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Compressed Sensing and Statistical Physics
Compressed sensing is triggering a major evolution in signal
acquisition that changes completely the way we think about
experiments and measurements. It indicates that most data, signals and
images, that are usually compressible and have redundancy, can be
reconstructed from much fewer measurements than what was usually
considered necessary, resulting in a drastic gain of time, cost, and
measurement precision. The idea consists in sampling a sparse signal
using some random projections, and later using computational power
for its exact reconstruction, so that only the necessary information
is measured. This has been applied to many situations, from medical
imagery and one-pixel-camera to confocal microscopy, acoustic
holography or DNA micro-array analysis in biology.
In this talk, I will start by a general instruction to compressed
sensing for physicists and discuss the state of the art reconstruction
algorithms. Currently used reconstruction techniques are however
limited to acquisition rates still higher than the true density of the
signal. By using a mapping to a statistical physics problem, and
motivated by the theory of crystal nucleation, I will introduce a new
algorithm, and new measurement protocols, that achieves exact
reconstruction of the signal even at measurement rates very close to
the lowest possible ones. The gain in efficiency obtained is confirmed
by numerical studies of several cases.
References :
http://arxiv.org/abs/1109.4424
http://nuit-blanche.blogspot.com/2011/09/stunning-development-in-breaking-donoho.html
http://nuit-blanche.blogspot.com/p/teaching-compressed-sensing.html
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