## 5 Pixel Based Scan Registration  
As discussed in the previous sections, the coherent SAS processing of the separate array data requires the knowledge of the underlying spatial transformations within reconstructed sonar images. The idea of using registration of multiple scans from different unknown sensor positions is based on our previous work on registration of noisy data with partial overlap including especially sonar [19; 20; 21]. This includes especially spectral registration methods [38; 39], which are capable of matching scans as an entire unit without a dependency on features within the scan representation. The registration method used here is a robust 2D Fourier-Mellin implementation, which determines rotation and translation in subsequent dependent steps. The resampling methodology shown in (27) is the 2D Fourier Mellin Invariant (FMI) [40]. It decouples transformation parameters (rotation/scale) from translation. Phase correlation of descriptor functions are generated according to (27) and yield unique Dirac peaks indicating all parameters of the underlying transformation. Once this registration is successful, which can be verified by a unique signal/noise ratio of the Dirac maximum, the registration is precise within the rendered pixel/voxel resolution. Even in the case of typical noisy, fuzzy sonar images, the correct position can be determined, although both peaks (FMI descriptor function plus FMI translational registration) appear to be smeared over a larger area. The experiments later on will demonstrate that sufficiently precise parameters (rotation, translation) can be determined, which allow a pixel alignment for our SAS. Since the transformation parameter of scale is not required, the FMI polar-logarithmically resampling (27) can be replaced by a simple resampling in polar coordinates. \[ \begin{split} u_{mk}&=\frac{\frac{N_{R}}{2}-1}{M-1}( M-1)^{\frac{m}{M-1}}\cos(\frac{\pi k}{K})+\frac{N_{R}}{2}\\ v_{mk}&=\frac{\frac{N_{R}}{2}-1}{M-1}(M-1)^{\frac{m }{M-1}}\sin(\frac{\pi k}{K})+\frac{N_{R}}{2}\\ m&=0,...,M-1;k=0,...,K-1\end{split}  \tag{27}\]  
_Image and Sensor-Platform Registration and Transformation_  
Although real spatial transformations are not required for the coherent SAS imaging (pixel-wise alignment), it is straightforward to determine the positions of the sensor-carrier from available imaging parameters. The following describes the relations between the sensor-platform, the range between the antenna-center and the generated image. The 3D rotation within a Cartesian coordinate system is defined as a result of a multiplication of the three matrices \(R(\alpha,\beta,\gamma)\in SO(3)\) corresponding to each axis \(R_{x}(\alpha)\ R_{y}(\beta)\ R_{z}(\gamma)\). Translation in meter, for example, for x/y-direction is denoted as \(x_{r}/y_{r}\). The resulting shift in pixel is then \(xy_{im}=xy_{r}\frac{N_{xy}}{S_{xy}}\). Here \(N_{xy}\) is the entire number of displayed pixels and \(S_{xy}\) is the corresponding image range in meter, which is covered in that direction. The corresponding translation matrix \(T\) in homogeneous coordinates is defined according to (28). In order to map the sensor-platform transformation directly to the image transformation, the image center needs to be shifted to the sensor-platform center. This is defined by the distance \(z_{Srange}\) from the sensor-platform center to the image center. The \(z\)-axis is defined perpendicular to the sensor platform resulting in a single shift. Finally, for a correct transformation in local image coordinates the origin of the pixel representation must be shifted to the center using \(I_{c}\) (29). Here, \(x_{c}\), \(y_{c}\) and \(z_{c}\) are defined as \(\frac{N_{xy,zz}}{2}\). Concretely, the complete transformation is defined according to (30) including all possible transformation parameters. The necessary translational shifts to the antenna-center \(z_{Srange}\) and to the image center are combined in the matrix \(I_{c}\) (29). The center in meter \(z_{Srange}\) is known by the signal runtimes.