Maps and Distances between Spaces - Mathematical Tools for Shape Analysis and Description - page 46

Figure 6.3: A Möbius transformation of an image (based on http://i.imgur.com/qz7P0.png ).
6.1.4 CONCEPTS IN ACTION
3D shape correspondence and symmetry detection via Möbius maps Finding correspondences
between a discrete set of points on two surface meshes is a fundamental problem of 3D shape
analysis, with applications in shape interpolation, attribute transfer, surface completion, statistical
shape modeling, symmetry analysis, shape matching, and deformable surface tracking. For many
of these applications, the input meshes represent different objects in different poses: they are often
approximately isometric, or are composed of large parts that are nearly isometric. In this case, the
problem is that of finding corresponding points such that the mapping between them is close to
an isometry (cf. section 6.1 ).
e authors of [ 128 ] developed an efficient algorithm for discovering dense sets of point
correspondences between surfaces that are approximately and/or partially isometric. e key ob-
servation is that isometries are a subset of the Möbius group, whose properties make computa-
tionally tractable the problem of looking for a correspondence map g between two genus zero
surfaces ² (Figure 6.4 ).
Indeed, any genus zero surface can be mapped conformally (with angles preserved) to the
unit sphere (maps 1 and 2 in Figure 6.4 ). erefore, any isometry g between genus zero surfaces
gives rise to a one-to-one and onto conformal map from the unit sphere to itself (the composition
of the inverse of 1 , g and 2 in Figure 6.4 ).
One-to-one and onto mappings of a sphere to itself are Möbius transformations. e
Möbius group is known to have six degrees of freedom, that is, fixing three distinct points on
each sphere defines a Möbius map uniquely. Moreover, the Möbius transformation that interpo-
lates any three points can be computed in closed-form. Finally, deviations from isometry can be
modeled by a transportation-type distance between corresponding points.
According to these observations, [ 128 ] developed a technique for finding point correspon-
dence between nearly isometric surfaces (Figure 6.5 ). e algorithm iteratively:
²Informally, a surface without donuts-like, passing holes; see chapter 3 for a formal definition.