טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentDan Raviv
SubjectInvariant Metrics for Non-Rigid Shapes
DepartmentDepartment of Computer Science
Supervisor Full Professor Kimmel Ron
Full Thesis textFull thesis text - English Version


Abstract

Many shape analysis methods in computer vision, graphics, and pattern recognition, model shapes as Riemannian manifolds. Differential geometry provides a broad arsenal of tools allowing to describe local and global invariant properties of the shape and based on them, compute similarity and correspondence between two shapes. Moreover, it is possible to represent metric structures in different spaces that are convenient to work with. Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity - by measuring the lack of isometry, deal with similarity - by multiplying the Euclidean arc-length by the Gaussian curvature,  and manage equi-affine transformations - by resorting to the special affine arc-length definition in classical affine geometry. Here, we propose a computational framework that is invariant to the affine group of transformations (similarity and equi-affine) and thus, by construction, can handle non-rigid shapes. Diffusion geometry encapsulates the resulting measure to robustly provide signatures and computational tools for affine invariant surface matching and comparison. In this thesis we consider the basic questions of comparison and correspondence from a metric point of view. We show how to construct new invariant differential forms and analytically prove their invariant properties. We justify our constructions with a set of numerical validations and experimental results.