Related papers: Sampling, splines and frames on compact manifolds
This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most…
We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of…
A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $R^{(m-1)N}$. The developed method depends on Wiener-Hopf…
We propose a Monte Carlo method to efficiently find, count, and sample abstract triangulations of a given manifold M. The method is based on a biased random walk through all possible triangulations of M (in the Pachner graph), constructed…
A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees…
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process.…
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo-…
We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…
Manifolds occur naturally as configuration spaces of robotic systems. They provide global descriptions of local coordinate systems that are common tools in expressing positions of robots. The purpose of this survey is threefold. Firstly, we…
This paper is concerned with a rate-distortion theory for sequences of i.i.d. random variables with general distribution supported on general sets including manifolds and fractal sets. Manifold structures are prevalent in data science,…
In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on…
We present here a set of lecture notes on tomography. The Radon transform and some of its generalizations are considered and their inversion formulae are proved. We will also look from a group-theoretc point of view at the more general…
We explore the use of tools from Riemannian geometry for the analysis of symmetric positive definite matrices (SPD). An SPD matrix is a versatile data representation that is commonly used in chemical engineering (e.g.,…
Representing complex shapes with simple primitives in high accuracy is important for a variety of applications in computer graphics and geometry processing. Existing solutions may produce suboptimal samples or are complex to implement. We…
We construct spherical wavelets based on approximate identities that are directional, i.e. not rotation-invariant, and have an adaptive angular selectivity. The problem of how to find a proper representation of distinct kinds of details of…
The class of special generic maps contains Morse functions with exactly two singular points, characterizing spheres topologically which are not $4$-dimensional and the $4$-dimensional unit sphere. This class is for higher dimensional…
The objective of this article is to describe a construction of Parseval bandlimited and localized frames on sub-Riemannian compact homogeneous manifolds.
This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by…
We analyze the detection and classification of singularities of functions $f = \chi_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated…
Spherical symmetry is ubiquitous in nature. It's therefore unfortunate that spherical system simulations are so hard, and require complete spheres with millions of interacting particles. Here we introduce an approach to model spherical…