Spherical Functions on Riemannian Symmetric Spaces

This paper deals with some simple results about spherical functions of type $\delta$, namely new integral formulas, new results about behavior at infinity and some facts about the related $C_\sigma$ functions.Comment: 15 page

The Abel, Fourier and Radon transforms on symmetric spaces

In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.Comment: 24 page

A duality in integral geometry; some generalizations of the Radon transform

First published in the Bulletin of the American Mathematical Society in Vol.70, 1964, published by the American Mathematical Societ

Quantum Mechanics on SO(3) via Non-commutative Dual Variables

We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.Comment: 29 pages; matches the published version plus footnote 7, a journal reference include

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Fast and Robust Femur Segmentation from Computed Tomography Images for Patient-Specific Hip Fracture Risk Screening

Osteoporosis is a common bone disease that increases the risk of bone fracture. Hip-fracture risk screening methods based on finite element analysis depend on segmented computed tomography (CT) images; however, current femur segmentation methods require manual delineations of large data sets. Here we propose a deep neural network for fully automated, accurate, and fast segmentation of the proximal femur from CT. Evaluation on a set of 1147 proximal femurs with ground truth segmentations demonstrates that our method is apt for hip-fracture risk screening, bringing us one step closer to a clinically viable option for screening at-risk patients for hip-fracture susceptibility.Comment: This article has been accepted for publication in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, published by Taylor & Franci