Maximum Entropy Encoding of Energy-Weighted Spherical Moments

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(-β r g(u)), where g(u) = 1 - b · u + uT Q u, whose directional probability and angular energy density are proportional to g-3 and g-4, respectively. When gmin > 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.