# How would you explain what compressed sensing is

Tags:
1. Feb 24, 2015

### Niaboc67

Hello,
I would like to know your definition of what compressed sensing is. As well as its applications/uses. It's a subject I hear thrown around a bit but I have not received a complete answer on what the area/focus is? As well as the concept of sparsing (parsing?) and it's use with compressed sensing.

Thank you

2. Feb 24, 2015

### Staff: Mentor

3. Feb 25, 2015

### jfizzix

Compressive sensing is a way to record all the relevant information in a signal without having to exhaustively sample every component.

In particular, compressive sensing is a way of recording a signal where the number of measurements you need depends not on the dimensionality of your signal, but only on the sparsity (i.e., approximate number of nonzero components).

The idea is that each of these randomized measurements tells you a little bit about every component of your signal.

With enough of these randomized measurements, it turns out you can uniquely find your signal by solving a complex optimization problem. That you can do this was shown by Donoho, Candes, Romberg, and Tao in 2004 (some of them won the Fields Medal for this, I believe).

This optimization problem is "Out of all possible signals that would give me these measurement results, which is the sparsest?"
Depending on the sparsity of your original signal, your signal will be the unique solution to the optimization problem.

There is much more to this, of course, and I recommend the online lecture notes of Justin Romberg. If you want a nice example you can sink your teeth into, I would look at the single pixel camera (one of the first groundbreaking inventions due to compressive sensing).

4. Feb 25, 2015

### jfizzix

oh, and its applications are for sensing signals that are highly compressible. Besides variations on the single pixel camera, I believe there's been some work on developing it for high resolution MRI scans, so that the same information can be recorded more quickly (especially when a patient needs to be sedated to stay still).

Rice University has a good website on compressive sensing too (at least as far as a list of resources go)
http://dsp.rice.edu/cs [Broken]

The Howell Research Group at the University of Rochester (which I work in) also uses compressive sensing to perform high resolution measurements at very low light levels (using single photon counters), where regular techniques would just be impractical (because of the time it would take to record enough photons hitting tiny pixels)

Last edited by a moderator: May 7, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook