Characterizing and Optimizing the Spatial Kernel of Multi Resolution Hash Encodings

TL;DR

The paper develops a physics-inspired PSF framework to analyze Multi-Resolution Hash Encoding (MHE), treating the linearized decoder as a kernel-based system and deriving how the PSF governs spatial resolution. It reveals grid-induced anisotropy and an optimization-driven broadening that makes the empirical FWHM scale with the average resolution $N_{ ext{avg}}$, rather than the finest $N_{ ext{max}}$, and shows hash collisions degrade SNR via speckle. The authors introduce Rotated MHE (R-MHE), applying per-level input rotations to reduce anisotropy without extra parameters, and they demonstrate PSF-guided hyperparameter selection that improves 2D regression tasks and maintains competitive 3D NeRF/SDF results. This physics-based perspective provides principled guidance for hyperparameter choice and a practical isotropy-enhancing modification, with potential applicability to related grid-based encodings.

Abstract

Multi-Resolution Hash Encoding (MHE), the foundational technique behind Instant Neural Graphics Primitives, provides a powerful parameterization for neural fields. However, its spatial behavior lacks rigorous understanding from a physical systems perspective, leading to reliance on heuristics for hyperparameter selection. This work introduces a novel analytical approach that characterizes MHE by examining its Point Spread Function (PSF), which is analogous to the Green's function of the system. This methodology enables a quantification of the encoding's spatial resolution and fidelity. We derive a closed-form approximation for the collision-free PSF, uncovering inherent grid-induced anisotropy and a logarithmic spatial profile. We establish that the idealized spatial bandwidth, specifically the Full Width at Half Maximum (FWHM), is determined by the average resolution, $N_{\text{avg}}$. This leads to a counterintuitive finding: the effective resolution of the model is governed by the broadened empirical FWHM (and therefore $N_{\text{avg}}$), rather than the finest resolution $N_{\max}$, a broadening effect we demonstrate arises from optimization dynamics. Furthermore, we analyze the impact of finite hash capacity, demonstrating how collisions introduce speckle noise and degrade the Signal-to-Noise Ratio (SNR). Leveraging these theoretical insights, we propose Rotated MHE (R-MHE), an architecture that applies distinct rotations to the input coordinates at each resolution level. R-MHE mitigates anisotropy while maintaining the efficiency and parameter count of the original MHE. This study establishes a methodology based on physical principles that moves beyond heuristics to characterize and optimize MHE.

Figures (11)

• Figure 1: Overview of MHE Characterization and Optimization.(a) The MHE architecture utilizes $L$ grid levels with resolutions growing by a factor $b$. The encoding $\mathbf{e}(\mathbf{x})$ is passed to an MLP $g_{\boldsymbol{\theta}}$. We characterize the system by optimizing for a point constraint and measuring the resulting Point Spread Function (PSF). (b) This analysis reveals inherent grid-induced anisotropy (narrower along axes) and optimization-induced broadening, establishing that the effective resolution (FWHM) scales with $1/N_{\text{avg}}$. (c) To mitigate anisotropy, we propose Rotated MHE (R-MHE), which applies distinct rotations at each resolution level, leading to a more isotropic PSF.
• Figure 2: Numerical Validation of the MHE PSF (2D). We analyze the empirical PSF (solid lines) compared to the broadened theoretical prediction (dotted lines, incorporating the total empirical broadening $\beta_{\text{emp}} \approx 3.0$). (a) Varying $L$ (fixed $b=1.5$). (b) Varying $b$ (fixed $L=10$). (Columns 1 & 2) Cross-sections along the Axis and Diagonal show characteristic anisotropy (broader on diagonal). The broadened theory accurately matches the empirical decay. (Column 3) Relative FWHM vs. angle confirms the B-spline anisotropy (narrower along axes, $\theta=0$). (Column 4) The empirical FWHM aligns well with the theoretical trends dictated by $N_{\text{avg}}$. Colors indicate the varied parameter ($L$ or $b$).
• Figure 3: Empirical Analysis of Two-Point Interactions (2D). We analyze resolution by optimizing for two nearby points, normalizing separation by $f=d/\text{FWHM}$. (a) The midpoint value between two constructive peaks drops significantly when $d \approx$ FWHM. (b) The empirically measured critical distance $d_{\text{crit}}$ scales linearly with the FWHM across various MHE configurations, confirming that FWHM ($N_{\text{avg}}$) dictates the practical resolution limit. (c, d) Constructive and destructive (dipole) interference profiles. The consistent shape confirms that the FWHM characterizes the spatial response.
• Figure 4: Quantitative Analysis of Collision Effects on SNR (2D). We analyze the SNR of the empirical PSF as a function of collision ratio and hash table size $T$. (a) Impact of varying the number of levels $L$ (fixed $b=1.5$). (b) Impact of varying the growth factor $b$ (fixed $L=10$). In all cases, SNR degrades rapidly at high collision ratios (low $T$). Higher $L$ or $b$ generally improves the achievable SNR for a fixed $T$. Colors indicate the varied parameter.
• Figure 5: R-MHE Validation: Isotropy and 2D Image Regression. We analyze the impact of increasing the effective number of rotations $M$. (a) Isotropy vs M. The Anisotropy Ratio decreases and then increases as $M$ increases, demonstrating a more isotropic PSF for moderate $M$. Colors indicate the amplitude level at which the anisotropy ratio is measured (e.g., 0.5 corresponds to FWHM). (b) Visualization of the PSF zoom for different $M$. The shape becomes more circular (isotropic) as $M$ increases. (c-e) Qualitative comparison of 2D image regression results (zoomed view). R-MHE improves reconstruction quality by mitigating artifacts arising from the anisotropic kernel.