cs.SDSep 18, 2026

Spherical Harmonic Sliced Wasserstein Displacement Interpolation for Acoustic Source and Reflection Density Modeling

Authors: Yuancheng Luo

Organizations: NuSpace Audio Cambridge, MA, USA

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Abstract

Spatial room impulse responses (SRIRs) capture directional distributions of acoustic sound-sources and their reflections. However, collecting SRIRs of moving sound-sources remains a challenge, requiring complex interpolations across measurements that account for multi-path spatial-temporal dynamics. This paper investigates the Wasserstein metric and displacement for evaluating interpolated SRIR echo densities in the spherical harmonic domain. We present novel sum-of-magnitude square expansions for efficiently fitting probability density functions, maximizing likelihood, inverse sampling, and computing spherical sliced Wasserstein interpolations. Experiments compare the Wasserstein displacements and metric to linear and geometric interpolations of SRIR image-source densities on a line-path, and demonstrate model-order reduction.

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Figure 1: SH density functions (max-order LC=6L_{C}=6 ) are estimated over directions ( N=100N=100 red circles) sampled from a uniform distribution over spherical coordinates. The projection kernels, spectral, and the maximum likelihood optimized densities span the range of bias-variance and computational cost trade-offs. For comparison, a uniform density with SH coefficients C=e1\bm{C}=\bm{e}_{1} (standard basis) obtains a log-likelihood of −253.1-253.1 .
Figure 2: The sample marginal CDF FΘ(θ∗)F_{\Theta}(\theta_{*}) for the maximum log-likelihood SH expansion in Fig. 1c integrates the PDF over the cap (0≤Θ≤θ∗, 0≤Φ<2π)(0\leq\Theta\leq\theta_{*},\,0\leq\Phi<2\pi) along the meridian semi-circle.
Figure 3: Wasserstein interpolations via Eqs. ( 17 ), ( 18 ) smoothly transport the SH expansion of squared exponential kernel (max-order L=8L=8 , ℓ=0.75\ell=0.75 ) of chordal distances [ 16 ] density (left column) to bifurcated and randomized densities (right column) across time-steps t∈{0, 0.25, 0.5, 0.75, 1}t\in\left\{{0,\,0.25,\,0.5,\,0.75,\,1}\right\} .
Figure 4: Ground-truth RIRs of an omni-directional sound-source to receiver are computed over a left-to-right line-path indexed by time-stamp tt , and their direct and image-sources expanded over the SH domain (top row). Source densities Dt∗\bm{D}_{t*} aggregate direct and image-source SH expansions over the RIR’s duration at each tt . Linear, product, and Wasserstein interpolated densities Dt\bm{D}_{t} at end-point supports ( t=0t=0 , t=1t=1 ) are compared to ground-truth densities, and shown for t=0.5t=0.5 (bottom row). Wasserstein interpolations achieve lowest SSW distances across the entire path.
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