Gauss Circle Lattices with Geometric Convolutions for Synthesizing High Dimensional Image-Source Room Impulse Responses

2026-06-02

AI Summary

This work addresses the prohibitive computational complexity—scaling as $O(k^N)$—of conventional image-source models when synthesizing room impulse responses (RIRs) in high-dimensional spaces, which severely limits scalability. The paper introduces a novel approach by reformulating the high-dimensional image-source counting problem as a Gaussian circle lattice point enumeration task. It proposes a geometric convolution-based dimensional recurrence scheme that establishes cross-dimensional correlations and integrates frequency-dependent and reflection-weighting mechanisms. This method dramatically reduces computational complexity to $O(N k^2 \log k)$, enabling efficient RIR generation in arbitrary integer-coordinate high-dimensional spaces. The study further provides rigorous error bounds, runtime analysis, and empirical validation of statistical properties, demonstrating both theoretical soundness and practical efficacy.

Abstract

The image-source model (ISM) is a widely adopted method for efficiently simulating acoustic room impulse responses (RIRs) under specular reflection assumptions. Acoustic paths between source and receiver are traced to lattice points computed from successive reflections over bounding planes of the room. Rectangular rooms bound the total number of image-sources to be polynomial in the RIR's duration or distance $k$ equivalent, with degree equal the number of room dimensions $N$. Direct ISM simulations are therefore compute upper-bound by $O \left ( k^N \right )$, and consider only cases of $N \leq 3$ for tractability and real-world applications. This work proposes an alternative computational method that lowers the asymptotic compute bound to $O \left ( N k^2 \log k \right )$ for integer coordinates and room dimensions via reducing ISM lattice point counting to the classic Gauss circle problem (GCP). We extend the lattice counting model to frequency-dependent and reflection weighted image-sources in higher dimensions, relating solutions between successive dimensions via the convolution operator. Two constructions for realizing RIRs are presented, along with time-frequency controls, error and run-time analysis, and RIR statistics.

Problem

• image-source model
• room impulse response
• high-dimensional
• computational complexity
• Gauss circle problem

Innovation

• Gauss Circle Problem
• Image-Source Model
• Geometric Convolution
• High-Dimensional RIR Synthesis
• Computational Complexity Reduction