Compressed Sensing What is it good for a

Compressed Sensing: What is it good for ? (a presentation on inimaginable wonders and how to make them work for you) Igor Carron Version 2 - draft

News #1: Everything is a linear system of equations o Most of your working life will be about solving linear systems of equations n n Get used to it ! If it’s nonlinear, then approximate it to make it a linear system of equations If it’s continuous, then discretize it to make it a linear system of equations you’re smart, you get my drift. If it’s blablablah, then …

Welcome to the world of Science Everything you know can be summarized as: y=Ax

Different Linear Systems of Equations o They come in different sizes: n n Overdetermined: Too many equations and too few unknowns (tall and thin matrix A) o Solution: least squares, remove equations There as many equations as there are unknowns (square matrix A) o n Sweet spot, you generally want to be there. Underdetermined: Too few equations and too many unknowns (fat and short matrix A) o This is what Compressive Sensing deals with

Warning (by the way don’t come and tell me square and overdetermined systems can be rank deficient. Yes they can be but if you have a problem like that, then you are doing something wrong)

Underdetermined System of Linear Equations n Too few equations and too many unknowns o -> Infinite number of solutions n n n n What The Hell ? How do we choose a solution among many ? Matrix A cannot be inverted as in the square case Solution x to y = A x is x_0 + x_1 with y = A x_0 and A x_1 = 0 Even if one finds x_0, which x_1 should we choose ? Too many choices, my head is going to explode Maybe we should look for a specific feature for x, what should that feature be ?

Within an infinite number of solutions some of these solutions are: o Cute (compressed sensing can’t help, yet)

Within an infinite number of solutions some of these solutions are: o Fractal (compressed sensing can’t help, yet)

Within an infinite number of solutions some of these solutions are: o Positive (compressed sensing can’t help, yet)

Within an infinite number of solutions some of these solutions are: o Bumpy (compressed sensing can’t help, not right away at least)

Within an infinite number of solutions some of these solutions are: o Blahblah….

Within an infinite number of solutions some of these solutions are: o Sparse (compressed sensing can help) n Compressed Sensing reconstruction techniques allows one to find a solution that is sparse i. e. has the property of having very few non-zeros elements (the rest of the elements are zeros).

Within an infinite number of solutions some of these solutions are: o Compressible (compressed sensing can help) n Compressed Sensing reconstruction techniques allows one to find a solution that is compressible.

Meaning of Compressed Sensing o o Compressed = Few Sensing = equations / linear combinations An instance of an underdetermined system of linear equation is a compressed sensing system. The recovery of a sparse solution to an underdetermined system of linear equations is performed using Compressed Sensing Reconstruction techniques /solvers.

Compressed Sensing o o Do all underdetermined systems of linear equations admit very sparse and unique solutions ? Ans 1: No Ans 2: Do you really care ? Ans 3: How do you find which one that do ? n Go to the next two slides

Do all underdetermined systems of linear equations admit very sparse and unique solutions ? o Some systems do, but they need to fulfill a condition (RIP, NSP, …. ) that cannot be checked (immediately).

Do all underdetermined systems of linear equations admit very sparse and unique solutions ? o Yes, if they recover sparse solutions following Donoho-Tanner phase diagram.

Summary o o Compressed sensing is about setting up underdetermined systems of linear equations. Compressed Sensing Reconstruction techniques are about finding the sparsest solution (out of an infinite number of solutions) of that system.

Summary (part II) o o The underdetermined systems that have sparsest solutions are the ones that have sparsest solutions. Yogi Berra would be proud!

Let’s try some Compressive Sensing o o o Solvers (mostly in Matlab): https: //sites. google. com/site/igorcarr on 2/cs#reconstruction Acceptable Matrix A: https: //sites. google. com/site/igorcarr on 2/cs#measurement Hardware/Sensors implementing A: https: //sites. google. com/site/igorcarr on 2/compressedsensinghardware

An example to wow your friends o http: //nuitblanche. blogspot. com/2011/11/howto-wow-your-friends-in-highplaces. html