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Coordinate mapping with rigid body transformation

Now we can deal with the misalignment or configuration error of each subaperture. Denote by gi the unknown configuration of the subaperture’s local frame with regard to the model frame. When nominally aligned, configuration gi is exactly the unit matrix I. The measuring points are then transformed into the model frame in homogeneous coordinates. We denote by fi the mapping from triplets wp to the coordinates in the model frame:

The mapping fi is a combination of object-image mapping and rigid body transformation.

The configuration of a rigid body is parameterized by elements in special Euclidean group SE(3), which is usual in robotics.57 Basically, other types of parameterization also work but result in different treatment in the subsequent stitching optimization. (Readers familiar with general robotics are encouraged to parameterize the configuration in their own way.)

Because all subaperture measuring points are transformed in the same frame, we use the coordinates to calculate the normal distances to the nominal surface in the model frame:

where hp is the homogeneous coordinates of the closest points on the nominal surface. The normal distances along with the normal vector and the closest points are related to the configuration. Accordingly, the overlapping deviation is represented by the difference between normal distances of the corresponding points in two subapertures. The objective of stitching is to make subaperture overlapping regions consistent in the global frame based on the LS principle:

where No is the total number of overlapping point pairs. The left superscript ik indicates the overlapping region between subapertures k and i.

 
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