Compressed Sensing

Ivan Cheung

Illustration: Gabriel Peyre

Overview

What is compressed sensing? How it works Applications

What is compressed sensing?

A paradigm shift that allows for the saving of time and space during the process of signal acquisition, while still allowing near perfect signal recovery when the signal is needed

Image: Windows Vista

Sparsity

The concept that most signals in our natural world are sparse

Diagram: Emmanuel Candès, Michael Wakin

How It Works

1) UndersampleA camera or other device captures only a small, randomly chosen fraction

of the pixels that normally comprise a particular image. This saves time and space.

Photos: Corbis; Image Simulation: Jarvis Haupt/Robert Nowak

How It Works

2) Fill in the dotsAn algorithm called l1 minimization starts by arbitrarily picking one of the

effectively infinite number of ways to fill in all the missing pixels.

Photos: Corbis; Image Simulation: Jarvis Haupt/Robert Nowak

How It Works

3) Add shapesThe algorithm then begins to modify the picture in stages by laying colored shapes over the randomly selected image. The goal is to seek sparcity, a

measure of image simplicity.

Photos: Corbis; Image Simulation: Jarvis Haupt/Robert Nowak

How It Works

4) Add smaller shapesThe algorithm inserts the smallest number of shapes, of the simplest kind, that match the original pixels. If it sees four adjacent green pixels, it may

add a green rectangle there.

Photos: Corbis; Image Simulation: Jarvis Haupt/Robert Nowak

How It Works

5) Achieve clarityIteration after iteration, the algorithm adds smaller and smaller shapes, always seeking sparsity. Eventually it creates an image that will almost

certainly be a near-perfect facsimile of a hi-res one.

Photos: Corbis; Image Simulation: Jarvis Haupt/Robert Nowak

Applications

Medical– Fast MRI Imaging

Artificial Intelligence– Baysian classifier

Facial recognition OCR

Applications

Distributed Networks– Data storage– Error correction codes– Sensor networks– Signal detection

Sensor Networks

A typical wireless sensor network contains a large number of wireless sensor nodes, which gather information and deliver it to a distant destination, termed a fusion center, where information is filtered and aggregated.

Diagram: Jarvis Haupt, Robert Nowak

Sensor Networks

processing and communication are combined into one distributed projection operation

it virtually eliminates the need for the sensor network to process and communicate

consistent signal estimation is possible with power and latency requirements growing sub-linearly in relation to the number of sensor nodes

Sources

Candès , Emmanuel J. and Michael B. Wakin. "An Introduction to Compressive Sampling." IEEE Signal Processing Magazine (March 2008): 21-30.

Candès , Emmanuel J. and Paige A. Randall. "Highly Robust Error Correction by Convex Programming." IEEE Signal Processing Magazine (July 2008): 2829-2840.

Ellenberg, Jordan. "Fill in the Blanks." Wired, March 2010, 62-67. Haupt, Jarvis, Waheed U. Bajwa, Michael Rabbat, and Robert Nowak. "Joint

Source-Channel Communication for Distributed Estimation in Sensor Networks." IEEE Transactions on Information Theory (October 2007).

Haupt, Jarvis, Waheed U. Bajwa, Michael Rabbat, and Robert Nowak. "Compressed Sensing for Networked Data." IEEE Signal Processing Magazine (March 2008): 92-101.

Rudelson, Mark, Roman Vershynin. "Geometric Approach to Error-Correcting Codes and Reconstruction of Signals." International Mathematics Research Notices (2005).

"Wikipedia." Wikimedia Foundation. http://en.wikipedia.org/wiki/Compressed_sensing (accessed April 9, 2010).

Thank You!

Questions?