# An ADS-Based Sparse Optimization Method for Sonar Imaging Sensor Arrays
## 3 Mills Cross Array Optimization Using ADS Method  
The Mills Cross array sensor is a kind of regular array, which is obtained by virtual array element technology. It has been widely investigated in radar and sonar imaging [19]. Figure 2 shows a Mills Cross array with nine transmit and nine receive elements, which form a transmit and receive array, respectively, arranged on the X and Y axes of the Cartesian coordinate system. According to the equivalent phase principle, the Mills Cross array can determine a \(9\times 9\) virtual array sensor under the condition of far field, as shown in Figure 3. Therefore, this will allow to greatly decrease the complexity of the imaging system compared with the common real array. However, there is still some degree of redundancy in the Mills Cross array sensor when the number of array elements reaches a certain scale. Because the Mills Cross array sensor is one two-dimensional planar array, there will be a problem of large computational complexity if the existing ADS optimization strategies are directly used [20]. For this reason, this paper proposes a different optimization strategy to avoid this problem. Firstly, the two-dimensional virtual array sensor from the Mills Cross array should be divided into several virtual linear arrays with equal length. As shown in Figure 3, the virtual circle represents the locations of virtual array elements, and each dashed frame is one virtual linear array of the same length. According to array theory [21], each virtual linear array can be regarded as a \"virtual large array element\". In this way, in the direction perpendicular to the virtual linear array, a linear array composed of virtual linear array can be formed. According to the expected beam pattern and the estimated _PSLL_, the linear array can be thinned by the ADS method to choose a virtual linear array, and then some virtual array elements are gained in the same way. As such, the sparse optimization of two-dimensional virtual array sensors can be realized from the azimuth X and Y. Finally, the position of these resultant sensors is used for reverse calculation to reserve and discard the array elements in the original Mills Cross array. As a result, a two-dimensional array with a more sparse and nonuniform \"cross\" shape is obtained. The optimization implementation is described in Figure 4.  
Figure 3: Virtual array form Mills Cross array in Figure 2.  
Figure 2: Mills Cross array with nine transmit and nine receive elements. Because the beam performance of the array is greatest when the sensors are spaced at half the wavelength of the transmitting sound wave [21], the virtual array elements from the Mills Cross array are also on the grid at half wavelength. Secondly, after the finite length linear array from the virtual array elements is thinned by the ADS method, the sensor arrangement of the sparse array is represented by a binary sequence, that is, 1 is represented as array element retention and 0 as array element abandonment. The obtained \(K\) array elements are sparsely distributed at \(N\) half-wave positions. Moreover, the advantages of the ADS method are shown by the low redundancy of fewer array elements. In this paper, 40 transmitting and 40 receiving elements were used in the original Mills Cross array. Because the thinned ratio in ADS research is usually 50%, that is, \(K\) is 20, the parameters (\(N\),\(K\),\(\Lambda\),\(t\)) in the ADS library are only (40, 20, 9, 10), and the corresponding original sequence is:  
\[D_{20}^{(0)}=(1,3,4,5,7,8,11,14,16,17,18,19,22,24,25,26,27,31,32,36)\]  
The \(PSLL\) range of the corresponding array expressed by this sequence is shown in Table 1.  
After the Mills Cross array with 40 transmitting elements and 40 receiving elements is further thinned by the (\(N\), \(K\), \(\Lambda\), \(t\))-ADS method, the element locations in the thinned array are determined, as  
Figure 4: Almost Different Sets (ADS) optimization procedure for a Mills Cross array. shown in Figure 5a.