Mario Micheli

Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks

This paper deals with the computation of sectional curvature for the manifolds of

A Linear Systems Approach to Imaging Through Turbulence

N

State estimation in stochastic hybrid Systems with sparse observations

landmarks (or feature points) in

Implementation of the Centroid Method for the Correction of Turbulence

D

Effects of curvature on the analysis of landmark shape manifolds

dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e., the cometric), when written in coordinates, is such that each of its elements depends on at most

Random Sampling of a Continuous-time Stochastic Dynamical System

2D

Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds

of the

The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature

ND

Matrix-valued kernels for shape deformation analysis

coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly nontrivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Marios formula). We apply such a formula to the manifolds of landmarks, and in particular we fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesics. The latter example gives insight into the geometry of the full manifolds of landmarks.

Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds@@@Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds

In this paper we address the problem of recovering an image from a sequence of distorted versions of it, where the distortion is caused by what is commonly referred to as ground-level turbulence. In mathematical terms, such distortion can be described as the cumulative effect of a blurring kernel and a time-dependent deformation of the image domain. We introduce a statistical dynamic model for the generation of turbulence based on linear dynamical systems (LDS). We expand the model to include the unknown image as part of the unobserved state and apply Kalman filtering to estimate such state. This operation yields a blurry image where the blurring kernel is effectively isoplanatic. Applying blind nonlocal Total Variation (NL-TV) deconvolution yields a sharp final result.