Registering sparse point sets

Sparse point sets arise after registering e.g. tree stems or other formations of the nature or 

built environment. There are problems with obstruction (some objects stay behind another 

for a long time, then appear again), failed registration, inexact placement of one representative point to each object, etc. These problems cause disparity between two

point sets, and one has to chase for subsets with more effort than normal registration algorithms allow. 

There are basically three similarity approaches to be used: 

• point-by-point. The Fig. above has five good matches between squares and circles.
• segment-by-segment. Four good matches above.
• triangle-by-triangle. One good match (*) depicted, also three half-matches (+) shown. 

First two approaches are rather trivial. The segment-by-segment error between points $p_1$ and $p_2$ can be e.g. the geometric mean $e_{12}=\sqrt{e_1e_2}$ of the point matching errors $e_1$ and $e_2$. There are also more specific segment matching, e.g. 
for penalizing more the case, when the errors at each end are opposite, diverting 
the segment alignment. 

The triangle-wise matching is more complex, and includes the categories of half-match (+) between three triangles and full match (*) between two triangles. If all three (or at least the best two of three) above approaches are employed cleverly, one has a good chance in matching two sparse point clouds with a lot of noise and missing point pairs.