Characterizing and Optimizing the Spatial Kernel of Multi Resolution Hash Encodings
Authors: Tianxiang Dai
Jonathan Fan
Venue: ICLR 2026 Poster (2026)
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
Primary area: applications to computer vision, audio, language, and other modalities
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.

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.

Published as a conference paper at ICLR 2026
CHARACTERIZING AND OPTIMIZING THE SPATIAL KERNEL OF MULTI RESOLUTION HASH ENCODINGS
Tianxiang Dai & Jonathan Fan" Department of Electrical Engineering
Stanford University Stanford, CA 94305, USA (txdai, jonfan)estanford.eee
ABSTRACT
Multi-Resolution Hash Encoding (MHE), the foundational technique behind In- stant 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 selec- tion. This work introduces a novel analytical approach that characterizes MHE by examining its Point Spread Function (PSF), which is analogous to the Green's 's function of the system. This methodology enables a quantification of the encod- ing's spatial resolution and fidelity. We derive closed-form approximation for the collision-free PSF, uncovering inherent grid-induced anisotropy and loga- rithmic spatial profile. We establish that the idcalized spatial bandwidth, specif- ically the Full Width at Half Maximum (FWHM), is determined by the average resolution, Navg- This leads to a counterintuitive finding: the effective resolution of the model is governed by the broadened empirical FWHM (and therefore Nawg). rather than the finest resolution Naxx 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 propose Signal-to-Noise Ratio (SNR). Leveraging these theoretical insights, Rotated MHE (R-MHE), an architecture that applies distinct rotations to the input coordinates each resolution level. R-MHE mitigates anisotropy while maintain- ing the efficiency and parameter count of the original MHE. This study establishes methodology based physical principles that moves beyond heuristics to char- acterize and optimize MHE.
INTRODUCTION
Multi-Resolution Hash Encoding (MHE) (Müller al.) (022), the central innovation underlying Instant Neural Graphics Primitives (Instant-NGP), has catalyzed significant advancements in neu- ral fields, enabling accelerated optimization and real-time rendering for applications such as Neural Radiance Fields (NeRF) (Mildenhall et al. 2020 and Signed Distance Functions (SDFs). This even extends beyond computer graphics and was applied to PINNs (Huang & Alkhalifah 20244 and physical designs (Dai et al., 2025). MHE offers compact and efficient parameterization; however, its behavior is critically dependent on hyperparameters, including the number of levels L,, growth factor b, resolutions (Nuaxx Nmiu), and hash table capacity T. These parameters typically lected using generalized heuristics. Despite extensive research into neural field architectures (Sun) al. 2022 Chen et al., 20222] Fridovich-Keil al. (023) and anti-aliasing techniques (Greer al.) 2021. substantial gap persists: the study of MHE, and NeRF models more generally, currently lacks rigorous analysis from physical systems perspective.
In this work, we introduce a novel methodology to characterize and understand the performance of MHE by analyzing its Point Spread Function (PSF). Analogous measuring the Green's 's function of a physical system, the PSF characterizes the model's response when optimized to represent an idealized point source (Figure [p). This approach permits the rigorous quantification of effective
"Corresponding author.

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Published as a conference paper at ICLR 2026
Measurement of PSF
Anisotropy & Broadening
Nmin
Point Constraint [(x)
Mave
b &
x Direction
P(x)
Fitting Optimizer
Rotated MHE
Nmax
ex)
(e()))
Observed PSF (x)
Hash Encoding
MLP Se
Increased Isotropy
Overview of MHE Characterization and Optimization. (a) The MHE architecture utilizes grid levels with resolutions growing by a factor The encoding e(x) is passed to an MLP ge. We characterize the system by optimizing for point constraint and measuring the resulting Point Spread Function (PSF). (b) This analysis reveals inherent grid-induced anisotropy (narrower along axes) and optimization-indueee
Figure 1:
broadening, establishing that the effective resolution (FWHM) scales with 1/Navg- (c) To mitigate anisotropy. we propose Rotated MHE (R (R-MH)) which applies distinct rotations at each resolution level, leading more isotropic PSF.
spatial resolution and the identification of performance issues that often contradict intuition derived from the architecture's ' specifications. We isolate the encoding by operating in a linearized decoder regime, framework motivated by kernel perspectives (Jacot et al.. 2018) and spectral analysis (Tan- cik et al. 2020).
Our analysis begins with the examination of the PSF of the idealized, collision free MHE. We derive a closed form approximation demonstrating that the PSF exhibits logarithmic radial decay and significant grid induced anisotropy, inherited from the underlying interpolation kernels (Keys) 1981). Theoretically, the idealized Full Width at Half Maximum (FWHM) is determined by the average resolution, Navg-
We confirm these trends through numerical experiments, which reveal that optimization dynamics induce significant spatial broadening compared to the idealized minimum norm prediction. This confirms that the effective two point resolution of the model is substantially lower then Nmax and governed by the broadened empirical FWHM (and thus Navg)-
We further investigate the impact of finite hash capacity, demonstrating how collisions introduce speckle-like side lobes and degrade the -ognal-to-Noise Ratio (SNR). Informed by our comprehen- sive PSF analysis, we demonstrate how these insights can be leveraged to improve reconstruction quality. We introduce Rotated MHE (R-MHE (Figure [b), an architecture that applies distinct ro- tations the input coordinates each resolution level. By utilizing the existing multi-resolutinn structure, R-MHE improves isotropy without requiring additional hash tables or parameters, main- taining the efficiency of the original MHE.
Contributions.
This work establishes new framework based on physical principles for analyzing MHE, providing several key advancements:
We derive closed-form approximation for the MHE Point Spread Function, rigorously character- izing its anisotropic and logarithmic spatial profile, and identifying the average resolution, Navg- the principal determinant of the idealized FWHM.
We reveal and characterize optimization optimization-induuedd spatial broadening, demonstrating theoretically and empirically that it arises from spectral bias.
We provide evaluation of the impact of hash collisions the Signal-to-Noise Ratio (SNR).
We introduce Rotated MHE (R-MHE), a novel, parameter-free modification that improves isotropy by applying distinct rotations at each resolution level.
We validate principled methodology guided by the PSF analysis for hyperparameter selection that demonstrably outperforms standard heuristics.
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2
BACKGROUND AND PRELIMINARIES
2.1 RELATED WORK
Our analysis draws upon and contributes to several interconnected areas of research.
Neural Fields, Encodings, and Spectral Analysis. The introduction NeRF established coordinate-based volumetric rendering (Mildenhall al. 2020). Positional encodings, including Fourier features (Tancik et al.] (020) and periodic activations (Sizzmann et al., 2020), are known to shape optimization dynamics and frequency bias (Rahaman al.] (018). Adopting Neural Tangent Kernel (NTK) viewpoint, where linearized training dynamics justify the analysis of an en- coding's induced kernel (Jacot al. 2018), utilize this theoretical lens to derive an explicit PSF for MHE (Müller et al. 2022).
Explicit Grids and Factorized Structures. Researchers have explored replacing or augmenting MLP decoders with explicit spatial representations, including voxel grids (Sun al. 2022 Peng 2022). tensor factorizations (Chen et al.] 2022), and planar factorizations (Fridovic- Keil
et al.
2023 While MHE is widely adopted, extensions such Dictionary Fields (Chen et al. have been proposed to improve expressivity. Our work is complementary; R-MHE improves the underlying grid structure and could potentially be integrated with these extensions. We aim to elucidate how MHE behaves spatially and provide principles for its optimization.
al.
2023
Interpolation Kernels and Anisotropy. The separable tent kernel underlying multilinear inter- polation (Keys) 1981 Thevenaz et al. 2000) inherently induces differences in effective blur along axes versus diagonals. We demonstrate that MHE inherits these anisotropies across multiple scales, resulting in direction-dependent FWHM and resolution limits even without hash collisions.
Hashing and Collisions. Spatial hashing has long history in computer graphics (Lefebvre & Hoppe 2006) and real-time reconstruction (NieBner et al. (003)). Our collision analysis formalizes how finite capacity hash tables translate into PSF speckle and SNR loss within MHE.
2.2 MHE ARCHITECTURE REVIEW
The MHE aims to learn a function f(x) = ge It utilizes resolution levels defined by Ni Nmin 6'. At each level I, features are retrieved from table F' of size T using spatial hash function H and multilinear interpolation. The interpolation kernel K (u) is constructed as the product of ID tent functions: ((u) = I2 max(0, 1 - (ul).. Consequently, the spatial kernel at level 1 is given by K7(x) K(Nx)).
The idealized spatial response of the encoding, averaged over all possible grid alignments, is char- acterized by the induced kernel (x),, which the auto-correlation of the interpolation kernel: (x (K₃ (x). This results in separable cubic B-spline kernel (Thevenaz et al. 2000).
CHARACTERIZING THE MHE SPATIAL KERNEL
We analyze the Point Spread Function (PSF) characterize the intrinsic spatial behavior of the MHE architecture, examining the system's 's response when optimized under sparse constraints. To isolate the properties of the encoding from the influence of the subsequent MLP decoder go, we assume the MLP can be approximated by its linearization, f(x) x We(x). This is justified by experiments regarding MLP depth (Appendix D.4)
3.1 THE IDEALIZED, COLLISION FREE PSF
We first consider the optimization process for a single point constraint C = (f(x)) - 2)2. In this idealized analysis, assume the absence of hash collisions (infinite 7). Under the linearized framework and the minimum norm assumption, the responsibility is distributed equally across all L levels. The resulting idealized PSF Piteat (x) is the average superposition of the normalized induced
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kernels (x) (the cubic B-spline, Section 2.2):
L-1 Σ 1=0
Pideal (x)
Bi(x Xo)
Generalized Closed-Form Approximation and Anisotropy. The induced kernel (x) (cubic B- spline) is separable but inherently anisotropic. We derive generalized closed-fomm form approximation by approximating the summation with an integral, utilizing the Taylor expansion of the B-spline kernel near the center (Appendix (.3).
Let vetv= = X be the normalized position. The closed-form approximation near the center can be expressed as:
Pideal (v) (inb) [-I[[[vv]]) + CD - Ap((v) + O(v²)
(2)
This expression reveals dominant logarithmic decay profile (-In([iv]])) modulated by an anisotropy factor Av(v) specific the B-spline kernel. As proven in Appendix A.3 the B-spline kernel is narrower along the axes.
FWHM and Average Resolution. The Full Width at Half Maximum (FWHM) is direction- dependent. We define the inherent broadening factor of the idcalized induced kernel as P The FWHM of the 1D cubic B-spline kernel is numerically calculated be fddcat 1.18. The FWHM of the composite PSF along the axes scales proportionally to the average resolution Navg (Appendix A.3
Axis, Ideal R Bideat /Nag R 1.18/Nwg
The idcalized spatial bandwidth is dictated by Nagg, while the FWHM along the diagonals is com- paratively wider.
3.2
EMPIRICAL VALIDATION AND OPTIMIZATIIO-NNDUEED BROADENING
We validate these theoretical results using customized 2D implementation based on the Instant- NGP framework. We configure MHE networks (varying L and 6) with a sufficiently large to minimize collisions, and optimize for single point objective L (f(0) 1)
Empirically, we observe that the realized PSF, P'mpirical (x,, is significantly broader than the ideal- ized minimum norm prediction Piacal (x). We characterize this additional broadening by introducing an optimization optimizztion-induueed spatial broadening factor Seps such that Premprrical (x) == Piteal Bomu). ). Be- cause the idealized B-spline model already accounts for the inherent anisotropy, the optimization- induced broadening Sepi can be accurately modeled as isotropic.
We define the total empirical broadening factor Bemp such that the empirical FWHM along the axis
is Axiss Emp = Pemp/Nwwe This combines the idealized broadening and the optimization-induced broadening:
Bemp
Biocal Pops
(4)
Spectral Bias. This optimization-indueed broadening (Beg > 1) occurs because the optimization process (e.g., using Adam) does converge the minimum-nome solution. Gradient- based op- timization exhibits implicit biases, often referred to as spectral bias (Rahaman et al.) 2018). where lower frequencies are learned preferentially. This leads the optimization trajectory to a solution where coarse features (low N)) are prioritized over fine features. This effective re-weighting towards lower frequencies results in the observed spatial broadening. We provide a theoretical derivation in Appendix D modeling the weights as x (Ni)- where y is the spectral bias exponent. We prove that Pape increases monotonically with y, and argue that y (and thus Bopp) increases with the spatial dimension D.
Characterizing the Broadening Factors. We consistently observe a total empirical broadening of Bamp 2 3.0 across various configurations of L and when using the Adam optimizer (Figure 2}. To understand the sensitivity of 3eps we conducted experiments varying the optimizer, MLP architecture, and training dynamics (Appendix D.4, D.5). We found that 8eg primarily dependent
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Axis
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10"
1.4
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0.0
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b
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 Semp 3.0). (a) Varying L (fixed 1.5). (b) Varying (fixed L 10). (Columns & 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, O = 0). (Column 4) The empirical
FWHM aligns well with the theoretical trends dictated by Navg- Colors indicate the varied parameter (T. or b).
on the optimizer choice (e.g., corresponding βemp R 3.0 for Adam variants in 21) and 3D, Fig- 10 but is remarkably stable across hyperparameters and insensitive to the MLP depth. This stability confirms that Sop is robust characteristic of the optimization dynamics for given setup.
Figure compares the empirical PSF with the theoretical predictions. We observe excellent agree- ment between the empirical results (solid lines) and the broadened theory (dotted lines, incorporating Semp Re 3.0). The characteristic anisotropy (broader on diagonals) predicted by the B-spline model clearly visible (Columns 1-3). Furthermore, the empirical decay confirms the logarithmic profile predicted by the closed-form approximation (Eq. D and the scaling of the FWHM with respect to
T. and aligns with the theoretical trends (Column 4).
3.3 RESOLUTION LIMITS AND TWO POINT INTERACTIONS
We analyze the system's behavior when optimized for two closely spaced point constraints, XA and xB, separated by distance d. Under the linearized assumption, the reconstruction R(x) the superposition of the individual PSFs.
Constructive Interference: Idealized vs. Empirical Resolution. The resolution limit deat (Rayleigh criterion) is the minimum distance such that dip exists between two peaks. In the idealized minimum-norm configuration (Appendix (A.5), the smoothness of the B-spline kernel im- plies the theoretical resolution limit is infinitesimal. However, empirical analysis reveals that the practical resolution limit is dictated by the broadened empirical PSF (Section 3.2). The ability resolve two points is therefore governed by the empirical FWHM, which scales with 1/Nawg.
Destructive Interference and the Dipole Response. For spatial dipole. R(x) = P(x- -1x-xA) XA) P(x xB). When d is small, R(x) d Px().. We find that the spatial behavior and extent of the dipole response are characterized by the empirical FWHM of the underlying PSF, due to the optimization induced broadening. When the separation is smaller than FWHM, the maximum values no longer appear at the points and artifacts appear.
Numerical Validation: Two-Point Interactions. We extend the 2D experimental setup, optimiz- ing for two point constraints while varying MHE parameters and separation d. For constructive
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Mid Value
b
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Constructive [L=9, b=1.2]
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f=d FWHM
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value
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f= / FWHM
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x/d
Figure 3: Empirical Analysis of Two-Point Interactions (2D). We analyze resolution by opti- mizing for two nearby points, normalizing separation by f = d 4/FWHM. (a) The midpoint value between two constructive peaks drops significantly when d = FWHM. (b) The empirically measured critical distance den scales linearly with the FWHM across various MHE configurations. confirm- ing that FWHM (Navg) dictates the practical resolution limit. (c, d) Constructive and destructive (dipole) interference profiles. The consistent shape confirms that the FWHM characterizes the spa- tial response.
interference (Figure 3 (a)), significant dip emerges when the separation is approximately equal to empirical FWHM. We empirically determine the critical distance derit across various configu- rations and find direct linear relationship with the FWHM (Figure (()).. This confirms that the practical two-point resolution limit is determined by the empirical FWHM (scaling with 1/Navg), not N max- For the dipole configuration (Figure d)), the profiles exhibit consistent shape charac- terized by the FWHM, illustrating that the spatial extent of the dipole response is also governed by Navg-
4 THE IMPACT OF FINITE HASH CAPACITY
Having characterized the inherent spatial properties of the MHE kernel, we now analyze the effects of practical memory constraints. In real implementations, the hash table capacity T is finite, leading to collisions where different spatial vertices map to the same entry in the feature table F
4.1 MODELING HASH COLLISIONS AND SPECKLE
When vertices collide, they share the same feature vector. For a single point objective at Xo. the optimized feature vector F becomes proportional to the sum of the interpolation weights of all colliding vertices C evaluated at the target point:
F & Σ Ki(v- Xo)
(5)
VEC!
vertex v near Xy might collide with spatially distant vertex v'. The optimized feature is now inadvertently activated when querying v'. This mechanism causes unintended "ghost" responses far from the center of the PSF, resulting in spurious side lobes or speckle patterns the spatial response. We model the resulting PSF as PColiisio (x) (x)eai(X) + n(x). The severity of the noise rxx) depends strongly on the collision ratio (load factor). As the collision ratio increases, the variance of n(x) increases, and the Signal to Noise Ratio (SNR) decreases.
4.2 NUMERICAL VALIDATION OF COLLISION EFFECTS
We investigate the impact of collisions experimentally by training 2D MHE network with a single point objective while systematically varying T, L, and b. Figure 4 summarizes the impact of these parameters on the SNR. Across all configurations, increasing the collision ratio eventually leads to a significant degradation in SNR. We observe that increasing L (Panel a) b (Panel b) generally im- proves the achievable SNR for a fixed capacity T. This suggests that distributing the representation across more levels or with greater separation enhances robustness, provided that T sufficient.
ROTATED MHE (R-MHE)

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SNR collision ratio
SNR vs log2(T)
b
SNR collision ratio
SNR log2(T)
30.0
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9NR (d8)
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22.5
25
25
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10
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10
20
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20
collision ratio
log2(T)
collision
log2(T)
Figure 4: Quantitative Analysis of Collision Effects on SNR (2D). We analyze the SNR of the empirical PSF a function of collision ratio and hash table size T. (a) Impact of varying the number of levels L (fixed 6 = 1.5). (b) Impact of varying the growth factor b (fixed L = 10). In all cases, SNR degrades rapidly high collision ratios (low T). Higher L generally improves the achievable SNR for fixed T. Colors indicate the varied parameter.
Our analysis identified that the reliance on axis-aligned grids in standard MHE leads inherent anisotropy (Section 3.1). This is undesirable in applications like NeRF where viewing angles vary continuously, or in image regression where features may not align with the axes. To address this limitation, propose the Rotated MHE (R-MHE) architecture.
5.1 MOTIVATION AND ARCHITECTURE
R-MHE leverages the existing multi-resolution structure of MHE. Instead of using single rotation for the entire encoding or requiring multiple independent hash tables, R-MHE applies a distinct rotation matrix R; to the input coordinates specifically at each resolution level 1. The encoding process at level is modified as follows:
(x) Interpolate (F,,((([))))
(6)
This model utilizes the same hash function 76 and feature tables F' as standard MHE, maintaining the exact memory footprint, parameter count, and computational efficiency.
5.2 ROTATION STRATEGIES
The key to R-MHE selecting set of rotations (Rt) that maximizes the diversity of grid orienta- tions across the levels. In 2D, we employ progressive rotation strategy. We define base rotation angle and set the rotation at level to be R, = Rot(I 0). This ensures that subsequent levels are oriented differently, maximizing the angular coverage over the L levels. We analyze the impact of the choice of Section In 3D, we aim for uniform sampling of the rotation space SO(3). We utilize the vertex orientations of regular polyhedra (tetrahedron, cube, octahedron, icosahedron) to define a set of canonical directions. The rotations {Rç} are chosen to align the grid axes with these directions, cycle through the vertices of one chosen polyhedron type across the levels L (Details in Appendix A.6
5.3
THEORETICAL BENEFITS: IMPROVED ISOTROPY
The R-MHE architecture offers significant advantage of improved isotropy derived directly from our PSF analysis.
The resulting idealized PSF of R-MHE is the superposition of the rotated induced kernels:
PR-MEE Σ 1 H Bi
xo))
I=0
where the rotations applied at each level. This averaging process across differently oriented levels effectively mitigates the angular dependencies inherent in the standard MHE, leading to a more uniform spatial resolution across different orientations.

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5.4 EXPERIMENTAL VALIDATION OF R-MHE
We validate the benefits of the R-MHE architecture using the established numerical framework and a practical 2D application. We analyze the impact of the base rotation angle 0. We define the strategy by the parameter M, such that 0 = 90 deg /M. M represents the effective number of unique orientations sampled within the first quadrant.
Quantifying Isotropy. We conducted experiments to measure the PSF for standard MHE (M = 1) and R-MHE with increasing M (up to 16). The objective was to quantify the improvement in isotropy (measured by the Anisotropy Ratio, the ratio of the maximum to minimum distance to center across directions at different levels). Analysis of this ratio and the kernels in S (a) and (b) reveals that for this averaging contibutes to better isotropy for moderate M. This trend suggests that while increasing M improves isotropy by diversifying orientations, excessively large M would cause L levels to be insufficient to effectively average very large number of unique orientations, potentially reducing the effectiveness of the rotation strategy. From M in (a())) (a) it evident that the anisotropy ratio of 1.17 proven in A.4 holds across many different levels.
Isotropy
PSF zooms
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1,20
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M-2
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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 more isotoopic PSF for moderate M. Colors indicate the amplitude level at which the anisotropy ratio is measured (c.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.
Application: 2D Image Regression. To demonstrate the practical advantages of R-MHE, we evaluated its performance 2D image regression task using three high-resolution images (Details in Appendix B.. We follow the standard configuration: L = 16 levels and F = features per level.
We employed our theoretical insight that the empirical FWHM (governed by Navg and Bemp) dictates the practical resolution limit (Section 3.3). Accordingly, we calculated the theoretical growth factor btheory such that the empirical FWHM along the axes direction, Bemp/ /Nug. matches the spatial extent of a single pixel. We used the validated total empirical broadening factor Semp = 3.0 (Section 3.2).
To validate this principled approach and find the empirical optimum, we performed experiments comparing bthewyy against neighboring values (Otheory + #0.2). The results (Appendix B.3, Ta- 5 show that the empirical optimum (bop)) consistently occurred near, but slightly below, byneary (specifically Duneory 0.1 or -0.2). This indicates the optimal effective kernel size is 2.5 pixels. This is physically intuitive, natural images possess spatial coherence and are rarely composed of pixel-perfect high-frequency noise; slightly broader kernel better matches the signal bandwidth and provides beneficial regularization.
We compare standard MHE (M = 1) and R-MHE (M E 22) using the respective empirically optimized bept for each configuration. All experiments were run with 5 random seeds. The results, summarized in Table [ and illustrated in Figure 5 c-e), highlight the improvement achieved by R- MHE. The standard MHE baseline achieves an average PSNR of 23.88 dB. R-MHE consistently improves performance, reaching a peak average PSNR of 24.82 dB at M = 8. an improvement of +0.94 dB. This gain demonstrates that the improved isotropy provided by R-MHE effectively miti- gates artifacts caused by the anisotropic kernel, at no additional cost in parameters computation.
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Table 1: 2D Image Regression performance (Average PSNR + Std Dev in dB) using the empirically optimized growth factor b. R-MHE significantly outperforms the standard MHE baseline (M 1)
with zero overhead. (L=16, F-2).
Method (Effective Rotations M)
Average PSNR (dB) t
Standard MHE (M=l)
23.88 = 0.02
R-MHE (M=2)
24.62 = 0.01
R-MHE (M=4)
24.69 = 0.01
R-MHE (M=8)
24.82 + 0.01
APPLICATIONS TO 3D NEURAL FIELDS AND SDFS
Having validated the benefits of R-MHE in 2D, we now evaluate its impact on complex 3D ap- plications: Neural Radiance Fields (NeRF) and Signed Distance Functions (SDFs). We aim to demonstrate the practical utility of our PSF analysis for hyperparameter selection and assess the performance of R-MHE in 3D using the polyhedral rotation strategy. We utilize customized im- plementation based on the Instant-NGP framework (Müller al. 2022).
6.1 EXPERIMENTAL SETUP
We follow the standard Instant-NGP configuration (L F 2). All models were trained for 20,000 steps using the Adam optimizer (Details in Appendix(C). We compare standard MHE against R-MHE using the 3D polyhedral rotation strategies (Tetrahedron, Cube, Octahedron, Icosahedron). All experiments were run with random seeds. We evaluate two strategies for selecting the growth factor b. The Baseline Heuristic utilizes the default approach in Instant-NGP. The PSF Guided (Theory) strategy is derived from our analysis (Section 3); is calculated such that the empirical FWHM (using Semp 3.0) matches the target spatial resolution.
6.2 RESULTS: NEURAL RADIANCE FIELDS (NERF)
We conduct experiments on the 8 scenes of the Synthetic NeRF dataset (Mildenhall et al. 2020 The average reconstruction quality (PSNR) is summarized in Table
Table 2: 3D NeRF reconstruction performance (Average PSNR + Std Dev in dB) the Synthetic NeRF dataset (8 scenes). We compare configuration strategies and R-MHE rotation types.
Configuration
Method
Average PSNR (dB)
Standard MHE
35.346 0.105
R-MHE (Tetra)
35.472 : 0.114
Baseline Heuristic (Optimized b)
R-MHE (Cube)
35.445 0.134
R-MHE (Octa)
35.449 + 0.115
R-MHE (Icosa)
35.479 + 0.134
Standard MHE
35.329 + 0.100
R-MHE (Tetra)
35.396 0.128
PSF Guided (Theory b)
R-MHE (Cube)
35.404 + 0.121
R-MHE (Octa)
35.409 0.139
R-MHE (Icosa)
35.440 + 0.119
The Baseline Heuristic (empirically optimized bapt) performs very similarly the PSF Guided (Theory) configuration (bthcory R 1.38). For Standard MHE, the theoretical configuration achieves 35.329 dB, matching the empirical optimum of 35.346 dB (Table 2). As visualized in the detailed parameter sweeps in Appendix C.2 the theoretical prediction (btheory RJ 1.38) consistently falls pre- cisely within the regime best performance across all scenes. This validates that our PSF analysis, successfully identifies optimal hyperparameters priori. R-MHE consistently maintains slightly improves the performance of standard MHE across both configuration strategies. For the Baseline configuration, the best R-MHE variant (Icosa, 35.479 dB) shows marginal improvement compared

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to Standard MHE (35.346 dB). These improvements often within one standard deviation of the baseline variance, and the quantitative performance gains in standard 3D benchmarks are statisti- cally marginal compared to the 2D regression tasks. Nonetheless, R-MHE provides these results at zero overhead while offering the theoretical benefit of improved isotropy.
6.3 RESULTS: SIGNED DISTANCE FUNCTIONS (SDF)
We further evaluate R-MHE on the task of learning SDFs. We utilize three standard benchmark meshes: Armadillo, Bunny, and Spot. We measure the Intersection over Union (IoU) quantity reconstruction quality (higher is better). The results for the optimal resolution configuration (6 Re 1.18) are summarized in Table 3 The results exhibit clear performance saturation. At the high spatial resolutions provided by the MHE configuration, the capacity of the hash grid is sufficient to resolve the geometry, yielding IoU scores exceeding 0.996 for all meshes across all methods, and the reconstruction error is dominated by the finite sampling resolution rather than the encoding's 's anisotropy.
Table 3: 3D SDF reconstruction performance (Intersection over Union, IoU f) on benchmark meshes. All methods achieve near-perfect reconstruction, indicating performance saturation.
Method
Armadillo
Bunny
Spot
Average IoU
Standard MHE
0.9994 + 0.0002
0.9966 + 0.0001
0.9998 + 0.0001
0.9986
R-MHE (Tetra)
0,9994 + 0.0002
0.9966 + 0.0001
0.9998 + 0.0001
0.9986
R-MHE (Cube)
0.9995 + 0.0001
0.9966 + 0.0001
0.9998 + 0.0001
0.9986
R-MHE (Octa)
0,9995 + 0.0001
0.9966 + 0.0001
0.9998 + 0.0001
0,9986
R-MHE (Icosa)
0.9994 + 0.0002
0.9966 + 0.0001
0.9998 + 0.0001
0.9986
7 CONCLUSION
This work addresses the deficiency of physical understanding in the study of Multi- RestiiiResottion Hash Encoding (MHE) by introducing an analysis based on the Point Spread Function (PSF). By treating the MHE model as a physical system, we provide novel framework to identify performance limita- tions and optimize the architecture. Our analysis reveals that the idcalized MHE PSF, characterized by the induced B-spline kernel, inherently possesses an anisotropic profile (broader diagonals), with its spatial bandwidth determined by the average resolution Navg- We demonstrated that opti- mization dynamics (spectral bias) lead significant spatial broadening (Bom), resulting in the crucial finding that the effective resolution is governed by the broadened empirical FWHM (and therefore Navg), not Nmax- Leveraging these insights, we introduced Rotated MHE (R-MHE), a parameter- free modification that mitigates anisotropy by applying distinct rotations at each resolution level.
Limitations of R-MHE While R-MHE provides a theoretical guarantee of improved isotropy, we observe that the quantitative gains in standard 3D benchmarks statistically marginal compared to the improvements seen in 2D tasks. We attribute this dampened effect to two primary factors inherent these 3D tasks. First, in volumetric rendering (NeRF), the integration of features along rays acts as a form of "view averaging," " which can effectively low-pass filter the high-frequency artifacts caused by grid anisotropy. Second, standard benchmarks often exhibit performance saturation the high spatial resolutions typically employed, where the grid capacity is sufficient to resolve geometry regardless orientation. Consequently, R-MHE is likely most critical regimes with strict memory constraints or sparse viewing angles where interpolation artifacts are less likely to be masked by averaging or high sampling rates.
Future Work and Broader Impact. The anisotropic behavior analyzed here is applicable to other grid-based encodings, such as multi-plane (TensoRF (Chen et al. 2022)) or planar factorizations (K- Planes (Fridovich- Keil et al.) 2023)). as they also rely on multilinear interpolation on axis-aligned structures. The R-MHE concept of applying rotations per level or plane could be directly transferred to these settings to improve isotropy. This study establishes a physics- methodology moving beyond heuristies to characterize and improve neural field models.
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ACKNOWLEDGMENTS
This work was supported by the Samsung Global Outreach Program and the National Science Foun- dation under Award Number 2103301.
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A EXTENDED DERIVATIONS
A.1 IMPACT OF GRID MISALIGNMENT AND THE INDUCED KERNEL
The analysis of the idealized PSF (Section 3.1) relies on the induced kernel framework. In the lin- earized regime, the spatial response of the encoding averaged over all possible alignments between the target point Xg and the grid structure is characterized by the auto-correlation of the interpolation kernel (x),, resulting in the induced kernel (x) (K₃ * (x) (the cubic B-spline). While the response for specific alignment (e.g., perfect alignment with a vertex, which yields the tent function) differs, the B-spline kernel correctly captures the general case, and the derived properties regarding anisotropy and FWHM scaling are robust.
A.2 CONTEXT: BEHAVIOR UNDER DENSE SUPERVISION
It is important to contrast the behavior of MHE under sparse supervision, which is the focus of our PSF analysis, with its behavior under dense supervision, such as image regression tasks. Our analysis. however, specifically targets the behavior relevant to scenarios with sparse constraints, where the assumption of linearized MI.P decoder most appropriate and the intrinsic spatial properties of the encoding isolated, and by our analysis, closely links the final performance on actual tasks with this idcalized results.
A.3 ANALYSIS OF THE INDUCED KERNEL: PROFILE, ANISOTROPY, AND FWHM
We analyze the properties of the idealized PSF (x) based on the induced kernel B(x) (the separable cubic B-spline kernel) and the multi-resolltion structure.
The 1D Cubic B-spline Kernel and FWHM (Batea). In 1D, the induced kernel is the normalized
cubic B-spline Bin( The piecewise definition for 0 < [t||
< is:
3
3
BιD(u) = 1
2 = +
|u]
(8)
4
We solve for the half-width / such that B (2) = 0.5. Solving this cubic equation numeri-
cally yields the relevant real root 21/2 0.5904. The FWHM is Bateal 201/2 /2 1.1808.
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Derivation of the Logarithmic Profile and Closed Form. We demonstrate that the superposition of B-splines results a dominant logarithmic profile near the center, justifying Eq. 2 We approxi- mate the summation with an integral and utilize the Taylor expansion of the B-spline near the center: (uD(u) = 1 uu²
We analyze the ID case. Let = 2 The effective upper limit of integration Lefs occurs approximately when the argument reaches the scale of the kernel support, bhertv 1. Thus Lefs so -In(r)/In()) Inbb)
phasss
3 02 0² dl 1 1.10 (1-
Pideal (v) ~ T.
T
Bin(b'
20
(9)
2
3u²
bess
Pidea ~
821
(10)
L 4ln
Substituting Leff and babsss 1 1/02:
3x2
Pideal (v)
Llnb - in(v)
+
(11)
This confirms the dominant - (I)) behavior near the center. The generalization to D dimensions yields the form in Eq. 2 where the anisotropy factor (n()) arises from the D-dimensional expan- sion and the specific properties of the B-spline kernel.
FWHM Scaling. The FWHM of the composite idealized PSF Pidcal (x) along the axes scales with the average resolution Nagg The idealized FWHM along the axis is approximately:
FWHMAsi. Ideal Fo Brdeat/ Navg so 1.18/Navg
(12)
A.4 GENERALIZED ANISOTROPY ANALYSIS IN D DIMENSIONS
We perform the anisotropy analysis of the induced kernel (cubic B-spline) on D dimensions using an analysis based on the Taylor expansion near the center. This analysis demonstrates that the kernel inherently narrowest along the primary and becomes progressively broader along directions that involve multiple components (diagonals).
D-Dimensional Kernel and Taylor Approximation. nel (u) is the separable product:
The D-dimensional normalized induced ker-
(u) = II BID(us)
(13)
1=1
We utilize the Taylor expansion of the 1D kernel near the origin. From the definition of the normal- ized cubic B-spline (Eq, 8), the second derivative "",(0) 3/2.
(0) = -3 We define C -
(u) ~ 1 Cu²
(14)
This quadratic approximation accurately captures the local curvature near the peak, which deter- mines the primary anisotropy trend.
Analysis of K-Sparse Directions. We analyze the anisotropy by comparing the Euclidean distance d from the center required to reach fixed amplitude A (e.g.. A 0.5 for FWHM) along different orientations. We consider a K K-sparse direction, where K components of the input vector are equal (u, for < i < K) and the remaining D K components
zero. This spans orientations from the primary axis (K = 1) to the main space diagonal (if K = D). The Euclidean distance is / Kr.
The response along this direction is:
D
Bi = (BID(x)) K
P(u) I BLD
II
(15)
[=11
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Setting the response to A:
BID(I)*= =A = BIO(E) = A2/K
(16)
Using the Taylor expansion:
- Cx² = A x²² 1 - A
(17)
C
The squared Euclidean distance d² is:
K(1 ((1-At/K (K) Kr²
(18)
C
Proof of Monotonicity. To determine how the width changes as the direction involves more com- ponents (moving from axis towards the main diagonal), analyze the monotonicity of d' (K). We analyze the function f(K) K(1 A for A E (0, 1).
We analyze the derivative ('( (treating K as a continuous variable for analysis):
f((K) - (1-A4/K) - K -AL/K In(A) (-K²) )= 1 A + A In(A)
1 K
(19)
Let = AIKK. Since A E (0,1) 1) and K 2 1, we have y E (0, Also In(A) = K KIn(y).
((( = 1 y + + In(y)
(20)
We examine the function g(y) = 11- y ln(y). The derivative is (v) = In +1 In(y). Since y 1, g ('() < 0. Thus, g(v) is monotonically decreasing. The minimum occurs the limit
y + 1, where g(1) = 0. Therefore, ('K > 0 for all A < 1.
This proves that f(K), and thus the distance d( (K), is monotonically increasing with K. The induced B-spline kernel is narrowest along the primary axes (K- 1) and broadest along the main space diagonal (where K is maximal).
Anisotropy Ratio and Dimensional Scaling. We define the Anisotropy Ratio RA (D) the ratio of the squared distances along the main diagonal (K = D) versus the axis (K = 1), using A = 0.5.
dD(D) d2(1
D O.5/P/C 2D11 0.5
RA(D)
(21)
(1-0.5)CC
D=2: (2) = 4(1 - 0.5 ~ 1.1716.
D=3: (3) 6(1 0.52/3) ~ 1.2378.
The Anisotropy Ratio (D) is also monotonically increasing with D (since F(D) is monotonic), demonstrating that the grid-induced anisotropy becomes more pronounced in higher dimensions. The limit D + 0o In(2) 1.386.
A.5 TWO-POINT RESOLUTION DERIVATION (CONSTRUCTIVE INTERFERENCE)
The resolution limit dent (Rayleigh criterion) the minimum separation such that R(midpoint) < R(peak). We analyze the idealized case for an axis-aligned configuration. The condition for dip the midpoint requires the second derivative of the reconstruction R(x) be positive at the center. Since the induced kernel (x) (cubic B-spline) is smooth (C2 continuous) and strictly concave near the peak, this condition is satisfied for any separation d > 0.
Therefore, in the idealized minimum-nore configuration using the induced kemel, the theoretical resolution limit is infinitesimally small. It is crucial to emphasize that this applies specifically to the idealized solution. As demonstrated by the empirical results in Section 3.3, practical optimization dynamics lead to a broader empirical PSE, where the actual resolution limit governed by the empirical FWHM (scaling with 1 /Nag))
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A.6 R-MHE 3D ROTATION STRATEGIES
We detail the 3D rotation strategies utilized in Section 5.2. We use the vertex orientations of regular polyhedra define canonical directions, aiming for uniform sampling of SO(3). The rotations R, are constructed by aligning the standard basis vectors with these normalized directions. We cycle through the following sets across the levels L.
Tetrahedron (4 vertices): (1,1,1), (-1,-1,1), (-1,1,-1), (1,-1-1).
Cube (8 vertices): All sign corners ( 1),, ordered lexicographically.
Octahedron vertices): (+11, 0, 0), (0, =1,0), (0, 0,,0,11).
Icosahedron (12 vertices): (0, =1, = (#1, 0), (#Ø, 01,, where Φ = (1+ is the golden ratio.
The Icosahedral strategy provides the most uniform distribution of orientations, which correlates with the superior empirical performance observed in Section
B EXPERIMENTAL DETAILS FOR 2D IMAGE REGRESSION
B.1 DATASET AND CONFIGURATION
We utilized three high-resolution images: "Mountain" (2473 x 3710), "City" (4000 x 6000), and "Forest" (3193 x 6016). To ensure fair comparison and remove potential anisotropy introduced by non-square aspect ratios during sampling, all images were center-cropped to the largest possible square (e.g., 2473 2473 for Mountain).
Configuration. We utilized L 16 levels, F = 2 features per level, and hash table capacity of 2 R-MHE configurations use the progressive rotation strategy (Section 5.2), with defining the base rotation angle 90 /M.
Training Details. The networks were trained for 5000 steps using the Adam optimizer with a leamning rate of 0.001. We used batch size of 27 (131072) randomly sampled pixels per iteration. All experiments were repeated across 5 random seeds.
B.2 PSF GUIDED PARAMETER SELECTION
We leveraged the relationship between the empirical FWHM along the axes direction and Navg- We determined the theoretical optimal growth factor bmeery by setting the target empirical FWHM (3emp/Nave based on the cropped image dimensions (1/shortest side). We used the empirically validated factor Semp 3.0. We fixed T, 16 and Nmin 16, and then solved numerically for the required growth factor b. The resulting theoretical growth factors Dibeorry are detailed in Table
Table 4: PSF Guided Hyperparameters for 2D Gigapixel Image Regression (Cropped Images).
Image
Resolution (Cropped) Target Empirical FWHM Growth Factor (htheory)
Mountain
2473 2473
8.09e-04
1.4614
City
4000 x 4000
5.00e-04
1.5078
Forest
3193 3193
6.26e-04
1.4863
B.3 VALIDATION OF THEORETICAL GROWTH FACTOR
The results (Table 5 summarize these results across the Standard MHE (M=1) and R-MHE con- figurations (M [2, 4, 8)). The analysis demonstrates that the best performance is consistently achieved near bawwory. specifically at bibear - 0.1 or -0.2. Since a lower growth factor corresponds to lower Navg and thus wider FWHM, this finding suggests that targeting resolution of exactly one pixel is slightly too aggressive. A kernel slightly broader than one pixel yields better reconstruc- tion, likely because natural images are band-limited and contain features larger than single pixel,
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whereas 1-pixel target may encourage overfitting aliasing quantization artifacts. The optimal FWHM is thus roughly 2.5 pixels, from this observation.
Table 5: Validation of PSF-Guided Growth Factor (b) Selection. PSNR (dB, Mean + Std Dev)
results comparing bheorry against neighboring values. Bold indicates the best performance for the configuration.
Image
M
biheory
0.2
buheory 0.1
bibeory
biheory + 0.1
Uteeoyy + 0.2
M=1
22.98
0.02
24.19 0.03
23.07
0.04
22.44 0.09
21.76 0.07
M=2
23.90 + 0.01
25.37 + 0.02
24.52 + 0.01
23.30 + 0.02
22.75 + 0.01
Mountain
M=4
24.09
14.09:0000 0.01
25.35 + 0.02
24.53
0.01
23.59 = 0.01
23.54 = 0.02
M=8
24.12 0.00
25.84 0.01
24.19
1.19:000 0.02
23.40 : 0.02
23.35 = 0.02
M=1
21.85
0.00
21.86
0.01
21.16 0.01
20.20 0.00
19.07 0.00
M-2
22.31
22.57 0.01
+
20.64
19.77 0,00
City
M=4
22.33
21.04 2.06
2..44
M=8
22.82 0.00
0.00
21.06 0.00
20.46
M=1
2.01
+:
24.76
24.46 0.03
24.12 4
M=2
2..56
25.92 0.01
25.19 2.00
24.66
Forest
M=4
25.55
26.02 0.00
25.38
2..85
M-8
25.79 + 0.01
25.78 0.00
2.03
25.04 0.01
25.02 0.01
B.4 DETAILED RESULTS
Table 6 provides the detailed PSNR results for each individual image across the different R-MHE configurations, utilizing the empirically validated optimal & (from Table 5
Table 6: Detailed PSNR (dB, Mean + Std Dev) results for 2D Gigapixel Image Regression using
the empirically optimized growth factor Bold indicates the best performance per row.
Image
M=1 (Baseline)
M=2
M=4
M=8
Mountain
24.19
25.37 + 0.02
25.35
0.02
25.84 0.01
City
21.86
22.57 + 0.01
22.71 0.00 0.00
22.82 + 0.00
Forest
25.59 + 0.01
25.92 + 0.01
26.02
0.00
25.79 + 0.01
Average
23.88 + 0.02
24.62 + 0.01
24.69 = 0.01
24.82 0.01
B.5 QUALITATIVE VISUALIZATION AND PSF ANALYSIS
Qualitative Comparison. Proviees a qualitative comparison of the reconstruction results.
PSF Visualization. Figure provides visualizations of the empirical PSF for R-MHE with varying M. As increases, the PSF profile becomes noticeably more symmetric and circular. For larger M. faint spiral pattern emerges, characteristic of the progressive rotation strategy employed in 2D R-MHE.
C EXPERIMENTAL DETAILS FOR 3D NEURAL FIELDS AND SDFs
C.I CONFIGURATION AND TRAINING
We utilized the standard Instant-NGP configuration (L=16, F=2, T 21.). Training utilized the standard Instant-NGP protocol and optimizer settings. All experiments were run with 5 random seeds.
C.2 NEURAL RADIANCE FIELDS (NERF) DETAILS
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Ground Truth
Baseline
M-2
M/4
M-8
$1.0
Figure 6: Qualitative Visualization of 2D Image Regression Results. Zoomed-in crops comparing
Ground Truth, Standard MHE (Baseline), and R-MHE (M-2, 4, 8). R-MHE demonstrates signifi- cant improvements in reconstruction quality. PSNR values indicated are for the specific runs shown.
M 16
Figure 7: Visualization of R-MHE Point Spread Function (PSF). Empirical PSF (zoomed view) for R-MHE with M=1 (Standard MHE), 2, 4, and 16.
We utilized all eight scenes from the Synthetic NeRF dataset. Figure g presents a comprehensive parameter sweep of the reconstruction quality (PSNR) as function of the growth factor b for all eight NeRF scenes.
These plots demonstrate that the theoretical growth factors derived from PSF analysis (hiseury 1.38, as listed Table ( consistently fall within the regime of best performance. As observed in the figure, the empirical peaks for scenes such as Lego, Mic, and Hotdog align precisely with the theoretical prediction of 1.38. This confirms that our physical systems approach accurately characterizes the optimal spatial resolution settings. allowing for precise hyperparameter selection without the need for exhaustive empirical grid searches.
C.3 SIGNED DISTANCE FUNCTIONS (SDF) DETAILS
Dataset. We utilized three standard benchmark meshes: Armadillo, Bunny, Spot. We used the baseline parameters provided in the Instant-NGP repository for SDF tasks.
Figure visualizes the IoU performance across the growth factor sweep for the SDF task.
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+
none
Mra
+
cuse
+ odto
lodda
Lego
Chair
Drums
Flcus
PSNR (dB)
30-
Mic
Hotdog
Materials
Ship
PSWR (68)
35-
30-
1.8
12
1.2
1.5
Figure 8: Detailed PSNR Sweep for Synthetic NeRF Experiments. We plot the Average PSNR (dB) vs. growth factor b across all scenes (Error bars indicate Std Dev over 5 seeds). The theo- retical optimum derived our analysis (buwerr ~ 1.38) consistently aligns with the empirical peak performance observed in these sweeps, validating the PSF-guided strategy. The R-MHE strategies (tetra, cube, octa, icosa) consistently track or exceed the performance of the Standard MHE baseline ("none").
Table 7: Growth Factors (b) for Synthetic NeRF Experiments. beherry is the value derived from the analysis framework. butt is the empirically optimized value used in the Baseline Heuristic configu- ration.
Scene
btheory
bopt (Baseline)
Chair
1.3767
1.3819
Drums
1.3872
1,3819
Ficus
1.3767
1.3819
Hotdog
1.3767
1.3819
Lego
1.3767
1.3819
Materials
1.3872
1,3819
Mic
1.3767
1.3819
Ship 1.3767
1.3819
D THEORETICAL DERIVATION AND EMPIRICAL ANALYSIS OF THE
BROADENING FACTOR
D.I MODELING OPTIMIZATION DYNAMICS VIA SPECTRAL BIAS
The idealized Point Spread Function (PSF), derived under the minimum 1.2-norm assumption (Eq. T. implies uniform weighting across all L resolution levels:
Pideal (x)
Σ (x)
(22)
L
1=0
This leads to the idealized FWHM factor Pideal RJ 1.18.
Empirical results demonstrate that gradient-based optimization methods (e.g., Adam) exhibit spec- tral bias, where lower frequencies are learned preferentially (Rahaman et al. 2018). In the context of MHE, this biases the optimization towards coarser grids (low Nj). We model the resulting empirical PSF as a weighted superposition with non-uniform weights wy:
L-1
PEimpirical (x) =
Σ wi (x)) 1=0
(23)
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Figure 9: Detailed ToU Sweep for 3D SDF Experiments. We plot the Intersection over Union (IoU) as a function of the growth factor b for the Armadillo, Bunny, and Spot meshes. The heavy overlap of the curves for all methods (Standard MHE and R-MHE variants) and the consistently high IoU values (> 0.996) illustrate the performance saturation discussed in the main text.
where Σw, wi 1. This leads to the total empirical FWHM factor Bemp.
We adopt a phenomenological model motivated by kernel methods, where the optimization-induced weights follow a power law relationship based on the resolution Ny:
Wi X (N))
(24)
Here, 7 2 0 is the spectral bias exponent. 7 O recovers the minimum-norm solution (Sideat), while y indicates bias towards coarser levels (Bemp Bideal).
D.2 RELATIONSHIP BETWEEN SPECTRAL BIAS BROADENING
We establish that the optimization cedimization-induce induced broadening factor (7) (where Semp -Baeaal Pops) is a monotonically increasing function of the spectral bias exponent y
Proof of Monotonicity of Bapt(1) (7). We want to show that d(FWHM)/d7 > 0. We analyze the empirical PSF value fixed spatial position x 0, P(y) Pempirical y): If we show that dP(r))/y > 0 (i.e., the tails of the PSF increase with 7), it implies the FWHM must also increase.
We express P(7) using normalized weights wy = (N:) (Nx)
P()) = B(( f())
(25)
Σx(Nx) g(7)
We analyze the derivative (*(() (8'g fg///92 We must show f'g fg 0.
f'( = Σ (n()))
(26)
g'(7) Σ (n(N))
(27)
We analyze the term fg'
f'g fg' ( ((N) ( (-(()))MMBCO)) (I(Narm) - -( (-X (Na)) In In
)
(-E(Na)
(28)
(NN) N₂ In N,)
(29)
We analyze this summation by grouping pairs of indices (1, k)) and (k,, where + k:
Pair = (N)) N₂- In No) + (N&) (N Ba N₂ - Nn
(30)
- Nin [B((z) Bx(xz)
(31)
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Published as a conference paper at ICLR 2026
Consider the case N& > Nr. The factor (In N₁. - -In Nt) is positive. Since N₁ > Nr (level & is finer), the kernel B& (a:) is narrower than (1))) For O, have Ba Bg (x). Thus, the second factor is also positive.
Every pair sum is positive, proving that l'g - 0. Consequently, dP(n)/d7 > 0, which implies do dr > O. A stronger spectral bias always leads to larger broadening.
D.3 EMPIRICAL ANALYSIS OF OPTIMIZER DEPENDENCE
investigated the dependency of broadening factor on the choice of optimizer. We trained the MHE network using various standard optimizers (Adam variants, SGD, RMSProp, etc.) and measured the resulting total empirical broadening Semp- Figure 10 summarizes the results.
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Figure 10: Sensitivity of Broadening Factor to Optimizer Choice. We measure the fitted total empirical broadening Bemp across different optimizers and MHE configurations (varying L and b)- (a) Results for 2D. (b) Results for 3D. Broadening depends significantly on the optimizer (Adam variants consistently yield Semp 3.0), but is robust across different MHE parameters and dimen- sions for a given optimizer.
We observe that the broadening is indeed dependent the optimizer. Adam and its variants (AdamW, NAdam, Adamax) consistently yield 9emp 3.0 (corresponding to Sopt As 2.54), with 3D results almost identical to 2D. Other optimizers exhibit different degrees of broadening, ferent optimization algorithms inherently possess different implicit regularization properties. For a given optimizer, the resulting Semp is highly robust across different MHE configurations (varying L and b) and dimensions (2D vs 3D), as shown in Figure TO Panels (a) and (b) show results for 2D and 3D respectively, both yielding similar broadening values for each optimizer, confirming that the broadening is primarily determined by the optimizer choice, rather than specific MHE parameters. This stability means that one-time calibration of Semp for a specific optimization setup sufficient for principled hyperparameter selection.
dif-
D.4 SENSITIVITY TO MLP ARCHITECTURR.
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Published as a conference paper at ICLR 2026
Axis
Diagonal
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FWHM vs parameter
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Figure 11: Impact of MLP Depth the Empirical PSF (2D). We analyze the PSF while varying the MI. depth (indicated by color). The results show that the empirical PSF profile and the FWHM largely insensitive the MLP architecture.
We investigated the impact of the MLP decoder depth on the empirical PSF Figure 11 shows the results of varying the MLP depth from (linear) to 3 layers. The empirical PSF profiles and the resulting FWHM are consistent across different depths, supporting the conclusion that the encoding structure and the optimization dynamics (spectral bias) primarily define the spatial response.
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Reviews:
[{"review_id": "NQ5gSG6HdQ", "reviewer": "Reviewer_4sPL", "summary": "This paper proposes rotated multi-resolution hash encodings, widely used for implicit neural representations in image regression and neural radiance fields. Previous hash encodings are dependent on axes, leading to anisotropy and hindering empirical performance. They fail to capture well frequency patterns like sinusoidal functions because hash encodings are implicitly aligned with the axes of Cartesian coordinates. In contrast, this paper proposes rotated multi-resolution hash encodings that allow us to be independent of Cartesian coordinates. It achieves this by using multiple rotation functions for input position $x$, resulting in an average of randomly sampled rotation functions. This approach mitigates the reliance on representation aligned with Cartesian coordinates and improves prediction performance without degradation in frequency-based patterns. One of the randomly sampled rotation functions may potentially match the directions of the frequency patterns. This paper demonstrates its effectiveness in theoretical and practical cases.", "questions": "- I’m curious about Figures 1, 2, and 3. The colors in these figures indicate different parameterizations and the value seems to be continuous in all of them. However, the color information is actually discrete because of the parameterization used in this experiment. Adopting a legend for continuous values makes it challenging to understand graphical explanations easily. \n\n- Figure 1 and 2 are particularly difficult to grasp the main messages from. They require more detailed explanations and concepts to understand.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "- This paper introduces a novel approach that explicitly uses multiple random rotation transforms. To maintain the total number of parameters, the paper splits the original feature dimensions into the number of rotation and new feature dimensions. Although it employs a smaller number of feature dimensions, the random rotation function for input $x$ enables it to effectively capture signals that do not align with Cartesian coordinates.\n\n- The paper demonstrates its effectiveness through both mathematical and practical cases. Specifically, it shows that the proposed approach consistently improves prediction performance in image regression and neural radiance fields, while maintaining the same total number of parameters across each case.", "weaknesses": "- I believe this paper falls short of meeting the standards of top-tier AI conferences like ICLR. The presentation lacks an effective and efficient way for readers to grasp the main contribution of the paper. While the mathematical contributions are included in the Appendix, and very abstracted equations are elaborated in the manuscript, I disagree that all the content in the manuscript must be abstracted. The Appendix is necessary to fully digest the mathematical details, but it doesn’t justify the need for the entire manuscript to be abstracted. I believe important equations (Eq 2 and 3) and their impact (e.g., FWMH) should be provided in the manuscript, but they aren’t. \n\n- Empirically, maintaining the total number of parameters is crucial for improving prediction performance. However, this paper artificially designs baselines and their parameterization. For instance, its baselines started with a feature dimension of 8, which is not the official parameterization of feature dimensions in NeRFs, which sets it to 2 (To my best knowledge). Therefore, it lacks the space to adjust and accommodate random rotation functions. In this case, the proposed approach naturally faces challenges in its practical implementation. \n\n- Following the second concern in Weakness, they should identify new cases where high-dimensional encodings are required to achieve accurate prediction performance. In that sense, they must demonstrate that rotated MHE not only maintains performance but also reduces the total number of parameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wfYfPXjzE1", "reviewer": "Reviewer_Rkrg", "summary": "This paper presents to provide understandings on multi-resolution hash encoding (MHE) originally presented from Instant-NGP.\n\nIn summary, the paper:\n- Frames MHE as a physical system via its point spread function (PSF); derives a closed-form showing logarithmic decay and grid-induced anisotropy. \n- Shows effective resolution follows the broadened empirical Full-width half-maximum (FWHM) = 1/N_avg (instead of N_max) due to optimization-induced broadening (beta=3). \n- Analyzes finite-capacity collisions causing speckle noise and SNR loss; proposes Rotated MHE (R-MHE) with independently hashed, rotated grids to improve isotropy and SNR.\nThe paper backs its claims with experimental results, through 2D and 3D evaluations.", "questions": "1. How are the hyperparameters calculated? Where does beta value come from?\n2. Why does performance gap between 2D and 3D MHE exist?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper provides interesting extension of multi-resolution hash encoding. It provides:\n- Clear theory–practice link: closed-form PSF, anisotropy factor, and FWHM = 1/N_avg give actionable design rules.\n- Comprehensive empirical validation: broadened PSF (beta=3) consistently matches measurements; two-point resolution aligns with FWHM, not N_max. \n- Practical payoff with fixed memory: R-MHE markedly improves isotropy and PSNR in 2D (+8.06 dB to 31.30 dB at M=8) and yields steady 3D gains up to +0.65 dB.", "weaknesses": "Despite its strong theoretical foundations, the paper contains following potential shortcomings.\n- Broadening factor of beta = 3 is largely empirical; sensitivity to optimizer/architecture/data is not exhaustively characterized in additional experiments.\n- PSF-guided hyperparameter (b) underperforms a tuned baseline in 3D, and R-MHE’s 3D gains are modest (+0.65 dB).\n- The paper fundamentally relies on MHE, of which multiple extensions such as dictionary fields has been proposed already. However, the paper does not compare with these extensions which limits its interpretability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bYaNtpNOxR", "reviewer": "Reviewer_6KVQ", "summary": "## Summary\n* This paper study the Multi Resolution Hash Encodings (MHE) proposed by instant-ngp, which is used to accelerate representation of 3D scene.\n* This paper characterize the PFS of MHE in collision free assumption, concluding both theoretically and empirically that the effective resolution of MHE is not controled by max resolution $N_{MAX}$ as described by instant-ngp, but by average resolution $N_{AVG}$.\n* Besides, this paper makes two practical contribution to HME when considering collision: a scheme for parameter selection and an improvement named R-MHE.", "questions": "* Is it possible to study the effect of anisotropy reduction and collision avoidance? Which one contributes mainly to the gain in PSNR?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "## Strength\n* The characterization of MHE using PFS is novel and interesting. The conclusion that FWHM is direction dependent and its width is determined by $N_{max}$ is a very useful insight that might inspire later works.\n* The empirical on 2D Image Regression is limited but convincing.", "weaknesses": "## Weakness\n* The proposed approach is said to have two advantages, avoiding anisotropy and avoiding hashing collision. Both of the two advantages improve the empirical result. However, either because it is not possible to study the two factors independently, or the authors have not done it, I find it hard to evaluate the contribution of avoiding anisotropy to practical results. In fact, the majority of this paper is about how MHE introduces anisotropy. And I am definitely more interested in how the anisotropy property harms the performance of MHE beyond toy examples. While the current empirical result is not strong enough to show it.\n* The MHE in its original paper (instant-ngp) is well evaluated with different tasks and contains a lot more formal comparsion with other methods. I do agree that this paper has its unique theoretical insights. However, implemented their proposed approach for other tasks and include more comparsion strengthen this paper a lot.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aEtyEfP0sT", "reviewer": "Reviewer_QjpZ", "summary": "This paper introduces a method to analyze multi-resolution hash encoding used in implicit neural representations. The authors observe that while MHE is widely adopted, its hyperparameters have been tuned empirically with little mathematical analysis of its behavior. Thus, this paper introduce a point spread function based theoreritcal framework to analyze how MHE operates. Under multi-resolution, they prove that the influence region of each point feature scales with 𝑁min; they also study differences that arise during optimization and propose a model to explain them. They further propose a speckle/noise approximation to model hash collisions, and introduce R-MHE to mitigate these issues. On high-resolution image regression and 3D NeRF training, they show that the analysis is consistent with practice and yields useful hyperparameter guidance.", "questions": "- The anisotropic behavior seems applicable to other encodings (e.g., multi-plane or multi-resolution plane encodings). Do the authors have a unified formulation or insights that transfer to those settings?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Motivation: While practitioners recognize certain behaviors of hash encodings, providing a mathematical model and using it to select hyperparameters is a meaningful contribution.\n\nClarity of modeling: The paper steps through the math cleanly, organizing the required formulas at each stage.", "weaknesses": "**3D experiment observations**\n- Even when values are similar, the baseline consistently outperforming the theory in 3D suggests some assumptions may not hold in 3D. Could this stem from the alignment assumption in the theory? A toy misalignment study (small perturbations) could demonstrate how trends evolve and help validate the proposed modeling more precisely.\n\n**Theoretical growth factor experiments.**\n- Results are shown by varying b with ±0.1 steps, but: NeRF training is stochastic; single runs are not sufficient—use N repeated trials to report robustness. Use wider resolution than 0.1 steps to confirm the monotonic/expected trend across denser b values.\n\n**Practical calibration of the broadening factor.**\n- Does the broadening factor depend on training iterations or optimizer choice? If training dynamics materially affect it—even if grids don’t—then practical calibration may be bottleneck for real applying this analysis on real hyperparameter tuning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)