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Neural Fields as Distributions: Signal Processing Beyond Euclidean Space
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Reinterprets neural field training as maximum likelihood over a distribution formed by convolving the signal with a filter, yielding a filter-aware loss that works on Euclidean and non-Euclidean (Lie group) domains without filtered ground truth.
Problem
Neural fields lack general, efficient filtering tools; applying filters at training or inference is costly or infeasible for sparse data on non-Euclidean domains such as ray manifolds and rotations.
Approach
Treat the signal as a sample distribution and the filter as a kernel distribution; the convolution becomes a random-variable addition, giving an O(1) per-sample MLE loss that trains the neural field to directly represent the filtered signal, extended to homogeneous spaces of Lie groups via group actions.
Key results
- On image filtering, the proposed method reaches 50.8 PSNR at the largest tested filter scale (σ=0.16), versus 46.6 for Monte Carlo and 46.1 for Nsampi et al., while inferring at ~660-700 MP/s versus ~100 MP/s for Nsampi et al.
- On Mip-NeRF 360 scenes trained with thin-lens depth of field, average test PSNR is 28.8 versus 29.2 for Mip-NeRF 360 when evaluated with pinhole rays, a loss of about 0.4 dB.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed filter-aware training loss instead of Monte Carlo filtering | Neural field image filtering · Image filtering, σ=0.16 | PSNR: 46.6 -> 50.8 | Helped |
| Use the proposed filter-aware training loss instead of Nsampi et al. | Neural field image filtering · Image filtering, σ=0.16 | PSNR: 46.1 -> 50.8 | Helped |
| Use the proposed filter-aware training loss instead of Monte Carlo filtering | Neural field image filtering · Image filtering, σ=0.02 | PSNR: 40.9 -> 38 | Hurt |
| Use the proposed filter-aware training loss instead of Nsampi et al. | Neural field image filtering · Image filtering, σ=0.02 | PSNR: 37.1 -> 38 | Helped |
| Train one network to apply a space of filter widths | Neural field image filtering · Filter spaces | PSNR: 43.5 -> 42.6 | Hurt |
| Train with thin-lens depth-of-field ray distributions | Mip-NeRF 360 · Mip-NeRF 360 test reconstruction, Average | PSNR: 29.2 -> 28.8 | Hurt |
Surprising findings
- At the smallest filter scale, Monte Carlo filtering outperforms the proposed method on PSNR (40.9 vs 38.0), although the proposed method is faster at scale.
Limitations
- Arbitrary filters unknown at training time cannot be applied at inference; zero-shot methods such as Nsampi et al. are better suited.
- Learning a continuous space of filters in one model degrades reconstruction as filter family complexity grows.
- Non-symmetric kernels require lifting the field to SE(3) with extra input, adding complexity.
Experimental setup
Datasets: Mip-NeRF 360 scenes image test set with filters
Benchmarks: Mip-NeRF 360 scenes
Metrics: PSNR SSIM LPIPS inference speed (MP/s)
Baselines: Monte Carlo filtering Nsampi et al. Mip-NeRF 360
Builds on: Mip-NeRF 360 Nsampi et al. filtering by repeated differentiation Light Field Networks Energy-based models
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