September 30, 2023
January-August 2012
Thematic Program on Inverse Problems and Imaging

Distinguished Lecture Series
May 7-9, 2012

Emmanuel Candes
Professor of Mathematics and of Statistics
Professor of Electrical Engineering (by courtesy)
Stanford University, Department of Mathematics


This event will be streamed live at using our new FieldsLive streaming system. Viewers with a webcam and access code can participate and ask questions remotely; access codes must be requested 24 hours in advance. See for details.

May 7, 2012 --3:30 p.m.
The Fields Institute, 222 College St, Room 230 (note revised location)

From compressive sensing to super-resolution

Compressive sensing is a novel theory which asserts that one can recover signals or images of interest with far fewer measurements or data bits than were thought necessary. The first part of the talk will introduce some of the theory and survey important applications which allow -- among other things -- faster and cheaper imaging. For instance, compressive sensing asserts that under sparsity constraints, one can recover or interpolate the whole spectrum of an object exactly from just a few randomly spaced samples by solving a simple convex program. In many applications, however, we cannot sample the spectrum at random locations; rather, one can only observe low-frequencies as there usually is a physical limit on the highest possible resolution. Is it then possible to extrapolate the spectrum and recover the high-frequency band? The second part of the talk will introduce recent results towards a mathematical theory of super-resolution -- a word used in different contexts mainly to design techniques for enhancing the resolution of a sensing system.

May 8, 2012 --3:30 p.m.
The Fields Institute, 222 College St, Room 230

Robust principal component analysis? Some theory and some applications

This talk is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. In the second part of the talk, we present applications in computer vision. In video surveillance, for example, our methodology allows for the detection of objects in a cluttered background. We show how the methodology can be adapted to simultaneously align a batch of images and correct serious defects/corruptions in each image, opening new perspectives.

May 9, 2012 --2:00 p.m.
The Fields Institute, 222 College St, Room 230

PhaseLift: Exact Phase Retrieval via Convex Programming

This talks introduces a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we present some novel theory showing that our entire approach may be provably surprisingly effective.

Speakers in the Distinguished Lecture Series (DLS) have made outstanding contributions to their field of mathematics. The DLS consists of a series of three one-hour lectures.
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