## 2 Image Reconstruction
### Coherent Image Reconstruction on a Cartesian Grid Using a Delay and Sum (D &S) Beamformer  
In the following, a precise phase-related (D&S) interpolation is described, which yields a Cartesian representation of the image reconstruction. A voxel-based beamforming (BF) [24] is employed here in order to avoid the need for image processing/conversions (Figure 1). We assume a baseband signal based on bandpass-sampling or baseband-shift, respectively. The processing of quadrature samples of the complex envelope of a bandpass signal is a common sonar processing/conversion technique [25]. This allows a precise (D&S) beamforming [26], which maps the signal coherence to a pixel/voxel representation. The coherent BF-recon  
Figure 1: The image reconstruction is directly calculated on a Cartesian grid. A single beam \(S_{bf}(z_{f})\) represents a perpendicular vector within the voxel-grid \(S_{bf}(x,y,z)\) in contrast to the representation in spherical-coordinates with azimuth/elevation angle. is described in the following. Vector \(\overrightarrow{V}(x_{p},y_{p},z_{f})\) points along the z-axis where \(z_{f}\) is an array of defined depth points and \(x_{p}\) and \(y_{p}\) are the remaining positions defining the entire reconstruction-grid. \(\tau_{r}(s,z_{f})\) describes the time-delay from each sensor \(s\) to the scattering-point point \(z_{f}\); it is the norm of the difference of the pointing vector and each receiver element \(\overrightarrow{w_{s}}(x_{r},y_{r},z_{r})\) (see (1)). For a precise interpolation of the array data, an SI-interpolation is used, which maps the corresponding spatial positions from the sampled pulse data. The argument for the 2D SI-function is calculated as in (2) with \(t(ri)\) is the corresponding time vector of the received array data \(M(s,t)\) and its corresponding sensor \(s\), where the sampled data in a single channel is also called stave-data. The following multiplication of the stave-data vector from the array matrix \(M(s,t)\) with this SI-matrix (4) interpolates the corresponding array-data on the desired spatial positions (5). This process has to be repeated for each sensor \(s\), which is the time consuming part of a (D&S) BF. The optimal step-width for the SI-interpolation function, which preserves the frequency-characteristics of the array data is a step-width one. This is a Dirac function with sub-pixel/voxel shift. Hence, a normalization is found, which meets this requirement \(\rho=(2\cdot L_{t})/(\xi_{max}-\xi_{min})\) where \(\xi_{max}\) and \(\xi_{min}\) are the maximum and minimum of (2) respectively. The scalar \(L_{t}\) is the length of time samples corresponding to the array data. \(S_{bf}(x,y,z)\) is interpolated according to (5) for a 1D vector along the z-axis. The aligned summation of all sensors yields the image reconstruction as given in Equation (6). This process is repeated for the remaining \(x_{p}\) and \(y_{p}\), which finally represents the voxel image as in (6). The resulting image reconstruction \(S_{bf}(x,y,z)\) is a function of the spatial positions \(x_{p}\), \(y_{p}\) and \(z_{f}\). The notation between \(z_{f}\) and \(x_{p}/y_{p}\) is different, in order to distinguish between \(z_{f}\) as a vector where each interpolation (Equations (1)-(5)) is carried out and the individual \(x/y\) spatial positions for which the process needs to be repeated. The spacing between \(x\), \(y\) and \(z\) is defined as \(r_{w}\) and it needs to be equidistant for all dimensions on a regular voxel grid. This factor plays a key role for the coherent processing and the interpolation of multiple scans; it will be discussed in more detail in Section 3. The computational load is comparable to general methods working in the near field [27]. An important requirement for this beamforming approach is the exact mapping between the signals and the voxels. The authors of [28] give a comprehensive description of the theory of the (D&S) BF describing this process as a spatial filter.