A novel robust approach of 3D CNN and SAE-based near-field acoustical holography relying on self-identity constraint data for Kalman gain

Abstract

For near-field acoustic holography, sparse array measurement for cost reduction can result in inaccuracy due to aliasing error. To attenuate it, there are data-driven methods based on artificial intelligence theories. Among these, the JTCSA-NAH method has not adopted measures for robustness enhancement despite its high accuracy in practice. In this work, the influence of measuring noise on JTCSA-NAH is analyzed followed by the principle of adding Gaussian noise for robustness improvement. Based on the relevant prior conditions, the ICCSA-NAH method, which relies on self-identity constraint data working as the Kalman gain is proposed. Subsequently, numerical example and experiment are carried out, and the results show that compared with JTCSA-NAH method, the mean errors of near-field vibration velocity reconstruction are theoretically and experimentally reduced from 15.19% and 23.64% to 6.03% and 12.45%, respectively, by the ICCSA-NAH method, which verifies the feasibility and superiority of the proposed method.

A. Appendix

The JTCSA-NAH method is composed of two parts. The first is the SAE module from original velocity to reconstructed velocity. The other part is the combined pressure to velocity (PV)-NN composed of holography feature extraction neural network (HFENN) and the decoder in the first part, which maps from the theoretical sparse pressure to the reconstructed velocity. Then we train the network to obtain the optimal parameter sets:

$${\varvec{\theta}}_{{{\text{opt}}}} = \mathop {\arg \min }\limits_{{\varvec{\theta}}} \left\{ {{\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}}}} \left[ {\left\| {f\left( {{\varvec{p}}_{{\varvec{s}}} |{\varvec{\theta}}} \right) - {\varvec{v}}} \right\|_{F}^{2} } \right]} \right\}.$$

(25)

When the measured pressure data includes noise n, the near-field velocity generalized by the trained network is:

$$\hat{{v}}_{{\varvec{n}}} = f\left( {{\varvec{p}}_{{\varvec{s}}} + {\varvec{n}}|\theta_{{{\text{opt}}}} } \right).$$

(26)

Assuming the network consists of N layers, $\hat{{v}}_{{\varvec{n}}}$ is the output of layer N. The output x_n of each layer is determined by x_n-1, and x_N = $\hat{{v}}_{{\varvec{n}}}$.

$${\varvec{x}}_{n} = {\text{diag}}\left[ {{\varvec{O}}_{n - 1} \left( \cdot \right)} \right] \otimes \left( {{\varvec{w}}_{n - 1} \cdot {\varvec{x}}_{n - 1} + b_{n - 1} } \right),$$

(27)

where O_n-1 is the function vector composed of activation functions of all neurons in layer n, ⊗ denotes generalized multiplication, representing the mapping of the function from its multiplied term. w_* and b_* are the weight vector and bias of each layer, respectively. There is no parameter in the upsampling layer, so it is not considered.

When the system is linear, the difference between generalized two different sound pressure data $\hat{{v}}_{{\varvec{n}}}$ and $\hat{{v}}$ is $\Delta \hat{{v}}$:

$$\left\{ \begin{aligned} \hat{{v}}\left( {\varvec{p}} \right){ = } & \prod\limits_{i = n}^{1} {k_{i} \cdot {\varvec{w}}_{i} \cdot {\varvec{p}}} + \sum\limits_{m = 1}^{n - 1} {\left( {\prod\limits_{j = m}^{n} {k_{j} } \prod\limits_{l = n}^{m + 1} {{\varvec{w}}_{l} \cdot b_{m} } } \right) + k_{n} b_{n} } \\ \Delta {\hat{v} = } & \hat{{v}}\left( {{\varvec{p}} + {\varvec{n}}} \right) - \hat{{v}}\left( {\varvec{p}} \right) = \prod\limits_{i = n}^{1} {k_{i} \cdot {\varvec{w}}_{i} \cdot {\varvec{n}}} \\ \end{aligned} \right..$$

(28)

In Eq. (25), the difference between the surface vibration velocity obtained by the generalization of the noise-free and the noisy data are substituted into the MSE loss function to obtain the final deference of loss function:

$$\left\{ \begin{aligned} L\left( {\hat{{v}}} \right)\varvec{ = } & {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}}}} \left( {\left\| {\hat{\varvec{v}} - {\varvec{v}}} \right\|_{F}^{2} } \right) \\ L\left( {\hat{\varvec{v}}_{{\varvec{n}}} } \right)\varvec{ = } & {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\left\| {\hat{\varvec{v}} - {\varvec{v}} + \Delta \hat{\varvec{v}}} \right\|_{F}^{2} } \right) \\ \Delta L\varvec{ = } & {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left[ {\left\| {\hat{\varvec{v}} - {\varvec{v}}} \right\|_{F}^{2} + 2 \cdot \left( {\hat{\varvec{v}} - {\varvec{v}}} \right) \odot \Delta \hat{\varvec{v}} + \left\| {\Delta \hat{\varvec{v}}} \right\|_{F}^{2} } \right] - {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}}}} \left( {\left\| {\hat{\varvec{v}} - {\varvec{v}}} \right\|_{F}^{2} } \right) \\ = & 2 \cdot {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\sum\limits_{q \in \Omega } {\left( {\prod\limits_{i = n}^{1} {k_{i} \cdot {\varvec{w}}_{i} \cdot {\varvec{n}}} } \right)_{q} \cdot } \left( {\hat{\varvec{v}} - {\varvec{v}}} \right)_{q} } \right) + {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left[ {\left\| {\prod\limits_{i = n}^{1} {k_{i} \cdot {\varvec{w}}_{i} \cdot {\varvec{n}}} } \right\|_{F}^{2} } \right] \\ \end{aligned} \right..$$

(29)

When the noise n on each pixel follows a Gaussian distribution with a mean value of 0 (${\varvec{n}}\sim \mathcal{N}\left(0,\delta \right)$), and there are enough samples with random additional noise, the mean of each pixel noise is 0, and the distribution of noise n and vibration velocity v is independent, and so the left term of ΔL is 0 in Eq. (29). Therefore, for linear systems, when there is measurement noise, the model trained by only clean data will magnify the reconstruction error. As for the PV-NN, although the activation function ReLU: y = (x + x·sign(x))/2 adopted in the network has a certain degree of linearity, the network framework is nonlinear globally.

For nonlinear systems, substituting the clean and noisy pressure into MSE loss function will yield:

$$\left\{ \begin{gathered} L\left( {\hat{\varvec{v}}} \right) = {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}}}} \left( {\left\| {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - {\varvec{v}}} \right\|_{F}^{2} } \right) \hfill \\ L\left( {\hat{\varvec{v}}_{n} } \right) = {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\left\| {f\left( {{\varvec{p}}_{{\varvec{s}}} + {\varvec{n}}|\theta_{opt} } \right) - {\varvec{v}}} \right\|_{F}^{2} } \right) \hfill \\ \Delta L = {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\left\| {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - \varvec{v + }\left. {\Delta f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right)} \right|_{{\Delta {\varvec{p}}_{{\varvec{s}}} = {\varvec{n}}}} } \right\|_{F}^{2} } \right) \hfill \\ \, - {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}}}} \left( {\left\| {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - {\varvec{v}}} \right\|_{F}^{2} } \right) \hfill \\ \, = 2 \cdot {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\sum\limits_{q \in \Omega } {\left( {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - {\varvec{v}}} \right)_{q} \cdot } \left. {\Delta f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right)} \right|_{{\Delta {\varvec{p}}_{{\varvec{s}}} = \varvec{n|}q}} } \right) \hfill \\ \, + {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left[ {\left\| {\left. {\Delta f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right)} \right|_{{\Delta {\varvec{p}}_{{\varvec{s}}} = {\varvec{n}}}} } \right\|_{F}^{2} } \right] \hfill \\ \end{gathered} \right.,$$

(30)

where the second term of ΔL (the last line of Eq. (30)) is larger than 0. For the remaining cross term (the penultimate line of Eq. (30)), when the statistical energy of noise is relatively low, and random combination samples are sufficient, the second-order infinitesimal term of it can be ignored, and therefore it turns to:

$$\begin{gathered} 2 \cdot {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left( {\sum\limits_{q \in \Omega } {\left( {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - {\varvec{v}}} \right)_{q} \cdot } \left. {\Delta f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right)} \right|_{{\Delta {\varvec{p}}_{{\varvec{s}}} = \varvec{n|}q}} } \right) \hfill \\ \approx 2 \cdot {\mathbb{E}}_{{\left( {{\varvec{p}}_{{\varvec{s}}} ,{\varvec{v}}} \right) \sim {\mathcal{D}},{\varvec{n}} \sim \mathcal{N}}} \left\{ {\sum\limits_{q \in \Omega } {\left[ {\left( {f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right) - {\varvec{v}}} \right)_{q} \cdot \left( {\frac{{\partial f\left( {{\varvec{p}}_{{\varvec{s}}} |\theta_{opt} } \right)}}{{\partial {\varvec{p}}_{{\varvec{s}}} }} \odot {\varvec{n}}} \right)_{q} } \right]} } \right\}. \hfill \\ \end{gathered}$$

(31)