Gauss Circle Lattices with Geometric Convolutions for Synthesizing High Dimensional Image-Source Room Impulse Responses
For acoustic simulation researchers, this provides a tractable method for high-dimensional RIR synthesis that was previously computationally infeasible.
This work reduces the computational complexity of simulating acoustic room impulse responses via the image-source model from O(k^N) to O(N k^2 log k) by reformulating lattice point counting as the Gauss circle problem, enabling efficient synthesis in high dimensions.
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.