Conference Agenda

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Session Overview
Session
MS149, part 1: Stability of moment problems and super-resolution imaging
Time:
Tuesday, 09/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)

 

Introductory talk: stability of moment problems and super-resolution imaging

Dmitry Batenkov
MIT Boston

I shall provide a brief overview of the minisymposium topics and the contributions.

 

Non-ideal Super-resolution and Variations on a Theme

Ayush Bhandari
Imperial College London

Super-resolution is a well studied topic and deals with recovery of spikes from low-pass projections in the Fourier domain. This is a common problem of interest that finds applications across several areas of science and engineering. When it comes to practice, however, this model may not be applicable as it is. Based on several motivating examples from experimental setups, in this talk, we cover the topic of non-ideal super-resolution. To this end, we discuss some new variations on the theme. This includes the question of (a) “essential bandwidth” selection or super-resolution with optimal number of trigonometric moments under noise, (b) super-resolution with time-varying pulses, (c) super-resolution with the unlimited sensing framework, and (4) a general theory of super-resolution that goes beyond the Fourier domain.

 

Clustered Super-Resolution

Gil Goldman
Weizmann Institute

Consider the problem of continuous super resolution, which is taken here as the reconstruction of a spike train signal (linear combination of shifted delta-functions), from noisy Fourier measurements limited to the band [−Ω,Ω]. We discuss some geometrical aspects of this problem, and the related problem of stability of Vandermonde matrices, in the case where the nodes form several clusters. For a single cluster of size h, we analyse the structure of the inverse image of a cube of size epsilon in the measurement space, which we call the epsilon-error set. It is shown that the inverse image has very different scaling along certain directions that depends mainly on the size of the super resolution factor SRF = 1/Ωh, and the noise level epsilon. This description is then extended to several clusters. We describe the effects of decimation (reducing of the sampling rate) on the geometry of the solution set. Specifically we examine aliasing and stability of such solutions. Joint work with: Yosef Yomdin and Dima Batenkov.

 

Geometry of Error Amplification in Spike-train Fourier Reconstruction

Yosef Yomdin
Weizmann Institute

We consider Fourier Reconstruction of spike-train signals (i.e. of linear combinations of delta-functions). In an important case when some of the nodes nearly collide, while the measurements are noisy, a dramatic error amplification may occur in the process of reconstruction. At least in part, this error amplification reflects the geometric nature of the problem itself, and does not depend on the choice of the solution method.

Our approach is based on the following observation: If the nodes near-collide, then the possible error-affected reconstructions are not distributed uniformly, but rather tightly follow certain algebraic-geometric patterns, known a priori (“Prony varieties").

We believe that understanding the geometry and singularities of the Prony varieties can improve our understanding of the geometry of the error amplification. In turn, this can be used in order to improve the overall reconstruction accuracy.

We plan to present some our results in case where all the nodes form a single cluster. Here the Prony varieties are defined by the initial equations of the classical Prony system. This fact strongly simplifies their algebraic-geometric study.

Next we plan to extend our results to the case of several “well-separated” node clusters. Here the one-cluster case serves as a model for the multi-cluster situation. Basically, here the relevant geometric-algebraic objects are the Cartesian products of the “local (at each cluster) Prony varieties”.