Technical Report MSC-2012-24

Title: Continuous Symmetries of Non-rigid Shapes
Authors: Carmi Grushko
Supervisors: Ron Kimmel
Abstract: Our world is known for its abundance of symmetric structures - in the fields of the animal kingdom, in astronomy, mathematics and chemistry, to name a few. The existence of symmetries in 3D shapes is of great interest when such applications as efficient storage, comparison and search are considered.

Traditionally, only symmetries which are a composition of rotations and reflections were studied. These symmetries, termed extrinsic, have limited use in non-rigid shapes, as they are easily lost when the shape is deformed. A different approach, treating 3D shapes as metric spaces, allows the definition of intrinsic symmetries, which offer a way for symmetry detection in deformable, non-rigid shapes.

In this thesis we present a method to compute intrinsic, continuous symmetries of 3D shapes, based on a general-purpose isometry-finding algorithm. We further explore non-Euclidean intrinsic continuous symmetries, and in particular, we demonstrate an algorithm for the detection of affine-invariant intrinsic continuous symmetry.

Finally, we develop an efficient algorithm to solve dense, low-dimensional linear min-max problems, which directly improves the performance of the previously mentioned general-purpose isometry-finding algorithm.

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