Resume

• 2013

  Society for Industrial and Applied Mathematics

• 2005

  Doctor of Philosophy (PhD)

  Mathematics

  Texas A&M University

• 2002

  Bachelor of Science (BS)

  Mathematics

  The University of Georgia

• 2000

  Mathematics

  College Bowl

  Math Team

  Student Government Association

  Math Lab Student Worker

  Dalton State College

• Medical Imaging

  LaTeX

  Harmonic Analysis

  ImageJ

  Computer Science

  Mathematica

  University Teaching

  Biomedical Engineering

  Inverse Problems

  Mathematical Modeling

  Python

  Research

  Partial Differential Equations

  Signal Processing

  Matlab

  Optimization

  Calculus

  Approximation Theory

  Maple

  Image Analysis

  Harmonic Singular Integrals and Steerable Wavelets in $L_2 (\\mathbb{R}^d)$

  Michael Unser

  Here we present a method of constructing steerable wavelet frames in $L_2(R^d)$ that generalizes and unifies previous approaches

  including Simoncelliʼs pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry

  whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis

  and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient.

  Harmonic Singular Integrals and Steerable Wavelets in $L_2 (\\mathbb{R}^d)$

  Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function (RBF) approximation. The purpose of this paper is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF networks on $R^d$. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding $L^p$ norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem.

  $L^p$ Bernstein inequalities and inverse theorems for RBF approximation on $R^d$

  Michael Unser

  The Riesz transform is a natural multi-dimensional extension of the Hilbert transform

  and it has been the object of study for many years due to its nice mathematical properties. More recently

  the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach

  however

  is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness

  decay

  and vanishing moments. In this paper

  we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants

  and as an application

  we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets

  we obtain steerable frames with rapid decay.

  Decay properties of Riesz transforms and steerable wavelets

  $L^p$ error estimates for approximation by Sobolev splines and Wendland functions on $\\mathbb{R}^d$

  It is known that a Green’s function-type condition may be used to derive rates for approximation by radial basis functions (RBFs). In this paper

  we introduce a method for obtaining rates for approximation by functions which can be convolved with a finite Borel measure to form a Green’s function. Following a description of the method

  rates will be found for two classes of RBFs. Specifically

  rates will be found for the Sobolev splines

  which are Green’s functions

  and the perturbation technique will then be employed to determine rates for approximation by Wendland functions.

  $L^p$ error estimates for approximation by Sobolev splines and Wendland functions on $\\mathbb{R}^d$

  John Paul

  Ward

  Texas A&M University

  EPFL

  North Carolina A&T State University

  University of Central Florida

  Orlando

  Florida

  University of Central Florida

  Texas A&M University

  College Station

  Texas

  Topic of research: Spline/RBF approximation \n\nTeaching duties: Undergraduate courses on \n 1) partial differential equations and asymptotic analysis \n 2) topics in contemporary mathematics.

  Visiting Assistant Professor

  Lausanne

  Switzerland

  General areas of interest: Applied harmonic analysis

  Functional analysis

  Approximation theory\n\nSpecific topics of research: Spline and wavelet approximation

  Adaptable frame representations

  Function spaces

  Stochastic processes

  Postdoctoral Researcher

  EPFL

  North Carolina A&T State University

  Texas A&M University

  College Station

  Texas

  Teaching duties: taught business mathematics; instructed Calculus students in the use of mathematical software (MATLAB and Maple); directed Calculus recitations.

  Research and Teaching Assistant