#### Milestone Year

### 2013

#### Bounded Distortion Deformation Spaces

Institute scientists developed a convex approach for representing
deformations of two and higher dimensional objects with desired
distortion guarantees.

In the fields of geometry modeling, computer graphics, and computer vision
it is often required to represent a deformation of a two or three-dimensional
objects with a flexible model that possess certain properties.
Two desired properties are that the deformation will not fold the space
onto itself (mapping two or more different points to the same target
location is not allowed), and that the deformation will not change
distances between pairs of nearby points too much
(i.e., bounded metric distortion).

Since the mathematical formulation of these constraints leads to complicated
non-convex equations, the standard solution to these problems was numerical
in essence. The model developed in the institute analytically characterizes
maximal convex subsets of the original non-convex constraints, opening
the door to better understanding of constrained deformation spaces.

Application of the new approach to existing challenges in graphics and vision
seems to provide best known algorithms. One example is the problem of image
matching that aims to automatically find corresponding points in different
images. Using this flexible yet constrained space of deformations leads
to robust algorithms for matching images. Another application of this convex
deformation model is in parameterization and registration of surfaces and
volumes that are used for comparison, analysis and visualization
of anatomical and artificial shapes.