### 2.1 Full Matrix Focused Data Acquisition

A-scan data is acquired and arranged in a matrix for imaging. Taking a linear array with *n* elements as an example, the 1st element is fired and all elements are received. The acquired data is named as *A*_{11}*, A*_{12}*, *...*, A*_{1n}. And the acquired data is placed in the first row. Then next element is excited in turn until all the *n×n* A-scan data are acquired as listed in Eq. (1).

$$ \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } & \cdots & {A_{1n} } \\ {A_{21} } & {A_{22} } & {A_{23} } & \cdots & {A_{2n} } \\ {A_{31} } & {A_{32} } & {A_{33} } & \cdots & {A_{3n} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {A_{n1} } & {A_{n2} } & {A_{n3} } & \cdots & {A_{nn} } \\ \end{array} } \right]. $$

(1)

### 2.2 Full Matrix Focusing Imaging Algorithm

Imaging algorithm according to the full matrix data is shown schematically in Figure 1. The imaging region below the transducers is discretized into many focus points in alignment. Taking focus point *P* as an example, the distances between the point *P* and two elements *i, j* can be calculated. The amplitude *A*_{ij}*(t*_{0}**)** in the signal *A*_{ij} can be determined according to the corresponding travel time *t*_{0}. The amplitudes from all A-scan signals is superimposed with the same rule. Thus the digital amplitude of the focus point *P* \(I_{{\text{P}}} \left( {x,z} \right)\) can be calculated according to Eq. (2).

$$ I_{{\text{P}}} \left( {x,z} \right) = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {A_{ij} \left( {t_{0} } \right)} } , $$

(2)

$$ t_{0} = \left( {\sqrt {\left( {x_{i} - x} \right)^{2} + z^{2} } + \sqrt {\left( {x_{j} - x} \right)^{2} + z^{2} } } \right)/c, $$

(3)

where *c* is the wave speed in the material.

### 2.3 Triangular Matrix Focusing Imaging Algorithm

The full-matrix focus imaging algorithm needs all A-scan data acquisition and superposition, so it is a time-consuming method. Considering the reciprocity principle of the multi-channel acoustic system, the transmit and receive channels are interchanged. A good consistency in A-scan amplitudes is maintained. The *A*_{ij} signal (the *i* element is transmitted, and the *j* element is received) should have a good consistency with the *A*_{ji} signal (the *j* element is transmitted, and the *i* element is received). Then, the full matrix data shown in Eq. (1) is a symmetric matrix. It is sufficient that the imaging algorithm only use the upper triangular matrix signal, as shown in Eq. (4). It is called as the triangular matrix focusing imaging algorithm according to Eq. (5).

$$ \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } & \cdots & {A_{1n} } \\ {A_{21} } & {A_{22} } & {A_{23} } & \cdots & {A_{2n} } \\ {A_{31} } & {A_{32} } & {A_{33} } & \cdots & {A_{3n} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {A_{n1} } & {A_{n2} } & {A_{n3} } & \cdots & {A_{nn} } \\ \end{array} } \right], $$

(4)

$$ I_{{\text{P}}} \left( {x,z} \right) = \sum\limits_{i = 1}^{N} {\sum\limits_{j \ge i}^{N} {A_{ij} \left( {t_{0} } \right)} } . $$

(5)

Comparing Eqs. (1) and (2) to Eqs. (4) and (5), the amount of data acquisition and calculation is reduced from *n×n* to *n×(n+1)/2*. Then the amount of both data transfer, storage and calculation will be almost reduced by half.

### 2.4 Trapezoidal Matrix Focusing Imaging Algorithm

In order to improve the computational efficiency, the signal energy weights of different channels to the focus point are also considered. Synthetic aperture focusing technique (SAFT) is very useful method. The data on the main diagonal of the matrix is used in SAFT. The transmitting element is same as the receiving element. For a given focus point as shown in Figure 2, the A-scan amplitude is weaken with the distance from the transmitter to the receiver element increasing. Taking path1, path2 and path3 in Figure 2 as an example, with the travelled distance increasing from the focus point, the A-scan amplitude of the receiving element *i*, *j* and *n* is weaken gradually.

It is easy to find that the data with larger energy weight are distributed near the diagonal of the triangular matrix. Therefore, the data near to the diagonal of the triangular matrix is reserved, and data far from the diagonal of the triangular matrix is ignored. Then, the trapezoidal matrix focusing imaging algorithm is determined as shown in Eqs. (6) and (7).

$$ \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } & \cdots & {A_{1n} } \\ {A_{21} } & {A_{22} } & {A_{23} } & \cdots & {A_{2n} } \\ {A_{31} } & {A_{32} } & {A_{33} } & \cdots & {A_{3n} } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {A_{n1} } & {A_{n2} } & {A_{n3} } & \cdots & {A_{nn} } \\ \end{array} } \right], $$

(6)

$$ I_{{\text{P}}} \left( {x,z} \right) = \sum\limits_{i = 1}^{N} {\sum\limits_{i \le j \le k}^{N} {A_{ij} \left( {t_{0} } \right)} } , $$

(7)

where \(i < k < N\).

After obtaining the superposition amplitude of each focusing point, the original image needs to be normalized for the digital display. The 256 colors bar is often used in the ultrasonic NDT field. The superposition amplitudes are normalized from − 127 to 128 using Eqs. (8) and (9).

If \(I_{{\text{P}}} \left( {x,z} \right) > 0,\)

$$ I_{{\text{P}}} \left( {x,z} \right) = \frac{{I_{{\text{P}}} \left( {x,z} \right)}}{{I\left( {x,z} \right)_{\max } }} \times 128, $$

(8)

If \(I_{{\text{P}}} \left( {x,z} \right) \le 0,\)

$$ I_{{\text{P}}} \left( {x,z} \right) = - \frac{{I_{{\text{P}}} \left( {x,z} \right)}}{{I\left( {x,z} \right)_{{{\text{min}}}} }} \times 127. $$

(9)