用几何卷积重构声学脉冲响应，计算效率提升至三维以上场景
Gauss Circle Lattices with Geometric Convolutions for Synthesizing High Dimensional Image-Source Room Impulse Responses
- 将声学镜像点计数转化为高维格点问题，通过几何卷积加速计算
- 计算复杂度从O(k^N)降至O(Nk²logk)，支持任意维度建模
- 适用于高维房间脉冲响应生成，适合声学仿真与空间音频研究
图像源模型（ISM）是基于镜面反射假设高效模拟声学房间脉冲响应（RIR）的常用方法。声学路径通过追踪源与接收器间的镜像点来建模，这些点由房间边界平面的连续反射计算得出。在矩形房间中，镜像源总数为时间或距离 $k$ 的多项式函数，其阶数等于房间维度 $N$，因此直接的ISM模拟计算上界为 $O(k^N)$，通常仅限于 $N \leq 3$ 维以保证可处理性与实际应用。本文提出一种新计算方法，将ISM格点计数问题转化为经典的高斯圆问题（GCP），使整数坐标下高维情况的渐近计算复杂度降至 $O(Nk^2\log k)$。进一步扩展该模型以支持频率相关和反射加权的镜像源，在不同维度间通过卷积算子建立联系。论文给出两种实现RIR的方法，包含时频控制、误差与运行时间分析及RIR统计结果。
原文摘要 · Abstract (English)
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.
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