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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Statistics Seminars: The Role of Quotient Spaces in Shape Analysis of Curves and Surfaces

Presented by Anuj Srivastava, Florida State University

28 June 2011 14:15 in CM103

This research seeks efficient techniques for analyzing shapes of 2D and
3D objects by considering their boundaries as curves and surfaces. An important
distinction is to treat boundaries not as point sets or level sets, as is commonly done, but
as parameterized objects. However, parameterization adds an extra variability
in the representation, as different re-parameterizations of an object do not change
it shape. This variability are handled by defining quotient spaces of object representations,
modulo re-parameterization and rotation groups, and inheriting a Riemannian metric
on the quotient space from the larger space. This last step requires the metric to be
such that the action of re-parameterization group is by isometries. For curves in Euclidean
spaces, we use an elastic Riemannian metric that can be viewed as an extension of the
classical Fisher-Rao metric, used in information geometry, to higher dimensions.
Furthermore, we define a specific square-root representation that reduces this
complicated metric to the standard L2 metric and, thus, greatly simplifying computations
such as geodesic paths, sample means, tangent PCA, and stochastic modeling of
observed shapes. For curves, we have proposed a similar square-root representation
and an elastic Riemannian metric, that allows parameterization-invariant shape analysis
of 3D objects. I will demonstrate these ideas using applications from computer vision,
biometrics and activity recognition, protein structure analysis, anatomical shape analysis,
and neuroscience spike train analysis.

(This research is done in collaboration with Eric Klassen, Sebastian Kurtek,
Ian Jermyn, Wei Wu, Jinfeng Zheng, and many others.)

Contact Ian Jermyn for more information