The goal of our work is to develop an efficient, automatic algorithm for discovering point correspondences between surfaces th at are approximately and/or partially isometric.

Our approach is based on three observations. First, isometries are a subset of the Mobius group, which has low-dimensionality -- six degrees of freedom for topological spheres, and three for topological discs. Second, computing the Mobius transformat ion that interpolates any three points can be computed in closed-form after a zero-error mid-edge flattening to the complex pl ane. Third, deviations from isometry can be modeled by a transportation distance between corresponding points in that plane.

Motivated by these observations, we have developed a Mobius Voting algorithm that iteratively: 1) samples a triplet of three r andom points from each of two point sets, 2) uses the Mobius transformations defined by those triplets to map both point sets into a canonical coordinate frame on the complex plane, and 3) produces ``votes'' for predicted correspondences between the mu tually closest points with magnitude representing their estimated deviation from isometry. The result of this process is a fuz zy correspondence matrix, which is converted to a permutation matrix with simple matrix operations and output as a discrete se t of point correspondences with confidence values.

The main advantage of this algorithm is that it can find intrinsic point correspondences in cases of extreme deformation. Dur ing experiments with a variety of data sets, we find that it is able to find dozens of point correspondences between different object types in different poses fully automatically.

@article{Lipman:2009:MVF,
   author = "Yaron Lipman and Thomas Funkhouser",
   title = "M{\"o}bius Voting for Surface Correspondence",
   journal = "ACM Transactions on Graphics (Proc. SIGGRAPH)",
   year = "2009",
   month = aug,
   volume = "28",
   number = "3"
}