Introduction to Compressive Sensing Aswin Sankaranarayanan system Is this system linear ?
system Is this system linear ? Given y, can we recovery x ? Under-determined problems measurements
signal measurement matrix If M < N, then the system is information lossy Image credit Graeme Pope
Image credit Sarah Bradford Super-resolution Can we increase the resolution of this image ? (Link: Depixelizing pixel art)
Under-determined problems measurements signal measurement matrix Fewer knowns than unknowns!
Under-determined problems measurements signal measurement matrix Fewer knowns than unknowns! An infinite number of solutions to such problems
Credit: Rob Fergus and Antonio Torralba Credit: Rob Fergus and Antonio Torralba Ames Room Is there anything we can do about this ?
Complete the sentences I cnt blv I m bl t rd ths sntnc. Wntr s cmng, n .. Wntr s hr Hy, I m slvng n ndr-dtrmnd lnr systm. how: ?
Complete the matrix how: ? Complete the image Model ?
Dictionary of visual words I cnt blv I m bl t rd ths sntnc. Shrlck s th vc f th drgn Hy, I m slvng n ndr-dtrmnd lnr systm. Dictionary of visual words
Image credit Graeme Pope Image credit Graeme Pope Result Studer, Baraniuk, ACHA 2012
Compressive Sensing measurements signal measurement matrix A toolset to solve under-determined systems by exploiting additional structure/models on
the signal we are trying to recover. modern sensors are linear systems!!! Sampling sampling
Can we recover the analog signal from its discrete time samples ? Nyquist Theorem An analog signal can be reconstructed perfectly from discrete samples provided you sample it densely.
The Nyquist Recipe sample faster sample denser the more you sample, the more detail is preserved The Nyquist Recipe
sample faster sample denser the more you sample, the more detail is preserved But what happens if you do not follow the Nyquist recipe ?
Credit: Rob Fergus and Antonio Torralba Image credit: Boston.com The Nyquist Recipe sample faster sample denser
the more you sample, the more detail is preserved But what happens if you do not follow the Nyquist recipe ? Breaking resolution barriers
Observing a 40 fps spinning tool with a 25 fps camera Normal Video: 25fps Compressively obtained video: 25fps Recovered Video:
2000fps Slide/Image credit: Reddy et al. 2011 Compressive Sensing Use of motion flow-models in the context of compressive video recovery 128x128 images sensed at 61x comp.
Nave frame-to-frame recovery single pixel camera CS-MUVI at 61x compression Sankaranarayanan et al. ICCP 2012, SIAM J. Imaging Sciences, 2015*
Compressive Imaging Architectures Scalable imaging architectures that deliver videos at mega-pixel resolutions in infrared visible image Chen et al. CVPR 2015, Wang et al. ICCP 2015
SWIR image A mega-pixel image obtained from a 64x64 pixel array sensor Advances in Compressive Imaging
Linear Inverse Problems Many classic problems in computer can be posed as linear inverse problems Notation Signal of interest measurement matrix
Observations measurement noise Measurement model
Problem definition: given , recover Linear Inverse Problems measurements
signal Measurement matrix has a (N-M) dimensional null-space Solution is no longer unique Sparse Signals measurements
sparse signal nonzero entries How Can It Work?
Matrix not full rank columns and so loses information in general
But we are only interested in sparse vectors Restricted Isometry Property (RIP) Preserve the structure of sparse/compressible signals K-dim subspaces
Restricted Isometry Property (RIP) RIP of order 2K implies: for all K-sparse x1 and x2 K-dim subspaces How Can It Work? Matrix
not full rank columns and so loses information in general Design so that each of its Mx2K
submatrices are full rank (RIP) How Can It Work? Matrix not full rank columns
and so loses information in general Design so that each of its Mx2K submatrices are full rank (RIP)
CS Signal Recovery Random projection not full rank Recovery problem: given find Null space Search in null space
for the sparsest (N-M)-dim hyperplane at random angle Signal Recovery Recovery: given
(ill-posed inverse problem) find (sparse) Optimization:
Convexify the optimization Candes Romberg Tao
Donoho Signal Recovery Recovery: given (ill-posed inverse problem)
find Optimization: Convexify the optimization
Polynomial time alg (linear programming) (sparse) Compressive Sensing Let.
If satisfies RIP with , Then
Best K-sparse approximation