1 / 13

Compressive Sensing & Applications

Compressive Sensing & Applications. Alissa M. Stafford Mentor: Alex Cloninger Directed Reading Project May 3, 2013. What is Compressive Sensing?. Signal Processing: Acquiring measurements of a signal and using these measures to recover the signal Compressive Sensing

fathi
Download Presentation

Compressive Sensing & Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Compressive Sensing & Applications Alissa M. Stafford Mentor: Alex Cloninger Directed Reading Project May 3, 2013

  2. What is Compressive Sensing? Signal Processing: Acquiring measurements of a signal and using these measures to recover the signal Compressive Sensing Acquiring a limited number of measurements

  3. What is Compressive Sensing? Signal Processing: Acquiring measurements of the brain and using these measures to recover an image of the brain Compressive Sensing Acquiring a limited number of measurements of the brain

  4. What is the Difference? ~1/2 measurements ORIGINAL

  5. Well, Where’s the Math? Ax=b y=Φx brain measurements NxN matrix

  6. Well, Where’s the Math? y=Φx brain measurements MxN matrix

  7. Is Everything Compressible? Sparse  Compressible K-Sparse K non-zero coefficients • Assume the brain is sparse

  8. Any More Assumptions? y=Φx The measurements depend on Φ What kind of Φ is needed so the measurements are an accurate representation of x?

  9. What kind of Φ? Φ satisfies Restricted Isometry Property (RIP) For all x that are K sparse, • If δsmall, same logic implies no two completely different measurements will give same image

  10. How Many Measurements? Φ is MxN When Φ satisfies RIP of order 2K with δ<sqrt(2)-1, M≥CKlog(N/K)

  11. How is Image of the Brain Recovered? Φ is MxNnot invertible Finding x is an optimization where z is in β(y) • Finds the sparsest x that is consistent with y • But 0-norm is nonconvex difficult to solve • 1-norm is convex

  12. Take-Homes Compressive sensing is signal processing, only with a limited amount of measurements y=Φx, where Φ is MxN and satisfies RIP M≥CKlog(N/K) Use the 1-norm to find the sparsest x

  13. References Baraniuk, Richard, Mark Davenport, Marco Duarte, ChinmayHegde, Jason Laska, Mona Sheikh, and Wotao Yin. An Introduction to Compressive Sensing. Houston: Connexions, 2011. Print. Kendall, James. "2010S JEB1433 Medical Imaging." wikipedia. N.p., 3 May 2010. Web. 30 Apr. 2013. <wiki.math.toronto.edu/TorontoMathWiki/index.php/2010S_JEB1433_Medical_Imaging>.

More Related