# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 3 Imaging Technology
### Total Focusing Method  
In 2005, [PERSON] and others [52] from the University of Bristol in the UK proposed TFM, which caused a sensation in the field of ultrasound imaging research. Under the evaluation coefficient criteria of the array performance indicator (API), the TFM exhibits excellent resolution compared with conventional ultrasonic arrays imaging, which is called the \"golden rule\" algorithm in ultrasonic arrays technology by some researchers. The data basis for TFM is full matrix capture (FMC). FMC is the most active and efficient of all data acquisition methods because it records the complete set of signals associated with all transmitting and receiving wafers in the array. The FMC process needs to excite each wafer one by one and record the signals received by all the wafers each time. This is shown in Figure 7.  
Figure 8 is a schematic diagram of FMC reconstruction imaging. After obtaining the full matrix data, a region of interest (ROI) is set in the workpiece and the ROI contains many pixels, i.e., the signal focus. The image amplitude at the point is obtained by delay matching and superposing the echo signal for each transmitting and receiving array element pair. By  
Figure 7: The schematic of the full matrix capture. traversing each focal point in the ROI, a 2-D reconstructed image is obtained. Taking the point \((x,z)\) as an example, the amplitude \(I(x,z)\) at this point should be expressed as:  
\[I(x,z)=\sum_{i=1}^{N}\sum_{j=1}^{N}e_{xi,xj}[t_{i,j}(x,z)] \tag{9}\]  
where \(e_{xi,xj}[t_{i,j}(x,z)]\) is the amplitude of the ultrasonic signal transmitted by the I element and received by the \(j\) element superimposed to the position \((x,z)\) and \(t_{i,j}(x,z)\) is the delay time required for the sound wave to be emitted from the array element \(i\) to the focal point \((x,z)\) and to return from that point to array element \(j\). \(t_{i,j}(x,z)\) can be expressed as  
\[t_{i,j}(x,z)=\frac{\sqrt{\left(x-x_{i}\right)^{2}+z^{2}}+\sqrt{\left(x-x_{j} \right)^{2}+z^{2}}}{c} \tag{10}\]  
In the formula, \(i_{x}\) and \(j_{x}\) are the abscissas of the transmitting array element and the receiving array element, respectively, and \(c\) is the longitudinal wave velocity of the ultrasonic propagating in the test block. Considering the amplitude attenuation and deflection of the acoustic beam during propagation, [PERSON] et al. [53] proposed two modified TFMs in 2007. The first algorithm for obtaining information about reflector orientation and the second algorithm for distinguishing between point-like reflectors that reflect uniformly in all directions and specular reflectors that have distinct orientations. Since then, [PERSON] et al. proposed a far-field approximate common-source method (CSM) based on sparse arrays [54,55]. The TFM algorithm has become the golden rule for post-processing algorithms [56,57,58]. How to effectively sparse optimize and perform weight assignment of ultrasound arrays to achieve better imaging and smaller data volume, as well as how to accelerate full-focus processing and computing, is the focus of current research by scholars in various countries. [PERSON] et al. [59] introduced the wavenumber algorithm from ultrasonic array radar to improve the computational efficiency of TFM from the frequency domain perspective. However, the wavenumber algorithm has stricter mathematical limitations than TFM, the implementation process is complex, and the selected frequency domain interpolation function can greatly affect the imaging quality. A sparse array FMC based on a particle swarm algorithm was proposed, which can significantly reduce the amount of imaging data while ensuring imaging quality [60]. In the literature [61], an imaging correction model was developed based on an array element directivity function based on a triangular  
Figure 8: The schematic of the total focusing method. matrix data acquisition and indexing technique for imaging acceleration algorithm to reduce the imaging time per frame to within 135 ms. [PERSON] et al.