Gauss Circle Lattices with Geometric Convolutions for Synthesizing High Dimensional Image-Source Room Impulse Responses
2026-06-03 • Sound
Sound
AI summaryⓘ
The authors study a common way to simulate how sound bounces in a room, called the image-source model (ISM). Normally, calculating these sound reflections gets very slow as the room size or time grows, especially in rooms with up to three dimensions. They propose a new math approach that makes this calculation much faster by connecting the problem to a famous math puzzle about counting points in a circle. Their method also works for more complicated cases involving sound frequency and reflection strength, and they analyze the performance and accuracy of their approach.
image-source modelroom impulse responsespecular reflectionlattice pointsGauss circle problemasymptotic complexityconvolution operatorfrequency-dependent reflectionrun-time analysis
Authors
Yuancheng Luo
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.