Spectral Prefiltering of Neural Fields

TL;DR

The paper introduces a fast, flexible framework for prefiltering neural fields by analytically modulating Fourier feature embeddings with a filter's frequency response. By training with a single symmetric Gaussian filter and supervising via a one-sample Monte Carlo estimate, the method generalizes to unseen filters such as Box and Lanczos, enabling continuous, kernel-agnostic smoothing in both 2D images and 3D SDFs without constraining network architectures. Key contributions include closed-form feature-space convolution for Fourier features, explicit kernel-magnitude formulas, and demonstrated performance gains over prior scale-aware neural-field methods across isotropic and anisotropic filtering. The approach offers practical benefits in rendering quality and multi-scale filtering, while noting limitations in speed, symmetry assumptions, and applicability to non-symmetric filters. Overall, the work provides a principled, efficient mechanism to control smoothing across filter families with strong empirical evidence of improved fidelity and generalization.

Abstract

Neural fields excel at representing continuous visual signals but typically operate at a single, fixed resolution. We present a simple yet powerful method to optimize neural fields that can be prefiltered in a single forward pass. Key innovations and features include: (1) We perform convolutional filtering in the input domain by analytically scaling Fourier feature embeddings with the filter's frequency response. (2) This closed-form modulation generalizes beyond Gaussian filtering and supports other parametric filters (Box and Lanczos) that are unseen at training time. (3) We train the neural field using single-sample Monte Carlo estimates of the filtered signal. Our method is fast during both training and inference, and imposes no additional constraints on the network architecture. We show quantitative and qualitative improvements over existing methods for neural-field filtering.

Figures (9)

• Figure 1: Method Overview. Our method rests on two key ideas: (i) using the analytic Fourier transform ($\mathcal{F}$) of a symmetric linear filtering kernel to modulate Fourier-feature embeddings, and (ii) using single-sample Monte Carlo (1 MC) estimates of the filtered signal for supervision. A neural field trained with one filter type $g_\text{train}$, generalizes to unseen filters $g_\text{test}$. Images from Adobe FiveK; © original photographers/Adobe.
• Figure 2: 1D Example. Given a discretely sampled 1D signal and a finite set of uniformly sampled Gaussian kernels, we train a neural field to encode the continuous scale space of the signal (top). By modulating the Fourier feature embeddings as per § \ref{['sec:prefilter_fourier']}, we prefilter the neural field with Box (middle) and Lanczos (bottom) filters with no additional supervision.
• Figure 3: Sensitivity Analysis. We evaluate how Fourier-feature embedding size and network architecture affect (1) reconstruction of the original signal, (2) isotropic smoothing, and (3) anisotropic smoothing. Embedding Size. While embedding size has relatively smaller effect on reconstructing the original signal, the model benefits substantially from larger embeddings in both isotropic and anisotropic smoothing tasks. Hidden Dimension. Increasing the hidden-layer width consistently yields higher PSNR across all tests. Network Depth. Adding more than three layers provides only marginal PSNR gains, indicating diminishing returns beyond a depth of three. The bottom row shows the PSNR range (highest minus lowest) observed over each hyperparameter sweep, illustrating the relative sensitivities.
• Figure 4: Effect of exact Fourier modulation. We show the impact of exact Fourier modulation on the Alien image. The inset on the right bottom represents the frequency spectrum of the image. Our model produces much smoother filtering results when encoding is exactly modulated to match the filter's Fourier transform.
• Figure 5: Comparisons against Neural Field Convolutions (NFC) nsampi2023neural and Neural Gaussian Scale-Space Fields (NGSSF) mujkanovic2024neural for image filtering across Gaussian, Box, and Lanczos kernels. Our model supports controllable smoothing across families and anisotropic covariances in a single forward pass. NFC parameterizes filters with Dirac impulses; it is reliable for isotropic/mild kernels but is capacity-limited for anisotropic and non-polynomial families. NGSSF is tuned for Gaussian smoothing; we recalibrated its encoding for each family, but the filter family cannot be switched explicitly at test time. Bottom-right insets show frequency spectra; top-right insets show mean error. See the supplemental and website for additional results. Images from Adobe FiveK; © original photographers/Adobe.