Maximum Entropy Encoding of Energy-Weighted Spherical Moments

Jiaze Sun

cs.GRAug 20, 2026

Maximum Entropy Encoding of Energy-Weighted Spherical Moments

Authors: Jiaze Sun

Organizations: Northwestern Polytechnical University

Abstract

We study how angular energy signals composed of non-negative Monte Carlo path samples can be compressed and reconstructed for irradiance using finite moments. Writing each sample as an energy-weighted directional feature $x = r u$ , we adopt total energy, the first directional moment, and the traceless second moment as $1+3+5$ linearly additive, rotationally covariant statistics. Under a fixed Lebesgue reference measure, the maximum-entropy closure yields $p(r,u) \propto \exp(-βr g(u))$ , where $g(u) = 1 - b \cdot u + u^T Q u$ , whose directional probability and angular energy density are proportional to $g^{-3}$ and $g^{-4}$ , respectively. When $g_{\min} > 0$ the closure is normalizable and the reconstruction is strictly positive. We further provide analytic moment matching, variance, inverse sampling, and closed-form diffuse response for the pure-dipole four-parameter subfamily, as well as the realizability domain, partition function, azimuthal algebraic integral, and LUT-oriented reconstruction form for the dipole-second-moment coaxial five-parameter subfamily. Experiments cover 981 Poly Haven HDRI 2K scenes and three Debevec probes. Five-parameter MaxEnt achieves a 78.7% per-scene win rate against stored QZH, with mean luminance RMSE reduced by 15.8%; the advantage is more pronounced in scenes with strong directionality. Both MaxEnt variants maintain zero negative irradiance across all scenes. Full second-order SH-2 yields the lowest overall error, while five-parameter MaxEnt ranks second and outperforms SH-2 in the high-directionality bucket; the coaxial subfamily shows systematic closure error on non-coaxial multi-source scenes.

Figures & tables

Figure 1: Empirical CDF of per-scene luminance RMSE across 981 Poly Haven HDRIs (logarithmic horizontal axis). The distributions of SH-2 and MaxEnt-5 lie below the other methods overall.

Figure 2: Median luminance RMSE bucketed by scene property. Left: first-order anisotropy

\rho

; Middle: axisymmetric second moment

\eta

; Right: second-band fraction of non-DC spherical harmonic energy.

Method	Median	p75	p90	p95	p99	Mean
SH-1	0.0186	0.0304	0.0363	0.0394	0.0425	0.0205
SH-2	0.0033	0.0067	0.0085	0.0094	0.0105	0.0041
Stored QZH	0.0081	0.0099	0.0123	0.0160	0.0219	0.0082
Curve-fit QZH	0.0112	0.0136	0.0161	0.0186	0.0273	0.0109
MaxEnt-4	0.0105	0.0135	0.0163	0.0193	0.0276	0.0107
MaxEnt-5	0.0058	0.0089	0.0122	0.0159	0.0219	0.0069

Table 1: Per-scene luminance RMSE percentiles and mean across 981 Poly Haven HDRIs

Figure 3: Per-scene comparison of MaxEnt-5 vs. stored QZH. Left: RMSE scatter of the two methods (dashed line = equality); Middle, Right: difference vs.

\rho

and second-band energy fraction.

Resolution	Mode	QZH median	$\Delta\ 32^{3}$	$\Delta\ 48^{3}$	$\Delta\ 64^{3}$
$1920\times 1080$	Random	279.3	$+0.9$	$-1.8$	$+164.1$
$1920\times 1080$	Coherent	127.8	$-8.2$	$-8.4$	$-8.1$
$1920\times 1080$	Boundary	151.2	$+6.3$	$+9.4$	$+9.9$
$2560\times 1600$	Random	562.9	$+1.2$	$-9.1$	$+331.3$
$2560\times 1600$	Coherent	229.8	$+10.9$	$+12.2$	$+14.4$
$2560\times 1600$	Boundary	296.1	$+30.3$	$+34.5$	$+36.7$

Table 2: GPU-only time in

\mu\mathrm{s}

per dispatch. Each sample is the mean of a dispatch batch; the table reports the median across samples. QZH column = absolute median; three

\Delta

columns = paired medians relative to stored QZH. Host encoding and submission cost approximately

175\,\mu\mathrm{s}

/round.

Domain	LUT	RMSE	p95	p99	p99.9	Max
Physically uniform	$32^{3}$	5.5e-3	1.3e-2	2.2e-2	5.2e-2	7.8e-2
Physically uniform	$48^{3}$	4.2e-3	8.9e-3	1.7e-2	4.4e-2	7.0e-2
Physically uniform	$64^{3}$	3.4e-3	6.8e-3	1.4e-2	3.9e-2	6.4e-2
Texture uniform	$32^{3}$	6.9e-3	1.6e-2	3.1e-2	4.9e-2	1.7e-1
Texture uniform	$48^{3}$	5.6e-3	1.3e-2	2.5e-2	4.3e-2	1.7e-1
Texture uniform	$64^{3}$	4.8e-3	1.1e-2	2.1e-2	3.9e-2	1.7e-1

Table 3: LUT back-response absolute error (R32F) on three sampling domains. Error decreases with resolution; maxima occur in the boundary degenerate region.

Figure 4: Empirical CDF of LUT error on the physically uniform domain.

Figure 5: LUT error on the $q\times u$ grid (in sample-index coordinates) at the $\rho=1-10^{-3}$ surface (logarithmic color scale). Error concentrates in the $q\to 1$ degenerate region.

Method	Mean RMSE	Median	Win rate vs. QZH
Stored QZH	0.00819	0.00807	–
Exact MaxEnt	0.00690	0.00580	78.7%
LUT $32^{3}$	0.00690	0.00580	75.8%
LUT $48^{3}$	0.00690	0.00580	77.2%
LUT $64^{3}$	0.00690	0.00580	78.5%

Table 4: End-to-end luminance RMSE and win rate relative to QZH across 981 Poly Haven scenes

Figure 6: Quality–performance Pareto: horizontal axis = GPU delta relative to QZH in 4K coherent mode, vertical axis = median benefit retention rate. Circles = all scenes, triangles =

\rho\geq 0.6

bucket. All three resolutions meet the 95% retention target.

Figure 7: Luminance RMSE of the six methods on the three HDR probes. SH-2 is lowest on this dataset; five-parameter MaxEnt is close to stored QZH.

Figure 8: Per-channel RMSE heatmap for the HDR probes. SH-2 is consistently best across all channels; five-parameter MaxEnt is close to stored QZH.

Figure 9: Average RGB negative-channel pixel fraction for each method on the HDR probes. SH-1 produces substantial negatives on Grace (5.6%) and Uffizi (13.7%); SH-2 has a small fraction only on Grace (0.5%); QZH and both MaxEnt variants are 0% on all tests.

Figure 10: Axisymmetric spotlight RMSE and negative pixel fraction as a function of the directionality parameter

\lambda

. Left: luminance RMSE (logarithmic vertical axis). Right: negative pixel fraction. The MaxEnt family has zero negative pixels at all four sharpness levels.

Figure 11: Luminance absolute error percentile curves for the three HDR probes. Horizontal axis = percentile (50th, 90th, 95th, 99th); vertical axis = corresponding normalized absolute error.

Feb 19, 2026

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Etienne Lempereur, Nathanaël Cuvelle--Magar, Florentin Coeurdoux +2

D´epartement d’informatique, ENS, Universit´e PSL, Paris, France · Capital Fund Management, Paris, France · Coll`ege de France, Paris, France +3

Generative Modeling Constrained Generative Modeling Diffusion Sampling

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May 11, 2026

Cross-Domain Lossy Compression via Constrained Minimum Entropy Coupling

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Nam Nguyen, Hassan Tavakoli, An Vuong +2

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Aug 20, 2026

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A pointwise-unbiased one-bit compressor reconstructs every real input in expectation while transmitting one bit. For a scalar source

P

with CDF

F

, mean

m

, and

\mathcal J(P)=\int_{\mathbb R}\sqrt{F(r)(1-F(r))}\,dr

, we prove that the infimum of the source-averaged reconstruction second moment over all public-coin one-bit codes unbiased on

\mathbb R

m^2+\mathcal J(P)^2

. For regular full-support sources, a distribution-centered random-threshold code attains this value; a converse over arbitrary randomized binary encoders and an equality analysis characterize every attaining code up to null sets, bit relabeling, and public-seed refinement. For the Gaussian location family

\mathcal N(μ,σ^2)

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bits per Gaussian-gradient query. On Kim's continuous quadratic hard family, the expected optimization guarantee matches the lower bound in its dependence on

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Key Laboratory of System Software (Chinese Academy of Sciences) State Key Laboratory of Computer Science Institute of Software, Chinese Academy of Sciences School of Computer Science and Technology, University of Chinese Academy of Sciences Beijing, China

Low-Bit Quantization Minimax Estimation Information-Theoretic Lower Bounds

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