Eric Klassen

Analysis of planar shapes using geodesic paths on shape spaces

For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic. Using the Surrey fish database, we demonstrate some applications of this approach: 1) interpolation and extrapolations of shape changes, 2) clustering of objects according to their shapes, 3) statistics on shape spaces, and 4) Bayesian extraction of shapes in low-quality images.

A Novel Representation for Riemannian Analysis of Elastic Curves in Rn

We propose a novel representation of continuous, closed curves in Rn that is quite efficient for analyzing their shapes. We combine the strengths of two important ideas-elastic shape metric and path-straightening methods - in shape analysis and present a fast algorithm for finding geodesies in shape spaces. The elastic metric allows for optimal matching of features while path-straightening provides geodesies between curves. Efficiency results from the fact that the elastic metric becomes the simple L2 metric in the proposed representation. We present step-by-step algorithms for computing geodesies in this framework, and demonstrate them with 2-D as well as 3-D examples.

Monte Carlo extrinsic estimators of manifold-valued parameters

Monte Carlo (MC) methods have become an important tool for inferences in non-Gaussian and non-Euclidean settings. We study their use in those signal/image processing scenarios where the parameter spaces are certain Riemannian manifolds (finite-dimensional Lie groups and their quotient sets). We investigate the estimation of means and variances of the manifold-valued parameters, using two popular sampling methods: independent and importance sampling. Using Euclidean embeddings, we specify a notion of extrinsic means, employ Monte Carlo methods to estimate these means, and utilize large-sample asymptotics to approximate the estimator covariances. Experimental results are presented for target pose estimation (orthogonal groups) and signal subspace estimation (Grassmann manifolds). Asymptotic covariances are utilized to construct confidence regions, to compare estimators, and to determine the sample size for MC sampling.

Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance

We consider the statistical analysis of trajectories on Riemannian manifolds that are observed under arbitrary temporal evolutions. Past methods rely on cross-sectional analysis, with the given temporal registration, and consequently may lose the mean structure and artificially inflate observed variances. We introduce a quantity that provides both a cost function for temporal registration and a proper distance for comparison of trajectories. This distance is used to define statistical summaries, such as sample means and covariances, of synchronized trajectories and Gaussian-type models to capture their variability at discrete times. It is invariant to identical time-warpings (or temporal reparameterizations) of trajectories. This is based on a novel mathematical representation of trajectories, termed transported square-root vector field (TSRVF), and the

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT2

mathbb{L}^2

Elastic shape matching of parameterized surfaces using square root normal fields

norm on the space of TSRVFs. We illustrate this framework using three representative manifolds---

Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds

mathbb{S}^2

2D Affine and Projective Shape Analysis

,

Gaussian Blurring-Invariant Comparison of Signals and Images

mathrm {SE}(2)

RNA global alignment in the joint sequence–structure space using elastic shape analysis

and shape space of planar contours---involving both simulated and real data. In particular, we demonstrate: (1) improvements in mean structures and significant reductions in cross-sectional variances using real data sets, (2) statistical modeling for capturing variability in aligned trajectories, and (3) evaluating random trajectories under these models. Experimental results concern bird migration, hurricane tracking and video surveillance.