## 4 Sampling the Aperture
### Compressed Sensing for Aperture Gap Recovery  
As already discussed in Section 4.1, in case exact desired sensor positions cannot always be achieved, a data extrapolation within a certain range is desirable. The 2D FT of the entire array data represents the fast- and slow-time spectrum, which originates from the slow movement along the synthetic aperture compared to the time-signal [1] (see Figure 8(a)). It is shown that a spatial gap recovery for small ranges leads to good results. Figure 8: Example null steering suppressing grating lobes occurring with overlap of only 20%. Compressed sensing (CS) [32] is a sampling reduction in order to avoid signal compression, which has been used before in the context of conventional stripmap SAR [33]. A requirement is a sparse signal representation in the parameter domain representing the chosen basis functions. This is usually the frequency domain in case the Fourier basis is chosen. As the 2D spectral array representation (Figure 8(a)) shows, the slow-time frequency content is a sparse representation concentrated at low frequencies. In case a successful scan registration in our approach yields sensor-platform positions detecting a gap between an ideal sequence of neighboring sensors, CS is a feasible way for recovering signal information. The suggested CS method is although only feasible for offline-processing. This is due to the fact that it requires full array signals from all antenna positions before image reconstruction. For more details to CS theory and applications we refer to corresponding references; note that it is currently an intensive research topic in many imaging areas far beyond sonar. For example, the work in [34] deals with multiresolution CS enabling the reconstruction of a low-resolution signal having insufficient CS samples. In many applications CS is tailored to the specific data representation and its sparsity [35; 36; 37]. Figure 9 shows the signal representation where the array manifold vector is multiplied with the phase shift matrix \(Y(s,z_{f})\) defined in (3). This representation corresponds to Equation (5) before the accumulation of the rectified channel information for the image reconstruction. Figure 8(b) shows  
Figure 9: Example of the application of Compressed Sensing (CS) on a larger gap of sensor information. This signal matrix corresponds to the rectified sensor-channel information where the array manifold vector is shifted with the time-delay/phase parameter for the corresponding direction/voxel information. the data matrix of two neighboring antenna signals where the gap corresponds to a space of 10 single sensor spacings. Hence, the CS signal recovering is applied at an interim result of the beamforming-reconstruction, which takes the sensor positions (array manifold vector) into account. So, CS is used to supplement missing samples, which are supposed to be reconstructed from basis functions where the corresponding weighting coefficients are approximated from an underdetermined linear system. The basis functions should fit to the specific problem; here a 2D Fourier basis is chosen. The subsampled signal is defined as a sliding window \(B(t,s)\) grabbing \(N_{t}\) samples along the time-segment and parts of the sensor channels. Hence, a signal matrix \(B(t,s)\) of size \(N_{s}\cdot N_{t}\) are the measurement inputs for CS. The number of missing sensor-channels describing the spatial gap is defined as \(N_{z}\). Concretely, the vector \(y_{v}\) defined in (17) defines the range of sensor indices describing the gap between neighboring sensor-platform positions. The window length along the time-segment overlaps to both sides with a length of one sample. The corresponding basis-functions are defined as a 2D Fourier basis (15). Matrix \(\Psi\) (16) contains as 1D vectors all possible combinations for the frequency vectors \(u\) and \(v\) with the defined time segments \(y_{v}\) and \(x_{v}\) given in (17).