).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| fit using Mean-SDF rather than Graff | R3 line reconstruction from 20 planes · Noise σ=0.5 | dmin: 0.2174 -> 0.1292 | Helped |
| fit using Mean-SDF rather than Graff | R3 line reconstruction from 20 planes · Noise σ=2.5 | dmin: 0.5946 -> 0.4599 | Helped |
| fit using Mean-SDF rather than Graff | R3 line reconstruction from 20 planes · Noise σ=2.5 | θ: 0.4817 -> 0.1374 | Helped |
| add median aggregation of Q* and r* instead of arithmetic-mean aggregation (ablation) | Mean-SDF · R3 line reconstruction from 20 planes, outliers | dmin: 15.0596 -> 0.3857 | Helped |
| add median aggregation of Q* and r* instead of arithmetic-mean aggregation (ablation) | Mean-SDF · R3 line reconstruction from 20 planes, no outliers | dmin: 0.0227 -> 0.0929 | Hurt |
| add median aggregation of Q* and r* instead of arithmetic-mean aggregation (ablation) | Mean-SDF · R3 line reconstruction from 20 planes, noise σ=0.5 | dmin: 0.1292 -> 0.1192 | Helped |
| add median aggregation of Q* and r* instead of arithmetic-mean aggregation (ablation) | Mean-SDF · R3 line reconstruction from 20 planes, noise σ=0.5 | θ: 0.0381 -> 0.0434 | Hurt |
Surprising findings
- Mean-SDF is not robust to outliers and at high outlier rates has worse displacement error than the iterative Grassmannian baseline.
- The Riemannian center on the affine Grassmannian varies with translation of the coordinate system, violating translation equivariance.
Limitations
- Fails when all input flats are points (k=0), since the summed field is isotropic.
- Mean-SDF is sensitive to outliers; the robust Median-SDF variant requires iterative Weiszfeld reweighting.
- Degenerate isotropic configurations leave the fitted flat non-unique.
Experimental setup
Datasets: synthetic flats generated from random target flats with Gaussian noise synthetic archery scene rendered from four cameras
Benchmarks: synthetic flat reconstruction in R^3 with noise and outliers running time over ambient dimensions 4-20 archery multi-view line reconstruction
Metrics: minimum distance d_min principal angle running time (ms)
Baselines: Graff (iterative optimization on affine Grassmannian) rank minimization of normal representations
Builds on: PCA Riemannian center of mass on Stiefel manifolds Weiszfeld algorithm Plücker coordinates
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