Date Speaker Topic
F Feb 25    
Feb 21 Presidents’ Day (University Holiday) No seminar
M Feb 28
F Mar 11 Eric Todd Quinto Microlocal analysis of Fourier Integral Operators in scattering tomography    

Abstract:    

We present a novel microlocal analysis of generalized Radon transforms that describe the integrals of functions of compact support over curves in the plane and surfaces of revolution of smooth curves in higher dimensional spaces. We show that the Radon transforms satisfy important properties that show that the reconstruction operator should image visible features of objects and not add artifacts. We introduce microlocal analysis, the math behind our results.

Our results apply to models in Emission Compton Scattering Tomography (ECST) and Bragg Scattering Tomography (BST). We show that the ECST and BST integration curves or surfaces in general satisfy the conditions to not produce artifacts. We provide simulated reconstructions from ECST and BST data to illustrate our points. Additionally, we give “sinusoidal” integration curves which do not satisfy the assumptions, and we provide simulations of the image artifacts. The observed artifacts in reconstruction are shown to align exactly with our predictions.

This is joint work with James Webber (Brigham and Women’s Hospital, formerly Tufts University).
F Apr 1 Kasso Okoudjou On a very fun and addictive problem: The HRT Conjecture
    

Abstract:    

Pick your favorite square integrable function g defined on the real line (you may assume that g is a Schwartz class function). Is the set \{ g(x), e^{2πix}g(x), g(x - 1), e^{2πibx}g(x - a) \} linearly independent in L^2(\mathbb{R}) for each (a, b) \in \mathbb{R}^2 \ \{ (0, 0),(0, 1),(1, 0) \}?This a special case of a conjecture put forth in 1996 by C. Heil, J. Ramanatha, and
P. Topiwala. The conjecture now known as the HRT conjecture asserts that the (finite) set G(g,\Lambda) = \{e^{2πib_k} \dot g( \dot - a_k) \}^N_{k=1} is linearly independent for any non-zero square integrable function g and subset \Lambda = {(a_k, b_k)}^N_{k=1 \subset \mathbb{R}^2}. In the talk I will give an overview of the conjecture, its connection to the zero-divisor conjecture in algebra, and my (failed) attempts at solving it. No background beyond curiosity
is required, and I promised you will leave wanting to work on the problem.
F May 6 Mostafa Maslouhi Some results and open problems on Frames and Probabilistic Frames
    

Abstract:    

In this talk we present a solution of two open problems posed in (Ehler and Okoudjou, 2013) related to the representation of Positive Operators Valued Measures (POVM) by means of tight probabilistic frames in \mathbb{R}^d and how far is the closest probabilistic tight frame from a given probability measure in \mathbb{R}^d, where the distance here is the quadratic Wasserstein metric W_2 used for optimal transportation problem for measures. For the first problem we characterize all the POVM representable by means of a tight probabilistic frames in \mathbb{R}^d and for the second, we solve the optimization problem
I(\mu):=\inf_{\nu\in \mathcal{T}(\R^d)}W_2^2(\mu,\nu), where \mathcal{T}(\mathbb{R}^d) is the set of all probabilistic tight frames in \mathbb{R}^d, and establish the uniqueness of its optimum and give an explicit expression of it.
We introduce also (if time allows it) a new probabilistic model for the occurrence of erasures in multiple channel data transmission. This new model uses Parseval frames to encode the information to be transmitted and a sequence of Bernoulli random variables. It turns out that, compared to existing models (Holmes and Paulsen, 2004; Casazza and Kovacevic, 2003; Leng et al., 2013; Li et al., 2018), our model gives better performance for recovering transmitted data.
The results presented in this talk are part of the published papers (Loukili and Maslouhi, 2020,
2022, 2018).
At the end of this talk, we present some open problems related to the subject of probabilistic frames.