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This event will be
streamed live at www.fields.utoronto.ca/live
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 www.fields.utoronto.ca/live 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.
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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.
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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.
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Distinguished
Lecture Series