工程科学与技术   2021, Vol. 53 Issue (4): 200-208

Adaptive Beamforming Algorithm for Ultrasonic Array with the Combination of Spatial Sampling and Coherence Factor
SONG Shoupeng, LI Qi
School of Mechanical Eng., Jiangsu Univ., Zhenjiang 212013, China
Abstract: In order to solve the problem of low computational efficiency of adaptive beamforming algorithms in ultrasonic imaging, an adaptive beamforming algorithm for ultrasonic array with the combination of spatial sampling and coherence factor was proposed. The maximum decimation factor with different numbers of array elements was deduced according to the beam pattern. The sparse echo data was obtained by spatially sampling the whole array element data using the maximum decimation factor. Therefore, the amount of data used for beamforming was greatly reduced. Taking the spatial sampling data as the input of a beamformer and constructing the covariance matrix as Toeplitz matrix, the adaptive weights of the sampling data were obtained according to the principle of minimum variance. Then, the adaptive weights were modified by introducing the coherence factor to highlight the effective information of the sampling data. Under the case of unequal data and spatial sampling data, the proposed algorithm, minimum variance algorithm and minimum variance algorithm combined with coherence factor were used to simulate the imaging of cracks and cross-drilled holes respectively. The results show that: for unequal data, the imaging quality of the proposed algorithm is between the other two algorithms; in terms of imaging time, compared with the other two algorithms, the average imaging time of the proposed algorithm is reduced by more than 85%. For the same spatial sampling data, the imaging quality of the proposed method is better than the other algorithms; in terms of imaging time, compared with the other two algorithms, the average imaging time of the proposed algorithm is reduced by more than 65%.
Key words: beamforming algorithm    spatial sampling    coherence factor    computational complexity

Sakhaei[19]描述了一种抽样MV波束形成器算法，首先结合接收波束的波束模式的分析对全部数据抽样，然后利用全部数据计算出加权系数并对抽样数据进行加权，最后在医学成像上验证了其可行性。在此基础上，Shamsian等[20]提出级联结构的快速波束形成方法，将低通滤波器与最小方差波束形成器级联。其中：第1阶段，通过低通滤波器去除回波信号中的离轴噪声，再对数据进行抽样；第2阶段，将抽样数据作为MV算法的输入以抑制轴上干扰，并提出抽取子波束的MV算法（decimated sub-beam MV，DSMV）。该方法实现了数据量和计算复杂度上的降低，但对阵元数据进行抽样这一过程不可避免会丢失部分信息，进而影响成像质量。

1 空域抽样与相干因子融合的自适应波束形成算法 1.1 阵列回波信号的空域抽样

 图1 阵列波束对比 Fig. 1 Contrast of the array beam

 $B(u) = \frac{{\sin \left( {\dfrac{{{\text{π}}uN}}{{2D}}} \right)}}{{\sin \left( {\dfrac{{{\text{π}}u}}{2}} \right)}}$ (1)

 ${D_{\max }} = \frac{1}{{{u_1}}}$ (2)

${u_1}$ 代入式（2）可以得到：

 $2D \cdot {D_{\max }}{\rm{ = }}N$ (3)

 $N \le {D^2}{\rm{ + }}D_{\max }^2$ (4)

 ${D_{\max }} \ge \sqrt {\frac{N}{2}}$ (5)

 图2 空域抽样数据原理示意图 Fig. 2 Schematic diagram of spatial sampling data

1.2 最小方差波束形成算法

 ${{y}}(k) = {{{w}}^{\rm{H}}}(k){{x}}(k)$ (6)

 ${{{w}}_{{\rm{opt}}}} = \frac{{{{{R}}^{{\rm{ - 1}}}}{{a}}}}{{{{{a}}^{\rm{H}}}{{{R}}^{{\rm{ - 1}}}}{{a}}}}$ (7)

 ${{R}}{\rm{ = }}\frac{1}{P}\sum\limits_{p = 1}^P {{{{x}}_p}(k){{x}}_p^{\rm{H}}(k)}$ (8)

 $\varepsilon {\rm{ = }}\gamma \cdot {\rm{tr}}\left\{ {{R}} \right\}$ (9)

 ${{y}}(k) = \frac{1}{P}\sum\limits_{p = 1}^P {{{w}}_p^{\rm{H}}(k){{{x}}_p}(k)}$ (10)

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{w}}_p}(k) = \frac{{{{({{{R}}_p}{\rm{ + }}\gamma \cdot {\rm tr}\left\{ {{{{R}}_p}} \right\}{{I}})}^{{\rm{ - 1}}}}{{a}}}}{{{{{a}}^{\rm{H}}}{{({{{R}}_p}{\rm{ + }}\gamma \cdot {\rm tr}\left\{ {{{{R}}_p}} \right\}{{I}})}^{{\rm{ - 1}}}}{{a}}}}$ (11)

1.3 相干因子计算

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C_{\rm{f}}}(k){\rm{ = }}\frac{{{S\!_{{\rm{xg}}}}}}{{{S\!_{{\rm{sum}}}}}}{\rm{ = }}\frac{{{{\left| {\displaystyle\sum\limits_{i = 1}^M {{{{x}}_i}(k{\rm{ - }}{\varDelta _i})} } \right|}^2}}}{{M\displaystyle\sum\limits_{i = 1}^M {{{\left| {{{{x}}_i}(k{\rm{ - }}{\varDelta _i})} \right|}^2}} }}$ (12)

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{y}}(k){\rm{ = }}\frac{{{C_{\rm{f}}}(k)}}{P}\sum\limits_{p = 1}^P {{{w}}_p^{\rm{H}}(k){{{x}}_p}(k)}$ (13)
2 协方差矩阵反演的计算过程与复杂度分析

 ${\;\;\;\;\;\;\;\;\;\;\;\left\{\!\!\!\! {\begin{array}{*{20}{c}} {{r_{{\rm{ - }}q}} = \dfrac{1}{{M - q}}\displaystyle\sum\limits_{i = 1}^{M - q} {{r_{i,(i + q)}}} ,} \\ {{r_q} = \dfrac{1}{{M - q}}\displaystyle\sum\limits_{i = 1}^{M - q} {{r_{(i + q),i}}} ,} \end{array}} \right.0 \le q \le M - 1}$ (14)

1）根据Zohar算法[22-23]，求解线性方程组：

 $\left\{\!\!\!\! {\begin{array}{*{20}{c}} {{{{R}}_{\rm{T}}}{{x}} = {{f}}{\rm{,}}} \\ {{{{R}}_{\rm{T}}}{{y}} = {{{e}}_M}} \end{array}} \right.$ (15)

2）根据Toeplitz矩阵定理， ${{R}}_{\rm{T}}^{ - 1}$ 的列向量 ${{{z}}_j}(j{\rm{ = }}1,$ $2, \cdots ,M)$ 满足如下关系[24]

 $\left\{\!\!\!\! {\begin{array}{*{20}{l}} {{{{z}}_M} = {{y}}{\rm{,}}} \\ {{{{z}}_j} = {{Q}}{{{z}}_{j + 1}} + {{{y}}_{M - j}}{{x}} - {{{x}}_{M - j}}{{y}}{\rm{,}}} \end{array}} \right.j = M - 1, \cdots ,2,1$ (16)

3 仿真与讨论

 图3 缺陷分布示意图 Fig. 3 Schematic diagram of the defect distribution

3.1 数据信息不对等时成像性能分析

 图4 数据不对等时裂纹缺陷成像效果对比 Fig. 4 Comparison of imaging effects of crack defects under unequal data

 图5 数据不对等时横通孔缺陷成像效果对比 Fig. 5 Comparison of imaging effects of cross-drilled hole defects under unequal data

 ${\rm API} = \frac{{{A_{{\rm{ - 6\;dB}}}}}}{{{\lambda ^2}}}$ (17)

3.2 使用空域抽样数据成像性能分析

 图6 空域抽样数据下裂纹缺陷成像效果对比 Fig. 6 Comparison of imaging effects of crack defects under spatial sampling data

 图7 空域抽样数据下横通孔缺陷成像效果对比 Fig. 7 Comparison of imaging effects of cross-drilled hole defects under spatial sampling data

 图8 4个通孔缺陷主、旁瓣分布 Fig. 8 Main and side lobes of four cross-drilled hole defects

4 结　论

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