# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 3 Imaging Technology
### Compressed Sensing of Array Imaging  
The widespread use of ultrasonic array 2-D transducers and the in-depth study of FMC-TFM increase the cost of data capture time and storage space. There is an urgent need to achieve less sampling and real-time detection of data while ensuring the accuracy and validity of the original information [75]. Compressed sensing (CS) is a method for recovering compressed signals from far fewer samples than required by the Nyquist sampling model. The theory was first proposed in 2004 by [PERSON] et al. [76; 77; 78; 79], who demonstrated in principle that subsampling can be performed when the signal is sparse and that the original signal can be efficiently reconstructed to reduce the amount of data to be acquired and stored. This method has become a major direction of research in ultrasound array imaging. Figure 9 shows the principal diagram of ultrasonic imaging based on CS. Taking the center of the array as the origin of the component coordinate system, assuming \(Nx\) and \(Ny\) are the number of scattering points in the horizontal \(X\) direction and vertical \(Y\) direction, respectively, the received signal can be obtained according to the relevant information as per [80; 81]. \[u(r_{0},t)=\sum\limits_{i=1}^{I}A(r,t)\cdot\sigma(r_{i})\]  
\[A(r,t)=\sum\limits_{i=1}^{N}p(r_{i},t)=\sum\limits_{i=1}^{N}je^{i}(\omega t- kr_{i})f(\theta_{i})=\sqrt{a^{2}+b^{2}}\cdot e^{j(\omega t+\psi)}=E\cdot e^{j( \omega t+\psi)} \tag{11}\]  
where \(I-N_{x}N_{y}\) is the total number of array elements, \(p(r,t)\) is the radiated sound field distribution of each array element \(E-\sqrt{(a^{2}+b^{2})}\), \(a=\sum\limits_{i=1}^{I}f(\theta_{i})\cos(kr_{i})\), and \(b=\sum\limits_{i=1}^{I}f(\theta)\sin(kr_{i})\)\(f(\theta)\) is the directivity function, \(\theta\) is the angle between the observation point and the \(Z\)-axis, \(r_{i}\) is the distance from the \(i\) th array element to the observation, \(\sigma(r)\) is the transmission coefficient at the midpoint p of the target object, \(\Psi\) is the initial phase, and \(A(r,t)\) is the sound field distribution at a point. \(u(r,t)\) is the distribution of the sound field received by the probe from the target object. For the case of the same received element, the echo function is discretized and sampled at different time nodes to obtain the scattered sound field from the target object:  
\[ \begin{pmatrix}u(r_{0},t_{1})\\ u(r_{0},t_{2})\\ \vdots\\ u(r_{0},t_{3})\end{pmatrix} = \begin{bmatrix}A(r_{1},t_{1})&A(r_{2},t_{1})& \dots&A\begin{pmatrix}r_{N_{x}N_{y}},t_{1}\\ A(r_{1},t_{2})&A(r_{2},t_{2})&\dots&A\begin{pmatrix}r_{N_{x}N_{y}},t_{2}\\ \vdots&\vdots&\ddots&\vdots\\ A(r_{1},t_{M})&A(r_{2},t_{M})&\dots&A\begin{pmatrix}r_{N_{x}N_{y}},t_{M}\\ \end{pmatrix}\end{bmatrix} . \begin{bmatrix}\sigma(r_{1})\\ \sigma(r_{2})\\ \vdots\\ \sigma\begin{pmatrix}r_{N_{x}N_{y}}\end{pmatrix}\end{bmatrix}  \tag{12}\]  
Constructing the observation matrix \(A(r,t)\) of Equation (12) as a non-singular matrix, the value \(\sigma(r_{1}),\sigma(r_{2})\cdots\sigma(r_{N_{x}N_{y}})\) after \(M\) modulations is obtained, that is, the image information. When \(M<N_{x}N_{y}\) Equation (12) can be simplified to:  
\[U=A\cdot\sigma \tag{13}\]  
where \(U\) represents the discretized received signal; \(\sigma\) represents the target scene information; and \(A\) is the measurement matrix. Considering the effect of noise \(e\) in the reception process, the ultrasound imaging model based on compression perception becomes:  
\[\min\lVert\sigma\rVert_{0}s.t.\lVert A\cdot\sigma-U\rVert_{2}^{2}\leq\varepsilon \tag{14}\]  
CS theory consists of three main elements: the sparse representation of the signal, the compressive measurement process, and the signal reconstruction. Since compressive sensing first transforms the signal to be recovered into a certain sparse domain, the sparse coefficients of the signal in the sparse domain directly determine the reconstruction effect.