Shape analysis of framed space curves
This work addresses shape matching and classification problems in computer vision and medical imaging by providing a theoretical framework for analyzing 3D curves, though it is incremental as it builds on existing planar methods.
The paper extends elastic shape analysis from planar curves to framed space curves by generalizing the square root transform using quaternions and the Hopf fibration, mapping them to an infinite-dimensional complex Grassmannian for explicit geodesic computation and enabling curve averaging via a novel flag means algorithm.
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. Averages of collections of plane and space curves are computed via a novel algorithm utilizing flag means.