Analytical Exploration of Spatial Audio Cues: A Differentiable Multi-Sphere Scattering Model

Siminfar Samakoush Galougah; Pranav Pulijala; Ramani Duraiswami

cs.SDMar 2, 2026

Analytical Exploration of Spatial Audio Cues: A Differentiable Multi-Sphere Scattering Model

Authors: Siminfar Samakoush Galougah, Pranav Pulijala, Ramani Duraiswami

Organizations: Perceptual Intelligence & Reality Lab, University of Maryland, College Park, MD, USA

Abstract

A primary challenge in developing synthetic spatial hearing systems, particularly underwater, is accurately modeling sound scattering. Biological organisms achieve 3D spatial hearing by exploiting sound scattering off their bodies to generate location-dependent interaural level and time differences (ITD/ILD). While Head-Related Transfer Function (HRTF) models based on rigid scattering suffice for terrestrial humans, they fail in underwater environments due to the near-impedance match between water and soft tissue. Motivated by the acoustic anatomy of underwater animals, we introduce a novel, analytically derived, closed-form forward model for scattering from a semi-transparent sphere containing two rigid spherical scatterers. This model accurately maps source direction, frequency, and material properties to the pressure field, capturing the complex physics of layered, penetrable structures. Critically, our model is implemented in a fully differentiable setting, enabling its integration with a machine learning algorithm to optimize a cost function for active localization. We demonstrate enhanced convergence for localization under noise using a physics-informed frequency weighting scheme, and present accurate moving-source tracking via an Extended Kalman Filter (EKF) with analytically computed Jacobians. Our work suggests that differentiable models of scattering from layered rigid and transparent geometries offer a promising new foundation for microphone arrays that leverage scattering-based spatial cues over conventional beamforming, applicable to both terrestrial and underwater applications. Our model will be made open source.

Figures & tables

Figure 1: Problem geometry: two rigid spheres

S_{2}

and

S_{3}

inside a semi-transparent sphere

S_{1}

. Spheres

S_{1}

and

S_{2}

are concentric. The exterior medium has density

\rho_{o}

and sound speed

c_{o}

; the interior medium has density

\rho_{i}

and sound speed

c_{i}

Figure 2: Three Spheres HRTF vs Source Phi (degrees) and Frequency (Hz)

Panel of Figure 3 — Figure 3: ILD vs Source $\phi$ (degrees) and Frequency (Hz).

Panel of Figure 4 — Figure 4: ITD vs source azimuth $\phi$ and frequency (Hz).

Figure 5: Loss landscape for the joint ILD–ITD objective as a function of source elevation and azimuth. Left: ILD component. Center: ITD component. Right: combined loss showing a single basin of attraction.

Panel of Figure 6 — Figure 6: Convergence of gradient-based localization for a single source at $(\theta^{\ast},\phi^{\ast})=(2.13,\;1.10)$ rad. (a) Angular error versus optimizer iteration. (b) Loss value and estimated $(\theta,\phi)$ versus iteration.

SNR	Mean err.	Median err.	Frac.	Frac.
(dB)	(deg)	(deg)	$<5^{\circ}$	$<10^{\circ}$
20	10.70	9.07	0.666	0.803
10	15.25	12.66	0.391	0.631
0	33.85	26.57	0.097	0.267

Table 1: Localization performance across 32 directions and 20 noise realizations per condition.

Panel of Figure 7 — Figure 7: Localization under measurement noise ( $\sigma_{\mathrm{ILD}}=0.5$ dB, $\sigma_{\mathrm{ITD}}=10~\mu$ s). (a) Angular error versus iteration. (b) Loss and parameter trajectories.

Figure 9: EKF angular tracking error over time.

Panel of Figure 10 — Figure 10: EKF tracking results for elevation (top) and azimuth (bottom).

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