Learning Compressed Sensing

Compressed Sensing is really a simple method for finding the sparsest solution to some underdetermined systems of linear equations. 
  • What sort of underdetermined systems are allowed? 
  • How can you find this sparsest solution ? 
Those are some of the questions that are being answered in some fashion or another in the following resources below (course note, videos) with varying degrees of difficulty. A second set of questions usually are then asked once some of these first issues are addressed:

  • Instead of the sparsest solution, can we find the most compressible solution ?
  • etc ...
Eventually, you might be interested in subscribing to the Nuit Blanche feed, There are also a Google+ Community, a CompressiveSensing subreddit, a LinkedIn Compressive Sensing group and a Matrix Factorization that you can join and post questions there. In this page, you will find some expository material aimed at various crowds, pick the one you feel most comfortable with:
  • Finally, here is a presentation that might provide some insight to engineers and other learned professionals but who may not be entirely familiar to what compressive sensing is. In particular, we try to avoid a unique reference to the L_1 norm and other deeper (and sometimes too narrow) mathematical statement while favoring the hardware/sensor issues:
  • A while back, I created a small video of a clown and a woman talking about compressed sensing, let me know if it helps better understand the subject:




  • Three videos presenting compressive sensing by Mark Davenport. It's short and to the point.

More in-depth explanation and teachings are provided below:

In terms of books, for less than $3.00 there is one the Kindle store: It can be read on the Kindle, iPad/iPod Touch and other tablets through the Kindle app:









Courses and Lecture Notes:
The following lectures were given at IAS in Princeton and provide a good introduction to CS and related issues:

You can also learn by doing, try any of these examples:

Webpages of  courses/classes given at different universities (undergraduate/graduate classes) can be found here and are listed below:

Emmanuel Candes was invited at the Centre for Mathematical Sciences in Cambridge, UK to give a series of lectures on compressed sensing. Here are the videos of these talks made at the LMS Invited Lecturer Series 2011:

Emmanuel Candes, Lecture 1: Some history and a glossy introduction










FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.21 GBViewDownload
Flash Video484x272   568.67 kbits/sec372.57 MBViewDownload
iPod Video480x270   506.21 kbits/sec331.65 MBViewDownload
MP344100 Hz125.0 kbits/sec81.70 MBListenDownload

Lecture 2: Probabilistic approach to compressed sensing


FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec992.84 MBViewDownload
Flash Video484x272   568.78 kbits/sec298.35 MBViewDownload
iPod Video480x270   506.27 kbits/sec265.56 MBViewDownload
MP344100 Hz125.02 kbits/sec65.38 MBListenDownload

Lecture 3: Deterministic approach to compressed sensing



MPEG-4 Video *640x360   1.84 Mbits/sec1.29 GB View Download
iPod Video480x270   506.19 kbits/sec353.32 MB View Download
MP344100 Hz125.0 kbits/sec87.05 MB Listen Download

Lecture 4: Incoherent sampling theorem



FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.11 GBViewDownload
Flash Video484x272   568.75 kbits/sec340.61 MBViewDownload
iPod Video480x270   506.25 kbits/sec303.18 MBViewDownload
MP344100 Hz125.02 kbits/sec74.67 MBListenDownload

 Lecture 5: Noisy compressed sensing/sparse regression




FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.18 GBViewDownload
Flash Video484x272   568.7 kbits/sec361.90 MBViewDownload
iPod Video480x270   506.2 kbits/sec322.13 MBViewDownload
MP344100 Hz125.01 kbits/sec79.35 MBListenDownload

Lecture 6: Matrix completion



FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.12 GBViewDownload
Flash Video484x272   568.75 kbits/sec345.26 MBViewDownload
iPod Video480x270   506.26 kbits/sec307.33 MBViewDownload
MP344100 Hz125.02 kbits/sec75.70 MBListenDownload

Lecture 7: Robust principal components analysis and some numerical optimization



FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.27 GBViewDownload
Flash Video484x272   568.66 kbits/sec391.37 MBViewDownload
iPod Video480x270   506.21 kbits/sec348.39 MBViewDownload
MP344100 Hz125.0 kbits/sec85.84 MBListenDownload

Lecture 8: Some Applications and Hardware Implementations



FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec1.11 GBViewDownload
Flash Video484x272   568.8 kbits/sec340.02 MBViewDownload
iPod Video480x270   506.2 kbits/sec302.66 MBViewDownload
MP344100 Hz125.0 kbits/sec74.54 MBListenDownload



Anders Hansen, Generalized sampling and infinite-dimensional compressed sensing



We will discuss a generalization of the Shannon Sampling Theorem that allows for reconstruction of signals in arbitrary bases. Not only can one reconstruct in arbitrary bases, but this can also be done in a completely stable way. When extra information is available, such as sparsity or compressibility of the signal in a particular bases, one may reduce the number of samples dramatically. This is done via Compressed Sensing techniques, however, the usual finite-dimensional framework is not sufficient. To overcome this obstacle I'll introduce the concept of Infinite-Dimensional Compressed Sensing.
FormatQualityBitrateSize
MPEG-4 Video *640x360   1.84 Mbits/sec829.14 MBViewDownload
Flash Video484x272   568.77 kbits/sec249.05 MBViewDownload
iPod Video480x270   506.27 kbits/sec221.68 MBViewDownload
MP344100 Hz125.02 kbits/sec54.55 MBListenDownload


Once you have graduated from any of these courses, you may want to take a peak at the Big Picture in Compressive Sensing that features some of the most recent measurement matrices and reconstruction solvers. You can also read the blog...
...or subscribe to the Nuit Blanche feed