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.

翻译：图像源模型（ISM）是一种在镜面反射假设下高效模拟声学房间冲激响应（RIR）的广泛采用方法。源与接收器之间的声学路径被追踪到由房间边界平面连续反射计算得到的格点。矩形房间将图像源的总数限制为RIR时长或等效距离$k$的多项式函数，其次数等于房间维度数$N$。因此直接ISM仿真的计算上界为$O \left ( k^N \right )$，且仅考虑$N \leq 3$的情况以保证可行性和实际应用。本文提出一种替代计算方法，通过将ISM格点计数问题简化为经典高斯圆问题（GCP），将整数坐标和房间维度的渐近计算下界降低至$O \left ( N k^2 \log k \right )$。我们将格点计数模型扩展至更高维度中具有频率依赖性和反射加权的图像源，并通过卷积算子建立相邻维度解之间的关联。本文提出了两种实现RIR的构造方案，同时给出了时频控制方法、误差与运行时间分析以及RIR统计特性。