## 2 Pattern-Coupled Bayesian Model for Sound Field Reconstruction  
### Sound Field Model  
The sound field model in the time domain for a reverberant environment has been extensively studied, and this paper adopts the approach established in previous research [34]. We introduce an alternative version designed for improved memory efficiency and programming convenience, as the dictionary used in the model typically requires the most memory. Our version utilizes a three-dimensional tensor dictionary instead of a two-dimensional matrix dictionary, reducing memory requirements by a factor of approximately \(2/N\), where \(N\) denotes the number of sampling points in the time domain, typically ranging from several hundred to tens of thousands. Additionally, we employ cross-correlation operations in lieu of convolution operations. These modifications, namely the use of a three-dimensional tensor dictionary and cross-correlation, simplify implementation in probabilistic program ming despite introducing slight complexity to the mathematical framework and potentially making it more challenging to comprehend. The task of reconstructing a sound field in a reverberant environment involves inferring the field from a limited number of measurements. Let \(\mathbf{x}_{m}\) (\(m=1,\cdots,M\)) represent the spatial locations where the sound field is measured, let \(\mathbf{x}_{sl}\) (\(l=1,\cdots,L\)) represent the locations of the virtual sources associated with the basis functions used in the reconstruction model, and let \(\mathbf{x}_{sk}\) (\(k=1,\cdots,K\)) represent the locations where the sound field will be reconstructed. In the time domain, the sound pressure is sampled at discrete time points \(t_{n}=(n-1)t_{s}\) (\(n=1,\cdots,N\)), where \(t_{s}\) is the temporal sampling interval. This framework provides the foundation for modeling the sound field and solving the reconstruction problem. The measured sound field, denoted as \(\mathbf{\widetilde{P}}=\left[\mathbf{\widetilde{p}}_{1}^{T},\cdots,\mathbf{ \widetilde{p}}_{M}^{T}\right]^{T}\in\mathbb{R}^{M\times N}\), can be expressed as the sum of the unknown sound field \(\mathbf{P}=\left[\mathbf{p}_{1}^{T},\cdots,\mathbf{p}_{M}^{T}\right]^{T}\in \mathbb{R}^{M\times N}\) and measurement noise \(\mathbf{E}\in\mathbb{R}^{M\times N}\), such that:  
\[\mathbf{\widetilde{P}}=\mathbf{P}+\mathbf{E}, \tag{1}\]  
here, \(\mathbf{\widetilde{p}}_{m}=\left[\mathbf{\widetilde{p}}_{m,1},\cdots, \mathbf{\widetilde{p}}_{m,N}\right]\in\mathbb{R}^{N}\) represents the sound pressure measured at location \(\mathbf{x}_{m}\), while \(\mathbf{p}_{m}=\left[p_{m,1},\cdots,p_{m,N}\right]\in\mathbb{R}^{N}\) represents the unknown sound pressure at the same location. The unknown pressure \(p_{m,n}\) is numerically approximated using a linear combination of basis functions through a cross-correlation operation, expressed as:  
\[p_{m,n}\simeq\sum_{l=1}^{L}\phi_{l,m,n}\star w_{l,n}=\sum_{l=1}^{L}\sum_{i=1}^ {N}\phi_{m,j,i+n-N}w_{l,j}. \tag{2}\]  
where \(\star\) represents the one-dimensional cross-correlation operator, \(\phi_{l,m,n}\) is the discrete representation of the basis function, specifically designed for spherical wave propagation [32; 34], and \(w_{l,n}\) represents the discrete form of the unknown temporal coefficients, determined by solving the inverse problem. The discrete basis function is given by:  
\[\phi_{m,l,n}=\frac{\sin\cbr{f_{s}(nt_{s}-\
orm{\mathbf{x}_{m}-\mathbf{x}_{sl} }_{l})}}{4\pi\
orm{\mathbf{x}_{m}-\mathbf{x}_{sl}}_{2}}, \tag{3}\]  
where \(f_{s}\) is the sampling frequency, and \(c\) is the speed of sound. A zero padding of size \(N-1\) is added to the left side of sequence \(\phi_{m,l,n}\), meaning \(\phi_{m,l,n}=0\) for \(2-N\leq n\leq 0\).