Diffusing the Error.

Pulli presents a registration method that minimizes the global error and avoids inconsistent scenes [20]. The registration of one scan is followed by registering all neighboring scans such that the global error is distributed. Other matching approaches with global error minimization have been published, e.g., [3,7]. Benjemaa et al. establish point-to-point correspondences first and then use randomized iterative registration on a set of surfaces [3]. Eggert et al. compute motion updates, i.e., a transformation $(\M R, \M t)$, using force-based optimization, with data sets considered as connected by groups of springs [7].

Based on the idea of Pulli we designed the relaxation method simultaneous matching[23]. The first scan is the masterscan and determines the coordinate system. It is fixed. The following three steps register all scans and minimize the global error, after a queue is initialized with the first scan of the closed loop:

1. Pop the first 3D scan from the queue as the current one.
2. If the current scan is not the master scan, then a set of neighbors (set of all scans that overlap with the current scan) is calculated. This set of neighbors forms one point set $M$. The current scan forms the data point set $D$ and is aligned with the ICP algorithms. One scan overlaps with another iff more than $p$ corresponding point pairs exist. In our implementation, $p=250$.
3. If the current scan changes its location by applying the transformation (translation or rotation) in step 2, then each single scan of the set of neighbors that is not in the queue is added to the end of the queue. If the queue is empty, terminate; else continue at step 1.

In contrast to Pulli's approach, our method is totally automatic and no interactive pairwise alignment has to be done. Furthermore the point pairs are not fixed [20]. The accumulated alignment error is spread over the whole set of acquired 3D scans. This diffuses the alignment error equally over the set of 3D scans [24].