Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS149, part 3: Stability of moment problems and super-resolution imaging
Time:
Thursday, 11/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)

 

Learning algebraic decompositions using Prony structures

Ulrich v. d. Ohe
University Genova

We propose a framework encompassing variants of Prony's reconstructiong method and clarifying their relations. This includes multivariate Hankel as well as Toeplitz variants for several classes of functions and relative versions.

 

Multidimensional Superresolution in Sonar and Radar Imaging

Annie Cuyt1, Wen-shin Lee2
1University Antwerpen, 2University of Stirling

Some sonar and radar imaging are essentially multidimensional exponential analysis techniques, consisting in identifying the linear coefficients αi and the distinct nonlinear parameters φi, i=1,...,n, in f(x)=Σi αi exp(<φi,x>) from samples f(xj) taken at regularly distributed points xj in the d-dimensional space. Exponential analysis is itself again connected to sparse interpolation from computer algebra, Padé approximation from rational approximation theory and tensor decomposition from numerical multilinear algebra.

We present a multidimensional generalization of a one-dimensional exponential analysis algorithm that: (1) requires the minimal number of (d + 1)n samples (through its connection with sparse interpolation), (2) validates the computed output for the φi (through its connection with Padé approximation), and (3) is robust against outliers. In addition, the samples may be collected at a rate below the classical Nyquist rate and the algorithm is easy to parallellize. We illustrate the working of the algorithm on some simulated examples taken from the engineering literature. The latter is joint work with Ferre Knaepkens and Yuan Hou from the Universiteit Antwerpen.

 

Recovery of surfaces and inference on surfaces: theory & applications to image recovery

Mathews Jacob, Qing Zou
University of Iowa

We introduce a sampling theoretic framework for the recovery of smooth surfaces and functions living on smooth surfaces from few samples. The proposed approach is as a nonlinear generalization of union of subspace models widely used in signal processing. This scheme relies on an exponential lifting of the original data points to feature space, where the features live on union of subspaces. The low-rank property of the features are used to recover the surfaces as well as to determine the number of measurements needed to recover the surface. The low-rank property of the features also provides an efficient approach which resembles a neural network for the local representation of multidimensional functions on the surface; the significantly reduced number of parameters make the computational structure attractive for learning inference from limited labeled training data.

 

Looking beyond Pixels: Continuous-domain Sparse Recovery with an Application to Radioastronomy

Martin Vetterli, Pan Hanjie
EPFL

We propose a continuous-domain sparse recovery technique by generalizing the finite rate of innovation (FRI) sampling framework to cases with non-uniform measurements. We achieve this by identifying a set of unknown uniform sinusoidal samples (which are related to the sparse signal parameters to be estimated) and the linear transformation that links the uniform samples of sinusoids to the measurements. It is shown that the continuous-domain sparsity constraint can be equivalently enforced with a discrete convolution equation of these sinusoidal samples. Then, the sparse signal is reconstructed by minimizing the fitting error between the given and the re-synthesized measurements (based on the estimated sparse signal parameters) subject to the sparsity constraint. Further, we develop a multi-dimensional sampling framework for Diracs in two or higher dimensions with linear sample complexity. This is a significant improvement over previous methods, which have a complexity that increases exponentially with space dimension. An efficient algorithm has been proposed to find a valid solution to the continuous-domain sparse recovery problem such that the reconstruction (i) satisfies the sparsity constraint; and (ii) fits the given measurements (up to the noise level). We validate the flexibility and robustness of the FRI-based continuous-domain sparse recovery in both simulations and experiments with real data in radioastronomy.