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MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation
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Proposes MetricGrids, a grid-based implicit neural representation that adds elementary metric grids (nonlinear distance metrics as Taylor-style high-order terms), hash-encoded per grid, plus a high-order extrapolation decoder, for 2D image, SDF, and NeRF fitting.
Problem
Standard grid-based implicit neural representations use linear interpolation over feature grids, which constrains latent features to a linear (degenerate) form and limits fitting accuracy for complex nonlinear signals at a fixed storage budget.
Approach
Replace linear-indexed feature grids with multiple elementary grids indexed by nonlinear distance metrics (powers of p-norms, sin/arcsin, sin/cos) that act as Taylor-expansion high-order terms, store them with per-grid hash encoding using shorter tables for higher orders, and extrapolate higher-order terms with a Hadamard-product decoder rather than storing them explicitly.
Key results
- On Kodak 2D image fitting with ~207K parameters, MetricGrids reaches 41.57 dB PSNR / 0.9735 SSIM versus 40.29 dB for SCONE and 37.06 dB for I-NGP.
- On Blender NeRF reconstruction at 30k steps, MetricGrids achieves 35.02 PSNR versus 34.62 for NeuRBF with 13.11M parameters.
- On Stanford 3D SDF reconstruction, MetricGrids reports the lowest NAE (3.09) among compared methods at 905K parameters, with IoU and Chamfer on par with baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace [sin(·), arcsin(·)] metrics with p-norm metrics [x2, x3] (ablation) | MetricGrids on Kodak · Kodak | PSNR: 41.57 -> 40.63 | Hurt |
| Use 3 nonlinear elementary metric grids in addition to the baseline representation (ablation) | I-NGP on Kodak · Kodak | PSNR: 37.06 -> 40.85 | Helped |
| Use 4 elementary metric grids rather than the baseline (ablation) | I-NGP on Kodak · Kodak | PSNR: 37.06 -> 41.91 | Helped |
| Increase restricted grid count from 2 to 3 (ablation) | MetricGrids on Kodak · Kodak | PSNR: 38.71 -> 38.95 | Helped |
| Increase restricted grid count from 3 to 4 (ablation) | MetricGrids on Kodak · Kodak | PSNR: 38.95 -> 38.98 | No clear effect |
| Add the high-order extrapolation decoder with hierarchical feature integration (ablation) | MetricGrids on Kodak · Kodak | PSNR: 40.23 -> 41.57 | Helped |
| Add the high-order extrapolation decoder with hierarchical feature integration (ablation) | MetricGrids on Stanford 3D Scanning Repository · Stanford 3D Scanning Repository | NAE: 3.29 -> 3.09 | Helped |
| Increase restricted grid count from 2 to 3 (ablation) | MetricGrids on Stanford 3D Scanning Repository · Stanford 3D Scanning Repository | NAE: 3.40 -> 3.35 | Helped |
| Increase restricted grid count from 3 to 4 (ablation) | MetricGrids on Stanford 3D Scanning Repository · Stanford 3D Scanning Repository | NAE: 3.35 -> 3.35 | No clear effect |
| Use MetricGrids instead of NeuRBF | Neural radiance field reconstruction · Blender | PSNR: 34.62 -> 35.02 | Helped |
| Use MetricGrids instead of NeuRBF | Neural radiance field reconstruction · Blender | LPIPSVGG: 0.034 -> 0.033 | Helped |
Surprising findings
- Nonlinear metrics [sin, arcsin] outperformed the p-norm metrics [x^2, x^3] on 2D image fitting, which the authors attribute to these metrics already containing multiple Taylor-order terms.
- Performance gains diminish when the number of explicitly stored higher-order grids exceeds three.
Limitations
- Training time roughly increases several-fold over the baseline (e.g., 1148s vs 195s in the 2D ablation).
- Gains from adding grids diminish beyond three explicitly stored higher-order grids.
- Choice of nonlinear metric is a design hyperparameter; the paper evaluates only a few metric pairs.
Experimental setup
Datasets: Kodak Gigapixel images Stanford 3D Scanning Repository Blender
Benchmarks: Kodak Stanford 3D Scanning Repository Blender
Metrics: PSNR SSIM MS-SSIM LPIPS IoU NAE Chamfer distance parameter count
Baselines: I-NGP NeuRBF SIREN WIRE SCONE NFFB GaussianImage NGLOD Mip-NeRF 360 TensoRF K-Planes NerfAcc
Builds on: Instant-NGP NeuRBF NerfAcc SIREN Multiplicative Filter Networks
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