Shape Analysis with Parametric Mixtures	Shape matching plays a prominent role in the analysis of medical and biological structures. We present a unifying framework for shape matching that uses mixture-models to couple both the shape representation and deformation. The theoretical foundation is drawn from information geometry where information matrices are used to establish intrinsic distances between parametric densities. When a parameterized probability density function is used to represent a landmark-based shape, the modes of deformation are automatically established through the information matrix of the density. We first show that given two shapes parameterized by Gaussian mixture models, the well known Fisher information matrix of the mixture model is a natural, intrinsic tool for computing shape geodesics.  We have also developed a new Riemannian metric based on generalized φ-entropy measures. In sharp contrast to the Fisher-Rao metric, our new metric is available in closed-form. Geodesic computations using the new metric are considerably more efficient.  (See Publications for related papers.  We acknowledge support from the National Science Foundation, NSF IIS-0307712.)
Wavelet Density Estimation   (point set)                  (estimated wavelet density)	Wavelet based density estimators have gained in popularity due to their ability to approximate a large class of functions; adapting well to difficult situations such as when densities exhibit abrupt changes. The decision to work with wavelet density estimators brings along with it theoretical considerations (e.g. non-negativity, integrability) and empirical issues (e.g. computation of basis coefficients) that must be addressed in order to obtain a bona fide density. We present a new method to accurately estimate a non-negative, density which directly addresses many of the problems in practical wavelet density estimation. We cast the estimation procedure in a maximum likelihood framework that estimates the square root of the density √p; allowing us to obtain the natural non-negative density representation (√p)². Analysis of this method brings to light a remarkable theoretical connection with the Fisher information of the density and consequently leads to an efficient constrained optimization procedure to estimate the wavelet coefficients.  (See Publications for related papers.  We acknowledge support from the National Science Foundation, NSF IIS-0307712.)
Sliding Wavelets for Indexing and Retrieval	Shape representation and retrieval of stored shape models are becoming increasingly more prominent in fields such as medical imaging, molecular biology and remote sensing. We present a novel framework that directly addresses the necessity for a rich and compressible shape representation, while simultaneously providing an accurate method to index stored shapes. The core idea is to represent point-set shapes as the square-root of probability densities expanded in a wavelet basis. We then use this representation to develop a natural similarity metric that respects the geometry of these distributions, i.e. under the wavelet expansion distributions are points on a unit hypersphere and the distance between distributions is given by the separating arc length. The process uses a linear assignment solver for non-rigid alignment between densities prior to matching; this has the connotation of ``sliding'' wavelet coefficients akin to the sliding block puzzle L'Âne Rouge.  (See Publications for related papers.  We acknowledge support from the National Science Foundation, NSF IIS-0307712.)