Introduction

Near-field acoustic holography (NAH) is a powerful tool for sound source identification and three-dimensional sound field visualization, with widespread applications in structural fault diagnosis, noise, vibration, and harshness (NVH) analysis, as well as acoustic design optimization1,2,3,4. However, conventional NAH implementations are fundamentally constrained by the Nyquist sampling theorem, which requires a dense spatial sampling grid to achieve high-resolution reconstructions. For a fixed hologram aperture determined by the physical extent of the source, achieving high spatial resolution necessitates an extremely small inter-sensor spacing, leading to a dramatic increase in the number of required measurement points. This significantly increases both time and economic costs associated with data acquisition and poses severe practical limitations for real-world engineering implementations.

Compressive sensing (CS) theory provides a promising framework to overcome the limitations imposed by the Nyquist theorem5,6. The core premise of this theory is that if a signal exhibits sparsity in a specific transform domain, the original signal can be reconstructed with high probability from a limited set of non-coherent measurements using appropriate optimization algorithms. By integrating CS with NAH, high-accuracy sound field reconstruction becomes achievable with significantly fewer sampling points than required by the Nyquist criterion, thereby enhancing the efficiency and applicability of NAH. The CS comprises three fundamental components: sparse representation of the underlying signal7, design of a measurement system with low coherence between sensing matrix and sparsifying bases, and a robust reconstruction algorithm capable of solving the resulting underdetermined inverse problem. Among these, the choice of reconstruction algorithm critically influences the trade-off between accuracy and computational complexity. To balance accuracy and efficiency, various CS reconstruction algorithms have been developed, including greedy algorithms8,9,10, convex optimization methods11,12, iterative thresholding algorithms13,14, and sparse Bayesian learning (SBL) algorithms15,16,17. Greedy algorithms, such as orthogonal matching pursuit (OMP)18,19, sequentially select atoms that best correlate with the residual signal. Despite their computational efficiency, these methods lack global optimality guarantees and are prone to suboptimal solutions due to myopic selection criteria. Convex optimization methods convert the non-convex sparsity-promoting problem into a convex formulation by replacing the ℓ0 norm with the ℓ1 norm, enabling sparse signal recovery via linear programming. Though these methods exhibit strong theoretical guarantees and superior accuracy, they suffer from high computational costs and slow convergence. Iterative thresholding algorithms strike a balance between efficiency and accuracy by iteratively applying thresholding operations to approximate the sparse solution. However, they still face challenges in achieving global convergence, often tending toward local optima. In contrast, SBL algorithms operates within a probabilistic framework, modeling sparsity via hierarchical priors and inferring the posterior distribution of sparse coefficients using Bayesian inference. Notably, they demonstrate robustness to coherent dictionaries and excel in identifying sparse solutions under highly undersampled conditions.

A critical limitation of conventional CS theory lies in its implicit assumption of randomly distributed non-zero coefficients, which neglects potential structured sparsity inherent in many practical signals. For instance, wavelet coefficients in images often exhibit tree-like dependencies, while multi-channel or multi-band signals display block-sparse characteristics, where non-zero entries cluster in predefined or unknown groups. Exploiting such structured sparsity can substantially improve reconstruction performance by incorporating prior knowledge about the spatial or spectral grouping of non-zero entries. Recent advances have led to the development of structured sparsity models, including block orthogonal matching pursuit (BOMP)20,21, mixed ℓ2/ℓ1 norm optimization22, group least absolute shrinkage and selection operator (Group LASSO) method23, structured orthogonal matching pursuit (StructOMP)24, and model-based compressive sampling matching pursuit (model-based CoSaMP)25. Within the Bayesian framework, further improvements in signal reconstruction performance have been achieved through algorithms that explicitly model intra-group correlations. However, in real-world applications, owing to the absence of a priori information regarding block partitioning, a mismatch might occur between the actual block structure and the predefined block configurations, which in turn leads to low reconstruction accuracy. Recently, a notable advancement has been the pattern-coupled sparse Bayesian learning (PC-SBL) algorithm26,27,28, which incorporates hyperparameter coupling across neighboring coefficients to encourage spatially clustered sparsity patterns. Building on this work, Xiao et al. proposed the pattern-coupled Bayesian compressive sensing (PCBCS) algorithm29, which introduces a fixed weighting parameter to link the prior distribution of each sparse coefficient element to its own hyperparameter and those of its neighbors. This encourages the aggregation of zero and non-zero elements into block-like structures without requiring prior knowledge of the block pattern.

However, in practical applications, the correlations between hyperparameters are often non-uniform and signal-dependent. To address the limitations of fixed coupling parameters, this paper proposes a data-adaptive pattern-coupled Bayesian compressive sensing (DA-PCBCS) algorithm for sparse sound field reconstruction. The proposed method integrates the equivalent source method (ESM) into a Bayesian inference framework, establishing a hierarchical Gaussian-Gamma prior model in which the sparse coefficients of the equivalent source strengths are inferred. Unlike PCBCS, which employs fixed coupling parameters, the proposed method introduces a set of data-adaptive coupling parameters that are iteratively updated alongside hyperparameters via the expectation-maximization (EM) algorithm, enabling accurate estimation of the maximum a posterior (MAP) solution for the sparse coefficients of equivalent source strengths. By dynamically characterizing the correlations between adjacent hyperparameters, these adaptive parameters more effectively guide the formation of block-like structures in the sparse solution, thereby improving reconstruction performance for signals with unknown block structures. Furthermore, the proposed algorithm achieves high reconstruction accuracy even when the noise variance is unknown, demonstrating robust performance and improved practical utility in engineering applications.

The remainder of this paper is organized as follows: Sect. 2 details the reconstruction theory of the proposed DA-PCBCS method. Section 3 evaluates the reconstruction performance via numerical simulations, including comparisons with Bayesian compressive sensing (BCS) and PCBCS to validate its superiority. Section 4 further verifies the method’s effectiveness through experimental results. Finally, Sect. 5 presents the conclusions.

Theory

ESM in the CS framework

In near-field acoustic holography based on the ESM, the radiated sound field of an actual sound source can be replaced by that generated by a series of equivalent sources30,31. By measuring the sound pressure data on the hologram plane, the source strengths of the equivalent sources are determined via inverse problem solving. Assuming that M microphones are distributed on the hologram plane (i.e., the measurement plane) and N equivalent sources are placed on the equivalent source plane, the sound pressure at the measurement points on the hologram plane can be expressed as

$${{\mathbf{P}}_{\text{h}}}{\mathbf{=}}{{\mathbf{G}}_{\text{h}}}{\mathbf{q+\varepsilon }}$$
(1)

where \({{\mathbf{P}}_{\text{h}}}\) is the sound pressure vector of M measurement points, \({\mathbf{q}}\) is the strengths vector of N equivalent sound sources, \(\bf \varepsilon\) is the noise vector of the zero-mean Gaussian distribution with a variance of \({\sigma ^2}\), \({{{{\mathbf{G}}_{\text{h}}}\left( {m,n} \right)=\exp \left( {ik\left| {{{\mathbf{r}}_m} - {{\mathbf{r}}_n}} \right|} \right)} \mathord{\left/ {\vphantom {{{{\mathbf{G}}_{\text{h}}}\left( {m,n} \right)=\exp \left( {ik\left| {{{\mathbf{r}}_m} - {{\mathbf{r}}_n}} \right|} \right)} {\left( {4\pi \left| {{{\mathbf{r}}_m} - {{\mathbf{r}}_n}} \right|} \right)}}} \right. \kern-0pt} {\left( {4\pi \left| {{{\mathbf{r}}_m} - {{\mathbf{r}}_n}} \right|} \right)}}\) is the transfer matrix between the m-th measurement point \({{\mathbf{r}}_m}\) and the n-th equivalent source point \({{\mathbf{r}}_n}\), and k is the wave number.

In the CS framework, the strengths of equivalent sources can be sparsely represented through an orthogonal basis matrix as

$${\mathbf{q=\Phi \mathbf{\theta} }}$$
(2)

where \({\mathbf{\theta }}\) is the sparse basis matrix of the equivalent source strengths \({\mathbf{q}}\), and \({\mathbf{\theta }}\) is the sparse coefficient vector of the equivalent source strengths.

The transfer matrix \({{\mathbf{G}}_{\text{r}}}\) between the equivalent source and the reconstructed plane can be decomposed via singular value decomposition as \({{\mathbf{G}}_{\text{r}}}{\mathbf{=US}}{{\mathbf{V}}^{\text{H}}}\), where \({\mathbf{U}}\) and \({\mathbf{V}}\) are left unitary matrix and right unitary matrix, respectively, \({\mathbf{S}}\) is a diagonal matrix composed of positive singular values, and the superscript “H” denotes the conjugate transpose.

Mathematically, the columns of the left singular matrix \({\mathbf{U}}\) form an orthogonal basis for the reconstructed sound pressure space; each column is a vector whose dimension equals the number of reconstruction points. This makes \({\mathbf{U}}\) suitable for representing or compressing the reconstructed pressure field. In contrast, the columns of the right singular matrix \({\mathbf{V}}\) constitute an orthogonal basis for the equivalent source-strength space, with each column’s dimension matching the number of equivalent sources. Since the core of the equivalent source method is to solve for the source strengths, and compressive sensing seeks a sparse representation in the source-strength space, \({\mathbf{V}}\) is the mathematically rigorous and natural choice for the sparse basis matrix \({\mathbf{\theta }}\). Physically, the column vectors of \({\mathbf{V}}\) correspond to acoustic radiation modes. These modes are ordered by descending radiation efficiency and exhibit inherent energy compaction, such that the first few modes capture the dominant acoustic energy, thus providing a physically meaningful sparse representation basis for the source strengths32,33. Compared with the left singular matrix \({\mathbf{U}}\) or other potential bases such as those adopted in compressed modal ESM (CMESM)12 or compressive-equivalent source method (C-ESM)34, \({\mathbf{V}}\) acts directly on the source-strength space, better preserves phase information, and maintains high accuracy over a broad frequency range6,29,35,36. Furthermore, derived automatically from the SVD of the transfer matrix, this basis ensures inherent adaptability to specific geometries and boundary conditions and provides built-in regularization, thereby improving algorithmic stability. Based on the above, the right singular matrix \({\mathbf{V}}\) is selected as the sparse basis matrix (i.e., \({\mathbf{\theta }}\) = \({\mathbf{V}}\)), given its demonstrated high reconstruction accuracy.

Substituting Eq. (2) into Eq. (1), the measured sound pressure on the hologram plane can be expressed as

$${{\mathbf{P}}_{\text{h}}}{\mathbf{=}}{{\mathbf{G}}_{\text{h}}}{\mathbf{V\theta +\varepsilon =H\theta +\varepsilon }}$$
(3)

where \({\mathbf{H=}}{{\mathbf{G}}_{\text{h}}}{\mathbf{V}}\) is the sensing matrix.

When the number of measurement points M is smaller than the number of equivalent sources N, the estimation of equivalent source strengths in Eq. (3) becomes an underdetermined problem. This can be solved by minimizing the ℓ₁ norm of the sparse coefficient vector \({\mathbf{\theta }}\) as

$$\arg\mathop {{\text{min}}}\limits_{{\mathbf{\theta }}} \left( {\left\| {{{\mathbf{P}}_{\text{h}}} - {\mathbf{H\theta }}} \right\|_{2}^{2}+\eta {{\left\| {\mathbf{\theta }} \right\|}_1}} \right)$$
(4)

Sound field reconstruction based on DA-PCBCS

As a probabilistic signal-processing method, sparse Bayesian compressive sensing does not naturally facilitate separate estimation of the real and imaginary components of complex sound pressure. Therefore, the complex-valued model in Eq. (3) is recast into a real-valued one by decomposing its real and imaginary parts as

$${\mathbf{P=GW+n}}$$
(5)

where \({\mathbf{P}}=\left[ {\begin{array}{*{20}{c}} {\operatorname{Re} \left( {{{\mathbf{P}}_{\text{h}}}} \right)} \\ {\operatorname{Im} \left( {{{\mathbf{P}}_{\text{h}}}} \right)} \end{array}} \right]\), \({\mathbf{G}}=\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\operatorname{Re} \left( {\mathbf{H}} \right)} \\ {\operatorname{Im} \left( {\mathbf{H}} \right)} \end{array}}&{\begin{array}{*{20}{c}} { - \operatorname{Im} \left( {\mathbf{H}} \right)} \\ {\operatorname{Re} \left( {\mathbf{H}} \right)} \end{array}} \end{array}} \right]\), \({\mathbf{W}}=\left[ {\begin{array}{*{20}{c}} {\operatorname{Re} \left( {\mathbf{\theta }} \right)} \\ {\operatorname{Im} \left( {\mathbf{\theta }} \right)} \end{array}} \right]\), \({\mathbf{n}}=\left[ {\begin{array}{*{20}{c}} {\operatorname{Re} \left( {\mathbf{\varepsilon }} \right)} \\ {\operatorname{Im} \left( {\mathbf{\varepsilon }} \right)} \end{array}} \right]\), \({\mathbf{P}} \in {{\mathbb{R}}^{2M}}\), \({\mathbf{G}} \in {{\mathbb{R}}^{2M \times 2N}}\), \({\mathbf{W}} \in {{\mathbb{R}}^{2N}}\), \({\mathbf{n}} \in {{\mathbb{R}}^{2M}}\), \(\operatorname{Re} ( \cdot )\) and \(\operatorname{Im} ( \cdot )\) denote the real and imaginary parts of taking the corresponding quantities, respectively.

The sparse Bayesian compressive sensing algorithm aims to determine the optimal sparse coefficient vector for the equivalent source strengths \({\mathbf{W}}\). In conventional Bayesian compressive sensing, the sparsity of equivalent source coefficients is governed by hyperparameters within a hierarchical Gaussian prior model, where each hyperparameter is assumed to be independent37. However, structured sparsity, such as block or tree sparsity, is prevalent in real-world signals, and traditional independent hyperparameter models often result in suboptimal reconstruction performance. To address this issue, a pattern-coupled Bayesian compressive sensing algorithm has been proposed to reconstruct sparse signals with unknown block structures. This algorithm posits that the hyperparameters controlling element sparsity are interdependent, and it employs a fixed parameter to regulate the correlation between adjacent hyperparameters. Nevertheless, in most practical scenarios, the correlations among different hyperparameters are non-uniform. Inspired by this observation, the algorithm developed in this paper adopts the concept of signal element coupling as its foundation. It introduces a novel parameter, termed the coupling parameter, to govern the correlation between adjacent hyperparameters, which is then incorporated into the Gaussian prior model. The data-adaptive coupling mechanism eliminates the need for predefined correlation constraints, while the Gaussian-Gamma prior provides a statistically rigorous foundation for modeling structured sparsity. The Bayesian hierarchy of the proposed data-adaptive pattern-coupled Bayesian compressive sensing method is depicted in Fig. 1.

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
Full size image

Graphical model of the data-adaptive pattern-coupled Bayesian compressive sensing.

In the pattern-coupled Bayesian compressive sensing, the prior for each coefficient depends on both its individual hyperparameter and neighboring hyperparameters. Formally, the prior over the sparse coefficient vector of equivalent source strengths \({\mathbf{W}}\) can be expressed as

$$\mathcal{P}\left( {{\mathbf{W|\boldsymbol{\gamma} }}} \right)=\prod\limits_{{i=1}}^{{2N}} {\mathcal{P}\left( {{{\mathbf{W}}_i}{\mathbf{|}}{\gamma _i},{\gamma _{i - 1}},{\gamma _{i+1}}} \right)}$$
(6)

in which

$$\mathcal{P}\left( {{{\mathbf{W}}_i}{\mathbf{|}}{\gamma _i},{\gamma _{i - 1}},{\gamma _{i+1}}} \right)=\mathcal{N}\left( {{{\mathbf{W}}_i}|{\text{0}},{{\left( {{\gamma _i}+\beta {\gamma _{i - 1}}+\beta {\gamma _{i+1}}} \right)}^{ - 1}}} \right)$$
(7)

where the parameter \(\beta ({\text{0}} \leqslant \beta \leqslant {\text{1}})\) quantifies the correlation between the signal element \({{\mathbf{W}}_i}\) and its adjacent elements \({{\mathbf{W}}_{i - 1}}\) and \({{\mathbf{W}}_{i+1}}\). As seen from from Eq. (7), when \(\beta ={\text{0}}\), the model reduces to the conventional sparse Bayesian framework. When \(\beta>{\text{0}}\), the sparsity of \({{\mathbf{W}}_i}\) is no longer determined solely by \({\gamma _i}\), but also influenced by \({\gamma _{i - 1}}\) and \({\gamma _{i+1}}\). Specifically, as \({\gamma _i}\) approaches infinity, \({{\mathbf{W}}_i}\) tends to zero, and \({\gamma _i}\) further affects the prior distributions of \({{\mathbf{W}}_{i - 1}}\) and \({{\mathbf{W}}_{i+1}}\). This hierarchical Gaussian model enables adaptive interactions in the sparsity of adjacent signal elements, thereby promoting the formation of block structures. However, in most practical scenarios, the correlations among different hyperparameters are non-uniform, and fixing a predetermined parameter to model these correlations is often inappropriate.

In the proposed algorithm, unknown coupling parameters are adopted to represent the correlation between adjacent hyperparameters, and these parameters are incorporated into the Gaussian prior model. The independent hyperparameters \(\boldsymbol{\gamma} = {\left[ {{\gamma _1},{\gamma _2}, \cdots ,{\gamma _{2N}}} \right]^{\text{T}}}\) undergo a projection transformation to yield correlated hyperparameters, i.e., \({\mathbf{\tilde {\gamma }}}={\left[ {{{\tilde {\gamma }}_1},{{\tilde {\gamma }}_2}, \cdots ,{{\tilde {\gamma }}_{2N}}} \right]^{\text{T}}}={\mathbf{T\gamma }}\), and the transformation matrix \({\mathbf{T}}\) contains non-zero elements solely along its main diagonal and first sub-diagonals can be expressed as

$${\mathbf{T}}=\left[ {\begin{array}{*{20}{c}} 1&{\beta _{2}^{1}}&{}&{}&{} \\ {\beta _{1}^{2}}&1&{\beta _{3}^{2}}&{}&{} \\ {}& \ddots & \ddots & \ddots &{} \\ {}&{}&{\beta _{{2N - 2}}^{{2N - 1}}}&1&{\beta _{{2N}}^{{2N - 1}}} \\ {}&{}&{}&{\beta _{{2N - 1}}^{{2N}}}&1 \end{array}} \right]$$
(8)

where \(\beta _{{i - 1}}^{i}\) and \(\beta _{{i+1}}^{i}\) denote the introduced parameter, referred to as the coupling parameter, which are used to quantify the influence of hyperparameters \({\gamma _{i - 1}}\) and \({\gamma _{i+1}}\) on \({\gamma _i}\). The hyperparameter \({\tilde {\gamma }_i}\) controlling the sparsity of elements in the proposed algorithm, can be expressed as

$${\tilde {\gamma }_i}={\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}},\begin{array}{*{20}{c}} {} \end{array}i=1, \cdots ,2N$$
(9)

Thus, the prior for the sparse coefficient element \({{\mathbf{W}}_i}\) can be rewritten as

$$\mathcal{P}\left( {{\mathbf{W|\tilde {\mathbf{\gamma}}}}} \right)=\prod\limits_{{i=1}}^{{2N}} {\mathcal{P}\left( {{{\mathbf{W}}_i}{\mathbf{|}}{{\tilde {\gamma }}_i}} \right)} =\prod\limits_{{i=1}}^{{2N}} {\mathcal{N}\left( {{{\mathbf{W}}_i}|{\text{0}},{{\left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right)}^{ - 1}}} \right)} ,\begin{array}{*{20}{c}} {} \end{array}i=1, \cdots ,2N$$
(10)

where \(\beta _{0}^{1}=\beta _{{2N+1}}^{{2N}}=0\).

Compared with the pattern-coupled sparse Bayesian algorithm, the proposed method replaces the fixed constant with a data-adaptive coupling parameter to capture correlations among neighboring hyperparameters. Thus, the correlations governing sparsity are entirely data-driven and iteratively derived, enabling adaptive structuring of neighboring sparse coefficients.

The noise \({{\mathbf{n}}}\) can be considered to follow a zero-mean Gaussian distribution with the precision \(\lambda ={1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}}}} \right. \kern-0pt} {{\sigma ^2}}}\),

$$\mathcal{P}\left( {{\mathbf{n}}|\lambda } \right)=\prod\limits_{{j=1}}^{{2M}} {\mathcal{P}\left( {{{\mathbf{n}}_j}|\lambda } \right)=\prod\limits_{{j=1}}^{{2M}} {\mathcal{N}\left( {{{\mathbf{n}}_j}|0,{\lambda ^{ - 1}}} \right)} }$$
(11)

According to sparse Bayesian learning, the second layer specifies a Gamma distribution as a hyperprior on the hyperparameter \({\mathbf{\gamma }}\) to adaptively promote sparsity in the solution as

$$\mathcal{P}\left( {\mathbf{\gamma }} \right)=\prod\limits_{{i=1}}^{{2N}} {\operatorname{\Gamma} \left( {{{\gamma} _i}|a,b} \right)=\prod\limits_{{i=1}}^{{2N}} {{{\varvec{\Gamma}}}{{\left( a \right)}^{ - 1}}{b^a}{\gamma} _{i}^{{a - 1}}{{\text{e}}^{ - b{\gamma _i}}}} }$$
(12)

where \({{\varvec{\Gamma}}}\left( a \right)=\int_{0}^{\infty } {{t^{a - 1}}{{\text{e}}^{ - t}}{\text{d}}t}\) is the Gamma function. Adopting the Gamma distribution as a hyperprior in Bayesian signal reconstruction induces sparsity by driving irrelevant coefficients to near zero while preserving significant non-zero components. Conventional Bayesian compressive sensing sets the hyperparameters a and b to near-zero values (e.g., \(a=b={10^{ - 4}}\)) for uninformative priors, yielding approximately uniform hyperparameter distributions. Conversely, the proposed algorithm assigns a a larger value (e.g., \(a=0.{\text{5}}\)), elevating the hyperparameter magnitude to actively promote sparsity in signal reconstruction.

Similarly, a Gamma hyperprior is placed on \(\it \lambda\) to estimate the noise variance as

$$\mathcal{P}\left( \lambda \right)=\operatorname{Gamma} \left( {\lambda |c,d} \right)={{\varvec{\Gamma}}}{\left( c \right)^{ - 1}}{d^c}{\lambda ^{c - 1}}{{\text{e}}^{ - d\lambda }}$$
(13)

where hyperparameters c and d are small values (e.g., \(c=d={10^{ - 4}}\)) to enforce non-informative priors, thereby preventing undue influence from predetermined assumptions.

Since the noise in the signal model follows a zero-mean Gaussian distribution with variance \({\lambda ^{ - 1}}\), the hierarchical Gaussian prior model yields \(\mathcal{P}\left( {{\mathbf{P}}|{\mathbf{W}};\lambda } \right)=\mathcal{N}\left( {{\mathbf{GW}},{\lambda ^{ - 1}}{\mathbf{I}}} \right)\), with \({{\mathbf{I}}}\) being the identity matrix. Consequently, the posterior density of \({{\mathbf{W}}}\) with given \(\left\{ {{\mathbf{\gamma }},\lambda ,{{\mathbf{\beta }}_1},{{\mathbf{\beta }}_2}} \right\}\) is then obtained as \(\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{P}},{\mathbf{\gamma }},\lambda ;{{\mathbf{\beta }}_1},{{\mathbf{\beta }}_2}} \right)=\mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)\), where

$${\mathbf{\mu }}=\lambda {\mathbf{\Sigma }}{{\mathbf{G}}^{\text{T}}}{\mathbf{P}}$$
(14)
$${\mathbf{\Sigma }}={\left( {\lambda {{\mathbf{G}}^{\text{T}}}{\mathbf{G}}+{\mathbf{D}}} \right)^{ - 1}}$$
(15)

where µ and Σ denote the mean and the covariance of this multivariate Gaussian distribution, respectively, and \({\mathbf{D}}={\text{diag}}\left( {{\mathbf{\tilde {\gamma }}}} \right)={\text{diag}}\left( {{\mathbf{T\gamma }}} \right)={\text{diag}}\left( {{\gamma _1}+\beta _{2}^{1}{\gamma _2}, \cdots ,{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}, \cdots ,{\gamma _{2N}}+\beta _{{2N - 1}}^{{2N}}{\gamma _{2N - 1}}} \right)\).

Given estimated hyperparameters \(\left\{ {{\mathbf{\gamma }},\lambda } \right\}\), the MAP estimate of \({{\mathbf{W}}}\) corresponds to the mean of the posterior distribution, i.e.,

$${{\mathbf{\hat {W}}}_{{\text{MAP}}}}={\mathbf{\mu }}=\lambda {\left( {\lambda {{\mathbf{G}}^{\text{T}}}{\mathbf{G}}+{\mathbf{D}}} \right)^{ - 1}}{{\mathbf{G}}^{\text{T}}}{\mathbf{P}}$$
(16)

To facilitate the description of the algorithm, let \({{\mathbf{\beta }}_1}=\left[ {\beta _{1}^{2}, \cdots ,\beta _{{i - 1}}^{i}, \cdots ,\beta _{{2N - 1}}^{{2N}}} \right]\) and \({{\mathbf{\beta }}_2}=\left[ {\beta _{2}^{1}, \cdots ,\beta _{i}^{{i - 1}}, \cdots ,\beta _{{2N}}^{{2N - 1}}} \right]\). The next step is to apply the expectation-maximization (EM) algorithm38 to estimate the parameter set \(\left\{ {{\mathbf{\gamma }},\lambda ,{{\mathbf{\beta }}_1},{{\mathbf{\beta }}_2}} \right\}\) by maximizing the posterior probability \(\mathcal{P}\left( {{\mathbf{\gamma }},\lambda |{\mathbf{W}};{{\mathbf{\beta }}_1},{{\mathbf{\beta }}_2}} \right)\). Define \({\mathbf{v }}=\left\{ {{\mathbf{\gamma }},\lambda } \right\}\) as the set of hyperparameters and \({\mathbf{\beta }}=\left\{ {{{\mathbf{\beta }}_1},{{\mathbf{\beta }}_2}} \right\}\) as the set of coupling parameters. The EM algorithm treats \({{\mathbf{W}}}\) as a hidden variable and maximizes the expectation of the complete log-posterior of \(\mathcal{P}\left( {{\mathbf{v }}|{\mathbf{P}},{\mathbf{W}};{\mathbf{\beta }}} \right)\), i.e., \({E_{{\mathbf{W}}|{\mathbf{P}},{\mathbf{v}},{\mathbf{\beta }}}}\left[ {\log \mathcal{P}\left( {{\mathbf{v }}|{\mathbf{P}},{\mathbf{W}};{\mathbf{\beta }}} \right)} \right]\), where the operator \({E_{{\mathbf{W}}|{\mathbf{P}},{\mathbf{v}},{\mathbf{\beta }}}}\left[ \cdot \right]\) denotes an expectation with respect to the posterior \(\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{P}},{\mathbf{v}},{\mathbf{\beta }}} \right)\). The EM algorithm iteratively produces a sequence of estimates \({{\mathbf{v}}^{(t)}}\) and \({{\mathbf{\beta }}^{(t)}}\) via two alternating sub-steps, referred to as the E-step and the M-step, where the superscript “(t)” denotes the t-th iteration.

E-step

Given the current estimate of the parameter set \(\left\{ {{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}} \right\}\) and the measurement data \({{\mathbf{P}}}\), calculate the expected value (with respect to the missing variables \({{\mathbf{W}}}\)) of the complete log-posterior of \(\left\{ {{\mathbf{v}},{\mathbf{\beta }}} \right\}\), which is also referred to as the Q-function and can be expressed as a summation of two terms.

$$\begin{gathered} Q\left( {{\mathbf{v }},{\mathbf{\beta }}|{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}} \right)=Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }},\lambda |{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}},{\lambda ^{(t)}}} \right) \\ =Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right){\text{+}}Q\left( {\lambda |{\lambda ^{(t)}}} \right) \\ ={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{W}},{\mathbf{\gamma }};{\mathbf{\beta }}} \right)} \right]+{E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{P}},\lambda |{\mathbf{W}}} \right)} \right] \\ \end{gathered}$$
(17)

Note that when calculating the Q-function with respect to \(\left\{ {{\mathbf{\gamma }};{\mathbf{\beta }}} \right\}\), \(\lambda\) is treated as a constant, and vice versa. Thus, the first term on the right hand side of Eq. (17) can be expressed as

$$\begin{gathered} Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right)={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{W}},{\mathbf{\gamma }};{\mathbf{\beta }}} \right)} \right] \\ ={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \left( {\mathcal{P}\left( {\mathbf{\gamma }} \right)\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{\gamma }};{\mathbf{\beta }}} \right)} \right)} \right] \\ =\log \mathcal{P}\left( {\mathbf{\gamma }} \right){\text{+}} \\ \frac{1}{2}\sum\limits_{{i=1}}^{{2N}} {\left[ {{\text{log}} \left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right) - \left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right)\int {\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{P}},{{\mathbf{\gamma }}^{(t)}},{{\mathbf{\beta }}^{(t)}}} \right)} {\mathbf{W}}_{i}^{2}{\text{d}}{\mathbf{W}}} \right]} \\ \end{gathered}$$
(18)

Since the posterior \(\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{P}},{{\mathbf{\gamma }}^{(t)}},{{\mathbf{\beta }}^{(t)}}} \right)\) is a multivariate Gaussian distribution with its mean and covariance matrix given by Eqs. (14) and (15), the expected value of partial terms in above functions can be calculated as

$$\int {\mathcal{P}\left( {{\mathbf{W}}|{\mathbf{P}},{{\mathbf{\gamma }}^{(t)}},{{\mathbf{\beta }}^{(t)}}} \right)} {\mathbf{W}}_{i}^{2}{\text{d}}{\mathbf{W}}={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{\gamma }}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}[{\mathbf{W}}_{i}^{2}]=\mu _{i}^{2}+{\Sigma _{i,i}}$$
(19)

where \({\mu _i}\) denotes the i-th entry of \({{\mathbf{\mu }}^{(t)}}\), and \({\Sigma _{i,i}}\) denotes the i-th diagonal element of the covariance matrix \({{\mathbf{\Sigma }}^{(t)}}\). With the specified prior in Eq. (12), Eq. (18) can eventually be written as

$$Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right)=\sum\limits_{{i=1}}^{{2N}} {\left( {a\log {\gamma _i} - b{\gamma _i}+\frac{1}{2}\log \left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right) - \frac{1}{2}\left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right)\left( {\mu _{i}^{2}+{\Sigma _{i,i}}} \right)} \right)}$$
(20)

Similarly, the second term on the right hand side of Eq. (17) can be expressed as

$$\begin{gathered} Q\left( {\lambda |{\lambda ^{(t)}}} \right)={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{P}},\lambda |{\mathbf{W}}} \right)} \right] \\ ={E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \left( {\mathcal{P}\left( \lambda \right)\mathcal{P}\left( {{\mathbf{P}}|{\mathbf{W}},\lambda } \right)} \right)} \right] \\ =\left( {M+c} \right)\log \lambda - \frac{\lambda }{2}{E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\left\| {{\mathbf{P}} - {\mathbf{GW}}} \right\|_{{\text{2}}}^{{\text{2}}}} \right] - d\lambda \\ \end{gathered}$$
(21)

where

$$\begin{gathered} {E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\left\| {{\mathbf{P}} - {\mathbf{GW}}} \right\|_{{\text{2}}}^{{\text{2}}}} \right]={{\mathbf{P}}^{\text{T}}}{\mathbf{P}} - 2E\left[ {{{\mathbf{W}}^{\text{T}}}{{\mathbf{G}}^{\text{T}}}{\mathbf{P}}} \right]+E\left[ {{{\mathbf{W}}^{\text{T}}}{{\mathbf{G}}^{\text{T}}}{\mathbf{GW}}} \right] \\ ={{\mathbf{P}}^{\text{T}}}{\mathbf{P}} - 2{\left[ {{{\mathbf{\mu }}^{(t)}}} \right]^{\text{T}}}{{\mathbf{G}}^{\text{T}}}{\mathbf{P}}+{\left[ {{{\mathbf{\mu }}^{(t)}}} \right]^{\text{T}}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}{{\mathbf{\mu }}^{(t)}}+{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right) \\ =\left\| {{\mathbf{P}} - {\mathbf{G}}{{\mathbf{\mu }}^{(t)}}} \right\|_{2}^{2}+{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right) \\ \end{gathered}$$
(22)

Then, Eq. (21) can eventually be written as

$$Q\left( {\lambda |{\lambda ^{(t)}}} \right)=\left( {M+c} \right)\log \lambda - \left( {d+\frac{1}{2}\left\| {{\mathbf{P}} - {\mathbf{G}}{{\mathbf{\mu }}^{(t)}}} \right\|_{2}^{2}+\frac{1}{2}{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right)} \right)\lambda$$
(23)

M-step

In the M-step, the updates of hyperparameters \(\left\{ {{\mathbf{\gamma }},{\mathbf{\beta }}} \right\}\) and \(\lambda\) can be obtained by alternately maximizing the Q-function, i.e.,

$$\begin{gathered} \left\{ {{{\mathbf{\gamma }}^{(t+1)}};{{\mathbf{\beta }}^{(t+1)}}} \right\}=\mathop {\arg \hbox{max} }\limits_{{{\mathbf{\gamma }};{\mathbf{\beta }}}} Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right) \\ =\mathop {\arg \hbox{max} }\limits_{{{\mathbf{\gamma }};{\mathbf{\beta }}}} {E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{W}},{\mathbf{\gamma }};{\mathbf{\beta }}} \right)} \right] \\ \end{gathered}$$
(24)
$$\begin{gathered} {\lambda ^{(t+1)}}=\mathop {\arg \hbox{max} }\limits_{\lambda } Q\left( {\lambda |{\lambda ^{(t)}}} \right) \\ =\mathop {\arg \hbox{max} }\limits_{\lambda } {E_{{\mathbf{W}}|{\mathbf{P}},{{\mathbf{v}}^{(t)}},{{\mathbf{\beta }}^{(t)}}}}\left[ {\log \mathcal{P}\left( {{\mathbf{P}},\lambda |{\mathbf{W}}} \right)} \right] \\ \end{gathered}$$
(25)

For the variance of the prior probability density of the signal in this algorithm, due to the involvement of the intersection of \({\gamma _i}\), \({\gamma _{i - 1}}\) and \({\gamma _{i+1}}\), the Q-function contains a cross-term in the form \(\log ({\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}+\beta _{{i+1}}^{i}{\gamma _{i+1}})\), making it difficult to obtain an explicit solution to Eq. (20). Therefore, a simple analytical sub-optimal solution is used instead of the optimal solution in the derivation. The specific process is as follows.

Set the first derivative of Eq. (20) with respect to \({\gamma _i}\), \(\beta _{i}^{{i - 1}}\), and \(\beta _{i}^{{i+1}}\) to zero, i.e.,

$$\begin{gathered} \frac{{\partial Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right)}}{{\partial {\gamma _i}}}=\frac{a}{{{\gamma _i}}} - b+\frac{1}{{2\left( {{\gamma _i}+\beta _{{i - 1}}^{i}{\gamma _{i - 1}}\beta _{{i+1}}^{i}{\gamma _{i+1}}} \right)}} - \frac{{{s_i}}}{2} \\ +\frac{{\beta _{i}^{{i - 1}}}}{{2\left( {{\gamma _{i - 1}}+\beta _{{i - 2}}^{{i - 1}}{\gamma _{i - 2}}+\beta _{i}^{{i - 1}}{\gamma _i}} \right)}} - \frac{{{s_{i - 1}}}}{2}\beta _{i}^{{i - 1}} \\ +\frac{{\beta _{i}^{{i+1}}}}{{2\left( {{\gamma _{i+1}}+\beta _{i}^{{i+1}}{\gamma _i}+\beta _{{i+2}}^{{i+1}}{\gamma _{i+2}}} \right)}} - \frac{{{s_{i+1}}}}{2}\beta _{i}^{{i+1}} \\ =0\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N \\ \end{gathered}$$
(26)
$$\frac{{\partial Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right)}}{{\partial \beta _{i}^{{i - 1}}}}=\frac{{{\gamma _i}}}{{2\left( {{\gamma _{i - 1}}+\beta _{{i - 2}}^{{i - 1}}{\gamma _{i - 2}}+\beta _{i}^{{i - 1}}{\gamma _i}} \right)}} - \frac{{{s_{i - 1}}}}{2}{\gamma _i}=0\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N$$
(27)
$$\frac{{\partial Q\left( {{\mathbf{\gamma }};{\mathbf{\beta }}|{{\mathbf{\gamma }}^{(t)}};{{\mathbf{\beta }}^{(t)}}} \right)}}{{\partial \beta _{i}^{{i+1}}}}=\frac{{{\gamma _i}}}{{2\left( {{\gamma _{i+1}}+\beta _{i}^{{i+1}}{\gamma _i}+\beta _{{i+2}}^{{i+1}}{\gamma _{i+2}}} \right)}} - \frac{{{s_{i+1}}}}{2}{\gamma _i}=0\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N$$
(28)

where \({s_i}=\mu _{i}^{2}+{\Sigma _{i,i}}\),\(i=1,2, \cdots ,2N\). It should be noted that \({\gamma _0}={\gamma _{2N+1}}=0\) and \({s_0}={s_{2N+1}}=0\).

Moreover, the hyperparameter \({\mathbf{\gamma }}\) and the coupling parameter \({\mathbf{\beta }}\) are all non-negative in the algorithmic model, it can be derived,

$$0<\frac{{{\gamma _i}}}{{{\gamma _{i - 1}}+\beta _{{i - 2}}^{{i - 1}}{\gamma _{i - 2}}+\beta _{i}^{{i - 1}}{\gamma _i}}}<\frac{{{\gamma _i}}}{{\beta _{i}^{{i - 1}}{\gamma _i}}}=\frac{1}{{\beta _{i}^{{i - 1}}}}$$
(29)
$$0<\frac{{{\gamma _i}}}{{{\gamma _{i+1}}+\beta _{i}^{{i+1}}{\gamma _i}+\beta _{{i+2}}^{{i+1}}{\gamma _{i+2}}}}<\frac{{{\gamma _i}}}{{\beta _{i}^{{i+1}}{\gamma _i}}}=\frac{1}{{\beta _{i}^{{i+1}}}}$$
(30)
$$0<\frac{{\beta _{i}^{{i - 1}}}}{{{\gamma _{i - 1}}+\beta _{{i - 2}}^{{i - 1}}{\gamma _{i - 2}}+\beta _{i}^{{i - 1}}{\gamma _i}}}<\frac{{\beta _{i}^{{i - 1}}}}{{\beta _{i}^{{i - 1}}{\gamma _i}}}=\frac{1}{{{\gamma _i}}}$$
(31)

Then, the lower and upper bounds of the hyperparameter \({\mathbf{\gamma }}\) and the coupling parameter \({\mathbf{\beta }}\) can be obtained respectively by

$$\frac{a}{{0.5\left( {{s_i}+\beta _{i}^{{i+1}}{s_{i+1}}+\beta _{i}^{{i - 1}}{s_{i - 1}}} \right)+b}}<{\gamma _i}<\frac{{a+1.5}}{{0.5\left( {{s_i}+\beta _{i}^{{i+1}}{s_{i+1}}+\beta _{i}^{{i - 1}}{s_{i - 1}}} \right)+b}}$$
(32)
$$0<\beta _{i}^{{i - 1}}<\frac{1}{{{s_{i - 1}}{\gamma _i}}}$$
(33)
$$0<\beta _{i}^{{i+1}}<\frac{1}{{{s_{i+1}}{\gamma _i}}}$$
(34)

With \(a=0.{\text{5}}\), and \(b={10^{ - 4}}\), the sub-optimal solution of the hyperparameter \({\mathbf{\gamma }}\) and the coupling parameter \({\mathbf{\beta }}\) can be expressed as

$$\gamma _{i}^{{(t+1)}}=\frac{{{\kappa _\gamma }}}{{0.5\left( {{s_i}+\beta _{i}^{{i+1}}{s_{i+1}}+\beta _{i}^{{i - 1}}{s_{i - 1}}} \right)+{{10}^{ - 4}}}}\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N$$
(35)
$${\left( {\beta _{i}^{{i - 1}}} \right)^{(t+1)}}=\frac{{{\kappa _{{\beta _1}}}}}{{{s_{i - 1}}{\gamma _i}}}\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N$$
(36)
$${\left( {\beta _{i}^{{i+1}}} \right)^{(t+1)}}=\frac{{{\kappa _{{\beta _2}}}}}{{{s_{i+1}}{\gamma _i}}}\begin{array}{*{20}{c}} {}&{\forall i=1,} \end{array} \cdots ,2N$$
(37)

where \({\kappa _\gamma } \in (0.5,2)\), \({\kappa _{{\beta _1}}} \in (0,1)\) and \({\kappa _{{\beta _2}}} \in (0,1)\).

Similarly, set the first derivative of Eq. (23) with respect \(\lambda\) to zero, i.e.,

\(Q\left( {\lambda |{\lambda ^{(t)}}} \right)=\left( {M+c} \right)\log \lambda - \left( {d+\frac{1}{2}\left\| {{\mathbf{P}} - {\mathbf{G}}{{\mathbf{\mu }}^{(t)}}} \right\|_{2}^{2}+\frac{1}{2}{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right)} \right)\lambda\)

$$\frac{{\partial Q\left( {\lambda |{\lambda ^{(t)}}} \right)}}{{\partial \lambda }}=\left( {M+c} \right)\frac{1}{\lambda } - \frac{1}{2}\left( {\left\| {{\mathbf{P}} - {\mathbf{G}}{{\mathbf{\mu }}^{(t)}}} \right\|_{2}^{2}+{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right)} \right) - d=0$$
(38)

Then, the optimal solution to Eq. (25) is given by

$${\lambda ^{(t+1)}}=\frac{{2(M+c)}}{{\left\| {{\mathbf{P}} - {\mathbf{G}}{{\mathbf{\mu }}^{(t)}}} \right\|_{2}^{2}+{\text{trace}}\left( {{{\mathbf{\Sigma }}^{(t)}}{{\mathbf{G}}^{\text{T}}}{\mathbf{G}}} \right)+2d}}$$
(39)

After the above iterative loops are completed, \({\mathbf{\mu }}\) can be obtained in Eq. (16), which provides the sparse coefficient vector for the equivalent source strengths \({\mathbf{\theta }}\) as

$${\mathbf{\theta }}={\mathbf{\mu }}\left( {1:N} \right)+{\text{i}}{\mathbf{\mu }}\left( {N+1:2N} \right)$$
(40)

For clarity, the framework of the DA-PCBCS algorithm is summarized in Table 1.

Table 1 Framework of data-adaptive pattern-coupled Bayesian compressive sensing (DA-PCBCS).
Full size table

Computational complexity analysis of DA-PCBCS

The computational load of the proposed DA-PCBCS algorithm primarily stems from updating the posterior mean µ, covariance Σ, hyperparameter γ, coupling parameters β1 and β2, and noise precision λ via Eqs. (14), (15), (35), (36), (37) and (39). Among these, the matrix inversion in (15) dominates the computational cost, requiring \(\mathcal{O}({N^3})\) operations. Fortunately, this complexity can be significantly reduced by exploiting the Woodbury matrix identity as

$${\mathbf{\Sigma }}={\left( {\lambda {{\mathbf{G}}^{\text{T}}}{\mathbf{G}}+{\mathbf{D}}} \right)^{ - 1}}={{\mathbf{D}}^{ - 1}} - {{\mathbf{D}}^{ - 1}}{{\mathbf{G}}^{\text{T}}}{\left( {{\lambda ^{ - 1}}{\mathbf{I}}+{\mathbf{G}}{{\mathbf{D}}^{ - 1}}{{\mathbf{G}}^{\text{T}}}} \right)^{ - 1}}{\mathbf{G}}{{\mathbf{D}}^{ - 1}}$$
(41)

A detailed breakdown of the steps and their associated costs is provided in Table 2.

Table 2 Computational complexity breakdown for computing Σ using the Woodbury matrix identity.
Full size table

The updates for γ, β1, β2, and λ involve vector operations and element-wise calculations with complexities of at most \(\mathcal{O}(M{N^2})\), which are subsumed by the dominant term from Step 6. Consequently, the computational complexity per iteration is \(\mathcal{O}(M{N^2}+{M^2}N+{M^3})\). In compressive sensing applications where \(M \ll N\), i.e., undersampled measurements, the term \(\mathcal{O}(M{N^2})\) is the dominant term. Thus, the computational complexity per iteration of the proposed DA-PCBCS algorithm is commonly approximated as \(\mathcal{O}(M{N^2})\).

Numerical simulations

The performance of the proposed method in sound field reconstruction is validated through numerical simulations. The simulated sound source is modeled as a simply supported steel plate with a thickness of 3 mm and dimensions of 50 × 50 cm2. A harmonic force with an amplitude of 1 N is applied to excite the center of the steel plate. The steel plate has a Poisson’s ratio of 0.28, a density of 7.85 × 103 kg/m3, and a Young’s modulus of 2.1 × 1011 Pa. The sound field generated by this plate is derived on the basis of Rayleigh’s integral. The measurement plane, with dimensions of 50 × 50 cm2, is positioned 5 cm away from the vibrating plate, and the sampling interval on the measurement plane is 2.5 cm in both the x and y directions. The simulated sound pressure is measured using a square random planar array consisting of 64 microphones. To mimic real-world conditions, Gaussian white noise with a signal-to-noise ratio (SNR) of 30 dB is added to the sound pressure measured on the hologram plane. The reconstruction plane, identical in size to the measurement plane, is located 2 cm from the vibrating plate. The equivalent source plane is located 5 cm from the vibrating plate plane, with a grid of 21 × 21 and a spacing of 2.5 cm, its size being equal to that of the reconstruction plane.

To quantify the reconstruction accuracy of sound pressure, the reconstruction relative error is defined as

$${\text{RE}}=\frac{{{{\left\| {{{\mathbf{P}}_{\text{r}}} - {{\mathbf{P}}_{{\text{th}}}}} \right\|}_2}}}{{{{\left\| {{{\mathbf{P}}_{{\text{th}}}}} \right\|}_2}}} \times 100\%$$
(42)

where \({{\mathbf{P}}_{\text{r}}}\) and \({{\mathbf{P}}_{{\text{th}}}}\) are the reconstructed and theoretical sound pressure vectors, respectively.

Figure 2 shows the comparison of reconstructed pressures obtained using 64 sampling points via BCS, PCBCS, and DA-PCBCS with the theoretical pressure at 400 Hz, 800 Hz, 1200 Hz, 1600 Hz, and 2000 Hz, respectively. At 400 Hz, BCS gives an error of 4.58%; PCBCS reduces it to 3.15%, whereas DA-PCBCS further lowers it to 2.64% and yields the contour closest to the theoretical distribution. For 800 Hz, the respective errors are 5.01%, 4.01% and 3.25%; although all methods capture the spatial pattern, DA-PCBCS provides the best amplitude agreement. At 1200 Hz, the BCS error rises sharply to 8.17%, while PCBCS and DA-PCBCS remain at 4.72% and 3.97%, confirming the superior robustness of the two coupled algorithms. When the frequency rises to 1600 Hz, the BCS error reaches 10.96%, while PCBCS and DA-PCBCS keep errors at 4.79% and 4.16%; notably, BCS shows obvious deviations from the theoretical pressure distribution, while DA-PCBCS remains highly consistent in both pattern and amplitude. At 2000 Hz, the errors of BCS, PCBCS, and DA-PCBCS are 8.99%, 5.77%, and 3.97%, respectively, and DA-PCBCS still demonstrates the best match with the theoretical result at this high frequency. Overall, across all five frequencies, DA-PCBCS consistently achieves the lowest error and thus outperforms both BCS and PCBCS. This superiority is more prominent at medium to high frequencies, where DA-PCBCS effectively enhances the reconstruction accuracy of sound field.

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Comparison of the reconstructed pressures using BCS, PCBCS and DA-PCBCS with the theoretical pressure with 64 sampling points at (a) 400 Hz, (b) 800 Hz, (c) 1200 Hz, (d) 1600 Hz and (e) 2000 Hz.

Figure 3 illustrates the reconstruction errors as a function of frequency for the BCS, PCBCS, and DA-PCBCS methods. It can be observed that the reconstruction error of BCS increases significantly with frequency, exhibiting a particularly steep rise above 1000 Hz and reaching over 8% in the high-frequency range. This indicates inadequate stability of BCS in high-frequency reconstruction. The error of PCBCS is lower than that of BCS but still shows a gradual increase with frequency. Furthermore, its error in the mid-to-high frequency range is noticeably higher than that of DA-PCBCS. In contrast, DA-PCBCS maintains a consistently low error level: below 600 Hz, the error is mostly around 2%, and even at 2000 Hz, it is only about 4%. DA-PCBCS also exhibits small overall fluctuations and optimal stability. In summary, DA-PCBCS exhibits superior reconstruction accuracy over both BCS and PCBCS across the entire frequency range, while also more effectively suppressing error growth in the mid-to-high frequency region. This demonstrates its greater frequency adaptability and enhanced reconstruction stability. This performance difference arises because traditional BCS fails to exploit the intrinsic structure of the signal to enhance solution sparsity, leading to a monotonic error increase with frequency. PCBCS improves sparse reconstruction performance by introducing coupled priors governed by a fixed parameter, which facilitates the respective aggregation of non-zero and zero elements. By contrast, DA-PCBCS further enhances sound field reconstruction performance through a data-adaptive coupled prior, producing a more realistic sparsity pattern and consistently maintaining the lowest reconstruction error across all frequencies. This result verifies the effectiveness and robustness of DA-PCBCS in achieving high-accuracy sound field reconstruction even under sparse spatial sampling at high frequencies.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Reconstruction errors of the three methods versus frequency.

To further investigate the impact of the number of sampling points on reconstruction performance, seven groups of sampling points with varying quantities were selected on the measurement plane. Three methods using BCS, PCBCS, and DA-PCBCS were used for sound field reconstruction. To eliminate the influence of frequency variations on the performance of these methods, a fixed frequency of 1100 Hz was selected for the experiment. Meanwhile, Gaussian white noise with an SNR of 30 dB was added to the measured sound pressures to simulate real-world working conditions. Figure 4 shows a comparison of the reconstruction results using the three methods under different numbers of sampling points. As shown in Fig. 4, when only 25 sampling points were used, all three methods achieved relatively accurate sound field reconstruction. Additionally, the reconstruction errors of the three methods decreased as the number of sampling points increased, indicating that a higher number of sampling points effectively improves reconstruction accuracy. Notably, under the same number of sampling points, the reconstruction error of the DA-PCBCS method was significantly lower than those of the BCS and PCBCS methods. These results demonstrate that the proposed DA-PCBCS method maintains high sound field reconstruction accuracy even with a significantly reduced number of sampling points, which means that this method can reduce measurement costs by decreasing the number of sampling points, thereby enhancing its feasibility in engineering applications.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Reconstruction errors of the three methods versus sampling points.

Subsequently, the impact of different SNRs on reconstruction performance was investigated. The reconstruction errors of BCS, PCBCS, and DA-PCBCS were compared within a frequency range of 200–2000 Hz (with a frequency interval of 200 Hz), using 64 sampling points and an SNR range of 5–50 dB (with an SNR interval of 5 dB). As shown in Fig. 5, at fixed frequencies, the reconstruction errors of all three methods decreased monotonically with increasing SNR. Notably, the DA-PCBCS method consistently exhibited smaller reconstruction errors than both the BCS and PCBCS methods. In low-frequency bands across various SNRs, DA-PCBCS achieved remarkably high reconstruction accuracy, with an error of approximately 10%. In contrast, the reconstruction errors for BCS and PCBCS reached 20%. This indicates the superior reconstruction performance of DA-PCBCS at lower frequencies. In high-frequency bands under low SNRs, the reconstruction errors for BCS and PCBCS exceeded 40%. Conversely, the errors for DA-PCBCS were significantly lower, with a maximum error below 35%. This result highlights the exceptional reconstruction performance of the DA-PCBCS method across the entire frequency range, even in low-SNR environments. Consequently, the DA-PCBCS method demonstrates superior noise resistance and robustness in sound field reconstruction, making it a highly effective solution for practical applications.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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Reconstruction errors versus SNR and frequency using (a) BCS, (b) PCBCS, and (c) DA-PCBCS.

Experimental studies

Experiments were conducted in a semi-anechoic chamber to further validate the performance of the proposed DA-PCBCS method. The experimental setup is shown in Fig. 6. A steel plate with dimensions of 0.5 × 0.5 m2 and a thickness of 3 mm was used as the sound source, whose center was excited by a vibrator. The excitation signal was composed of sinusoidal signals within a frequency range of 100–1500 Hz with a frequency interval of 100 Hz, which was input through a power amplifier. In the experiment, a linear array consisting of 11 microphones was used to scan the sound pressures on two planes, each with a size of 0.5 × 0.5 m2. These two planes were located 0.09 m and 0.05 m above the steel plate, corresponding to the hologram plane and the reconstruction plane, respectively, as shown in Fig. 7. Each plane adopted an 11 × 11 grid (with a grid spacing of 0.05 m), and a total of 121 points were measured. During the scanning process, the excitation signal served as the reference for sound pressure phase correction. The sound pressure signals were acquired at a sampling frequency of 10.24 kHz for a duration of 5 s. After applying the fast Fourier transform to the measured sound pressure signals, the phase of the reference was subtracted from all frequency-domain sound pressure data to achieve synchronous measurement with the linear array. An equivalent source plane with an 11 × 11 grid was set 0.05 m behind the steel plate, uniformly distributing 121 equivalent sources. Sixty-four sampling points were randomly selected from the 121 points on the hologram plane as input data for each method to reconstruct the sound pressure at 121 sampling points on the reconstruction plane. The measured sound pressure at the 121 points on the reconstruction plane was used as the reference sound pressure to evaluate the performance of the three reconstruction methods.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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The experimental setup.

Fig. 7
Fig. 7The alternative text for this image may have been generated using AI.
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Schematic of the experimental spatial configuration.

Figure 8 compares the reconstructed sound pressure distributions of the three methods with measured results at 300 Hz, 700 Hz, 1200 Hz, and 1500 Hz. As shown in Fig. 8, the sound pressure distributions reconstructed by all three methods at 300 Hz and 700 Hz closely match the measured distributions. Notably, the DA-PCBCS method yields the smallest reconstruction errors at these frequencies, with values of 3.83% and 5.88%, respectively. At 1200 Hz, the reconstruction error for the BCS method increases to 10.66%. In contrast, both PCBCS and DA-PCBCS maintain errors below 10%, with the DA-PCBCS method achieving an error of 7.11%, significantly superior to the others. At 1500 Hz, BCS and PCBCS exhibit errors of 12.03% and 10.51%, respectively, whereas DA-PCBCS reduces the error to 9.08%. These results demonstrate that DA-PCBCS consistently surpasses the BCS and PCBCS methods in reconstruction performance across all tested frequencies. Furthermore, it achieves higher-accuracy sound field reconstruction using fewer measurement points over the entire frequency range.

Fig. 8
Fig. 8The alternative text for this image may have been generated using AI.
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Comparison of the reconstructed sound pressure distributions of the three methods with measured results at (a) 300 Hz, (b) 700 Hz, (c) 1200 Hz, and (d) 1500 Hz.

Figure 9 shows the comparison of reconstruction errors of three methods using 64 sampling points within the frequency range of 100–1500 Hz (with a frequency interval of 100 Hz). As shown in Fig. 9, reconstruction errors generally increase with frequency. Specifically, the BCS method exhibits reconstruction errors up to 12% at higher frequencies, while the PCBCS method constrains errors within 10%. In contrast, the DA-PCBCS method demonstrates superior consistency by maintaining errors below 8% across the entire range, highlighting its enhanced robustness for high-frequency sound field reconstruction. While all three methods achieve acceptable reconstruction accuracy, DA-PCBCS consistently yields the lowest errors across all frequencies. This result validates its enhanced reconstruction capability and confirms its reliability for practical sound field reconstruction.

Fig. 9
Fig. 9The alternative text for this image may have been generated using AI.
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Reconstruction errors versus frequency using three methods with 64 sampling points.

To further validate the performance of the proposed method under different numbers of sampling points, seven groups of sampling points (25, 36, 49, 64, 81, 100, and 121) were selected on the measurement plane for sound field reconstruction. A fixed frequency of 1000 Hz was employed to eliminate frequency-dependent effects on the performance of the three methods. Figure 10 shows the comparison of reconstruction errors of the three methods under different numbers of sampling points. As shown in Fig. 10, the reconstruction errors of all methods exhibit a declining trend with increasing sampling points. At the same number of sampling points, the DA-PCBCS method consistently achieves lower reconstruction errors than the BCS and PCBCS methods, indicating its superior accuracy in sound field reconstruction. The experimental results further confirm that the proposed DA-PCBCS method can still demonstrate excellent sound field reconstruction performance even with a small number of sampling points, demonstrating its potential to reduce acoustic testing costs while preserving reconstruction fidelity.

Fig. 10
Fig. 10The alternative text for this image may have been generated using AI.
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Reconstruction errors of the three methods under different numbers of sampling points at 1000 Hz.

Conclusion

This paper proposes a data-adaptive pattern-coupled Bayesian compressive sensing method for sparse sound field reconstruction. The method utilizes near-field acoustic holography based on the equivalent source method, and achieves sound field reconstruction under the BCS framework by solving for the sparse coefficient vector of equivalent source strengths. The method constructs a pattern-coupled prior by introducing a projection transformation matrix into the hyperparameter space that controls the sparsity of equivalent source strengths. This ensures that each element of the sparse coefficient vector is not only governed by its own hyperparameter but also influenced by the hyperparameters of adjacent elements. To further enhance the adaptability of the prior model, the proposed algorithm introduces a set of new adaptive coupling parameters by performing a learnable transformation on the initially independent hyperparameters to regulate their mutual correlations, ultimately establishing a hierarchical Gaussian-Gamma prior model with data-adaptive hyperparameter dependencies. During the Bayesian inference process, the coupling parameters controlling hyperparameter correlations and the hyperparameters governing the sparsity of the equivalent source strengths undergo data-driven iterative updates, thereby enabling the sparsity patterns of elements within the sparse coefficient vector to adaptively influence each other. By adaptively coupling the sparsity of neighboring elements, the designed prior model effectively promotes the clustering of non-zero coefficients into blocks and the concentration of zero coefficients. Consequently, it induces a physically meaningful block-sparse structure that accurately reflects the spatial continuity of actual sound sources, thereby improving the reconstruction accuracy of the sparse signal (i.e., the equivalent source strength coefficients). Finally, the method employs the expectation-maximization algorithm to estimate the maximum a posterior of hyperparameters and the posterior distribution of the sparse coefficient vector. Its core advantage lies in fully leveraging the intrinsic statistical correlations between elements of the sparse coefficient vector. This allows for the accurate recovery of solutions exhibiting block-sparse characteristics without relying on predefined block segmentation (i.e., prior knowledge of block structure), significantly enhancing the accuracy of sound field reconstruction under undersampled conditions.

Numerical simulations demonstrate that the proposed DA-PCBCS method achieves lower reconstruction errors across the entire frequency range compared to BCS and PCBCS, confirming its superiority in sound field reconstruction. When the number of sampling points is limited, DA-PCBCS attains higher accuracy than the two benchmark methods, potentially reducing measurement costs in practical engineering applications. Moreover, DA-PCBCS maintains low reconstruction errors in low-SNR environments, indicating enhanced noise robustness. Experimental results further validate the effectiveness and reliability of DA-PCBCS in real-world scenarios, consistently yielding reconstruction errors below 10%. Even with fewer measurement points, DA-PCBCS significantly outperforms BCS and PCBCS by exploiting underlying block-sparse structures through the introduction of coupling parameters, without requiring prior knowledge of block partitioning.

Although the proposed DA-PCBCS method significantly improves the accuracy of sound field reconstruction, several limitations remain. First, the introduction of learnable transformation matrices and adaptive coupling parameters inevitably increases computational complexity and the computational burden per iteration, potentially limiting its applicability to large-scale engineering problems requiring rapid or real-time processing. Second, the algorithm employs suboptimal solutions for hyperparameter and coupling parameter updates, which theoretically limits its potential to achieve optimal performance and may render the method sensitive to initial values with the risk of converging to local extrema. Additionally, regarding numerical stability, regularization techniques, stable decompositions, and error control measures have not been implemented to ensure numerical robustness. To address these limitations of the DA-PCBCS method, future improvements will focus on: (i) developing fast implementation schemes to reduce computational complexity; (ii) investigating parameter update rules with theoretical convergence guarantees to replace the current suboptimal strategies; (iii) enhancing numerical robustness by introducing techniques such as stable matrix decompositions. Future work will also explore more flexible structured priors that go beyond the fixed block-sparsity assumption, and integrate with deep learning frameworks to data-drive the learning of optimal coupling relationships.