Title: Characterizing and Optimizing the Spatial Kernel of Multi Resolution Hash Encodings
Primary area: applications to computer vision, audio, language, and other modalities
Keywords: multi-resolution hash encoding, implicit neural representations, neural fields, point spread function, spatial kernel analysis, anisotropy, resolution limit, FWHM, hash collisions, signal-to-noise ratio, NeRF

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 crucial, 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 $N_{\max}$. 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 two main advancements. First, we establish and validate a principled methodology for hyperparameter selection guided by our PSF analysis, which demonstrably outperforms standard heuristics by accounting for the analytically derived effective resolution and optimization induced broadening. Second, we introduce Rotated MHE (R-MHE), an architecture that employs rotated and independently hashed grids to mitigate anisotropy and average collision noise. This study establishes a methodology based on physical principles that moves beyond heuristics to characterize and optimize MHE.

TL;DR: We analyze Multi-Resolution Hash Encoding (MHE) using its Point Spread Function (PSF) to reveal that effective resolution is governed by average, not finest, resolution, and introduce Rotated MHE to mitigate inherent anisotropy and collision noise.