Conference Agenda

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Session Overview
Session
MS149, part 2: Stability of moment problems and super-resolution imaging
Time:
Wednesday, 10/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-111
30 seats, 56m^2

Presentations
10:00am - 12:00pm

Stability of moment problems and super-resolution imaging

Chair(s): Stefan Kunis (University Osnabrueck, Germany), Dmitry Batenkov (MIT Boston)

Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model.

Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

The condition number of Vandermonde matrices with clustered nodes

Dominik Nagel
University Osnabrueck

The condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is important for the stability analysis of algorithms that solve the trigonometric moment problem, e.g. Prony's method. In the univariate case and with well separated nodes, the condition number is well studied, but when nodes are close together, it gets more complicated. For this setting, there exist only few results so far. After providing a short survey over recent developments, our own results are presented.

 

Prony's problem and the hyperbolic cross

Benedikt Diederichs
University Passau

Multivariate extensions of Prony's problem have been actively investigated over the last few years. One problem has been that it is less clear than in the univariate case what sampling sets are optimal. One choice, proposed by Sauer, are sets linked to the hyperbolic cross. We state how and why the hyperbolic cross emerges and give an even smaller sampling set than Sauer, but without an efficient algorithm. Then we derive small sampling sets for multivariate extensions of MUSIC and ESPRIT. Using these sets, the algorithms need significantly less samples compared to the full grid.

 

Reconstruction of generalized exponential sums

Markus Wageringel
University Osnabrueck

Exponential sums, as used in signal processing, are functions that can be considered to encode the moments of measures supported at finitely many points. Algebraic techniques, such as Prony's method, are used to recover the underlying data of such measures from moments. After introducing these notions, we provide generalizations of the concept of an exponential sum. We follow an algebraic and geometric approach by associating algebraic varieties to these generalized objects and investigate the problem of parameter recovery in this setting.

 

Phase retrieval of sparse continuous-time signals by Prony's method

Robert Beinert
University Graz

The phase retrieval problem basically consists in recovering a complex-valued signal from the modulus of its Fourier transform. In other words, the phase of the signal in the frequency domain is lost. Recovery problems of this kind occur in electron microscopy, crystallography, astronomy, and communications. The long history of phase retrieval include countless approaches to find an analytic or a numerical solution, which is generally challenging due to the well-known ambiguousness.

In order to solve the phase retrieval problem nevertheless, we assume that the unknown continuous-time signal is sparse in the sense that the signal is a superposition of shifted Dirac delta functions or can be represented by a non-uniform spline of certain order. The main question is now: can we always recover the parameters of the unknown signal from the given Fourier intensity?

Using a constructive proof, we show that almost all sparse signals consisting of finitely many spikes at arbitrary locations can be uniquely recovered up to trivial ambiguities - up to rotations, time shifts, and conjugated reflections. An analogous result holds for spline functions of arbitrary order. The proof itself consists of two main steps. Exploiting that the autocorrelation function of the sparse signal is here always an exponential sum, we firstly apply Prony's method to recover the unknown parameters (coefficients and frequencies) of the autocorrelation. In a second step, we use this information to derive the unknown parameters of the true signal. Finally, we illustrate the proposed method at different numerical examples.