Compressive Sensing
Aishwarya Sreenivasan
Department of Biomedical Engineering
Project proposal
ECE 533-Digital Image processing
Prof Yu Hen Hu
Background:-
There is an increasing pressure on image processing algorithms to support higher resolutions that requires denser sampling. At present the standard approach to image acquisition is to measure the complete copy of the image and then process it to extract the important information. A good example is that we measure an image on a pixel basis and the apply something like JPEG compression to represent it in a better way.
New method:-
The new method of compressive sensing instead collects a smaller number of carefully chosen linear measurements that condense the information in the signal. “Sparse signal statistics can be recovered from a small number of nonadaptive linear measurements”.
The basic principle
The signal/image X is K sparse in basis
Replace it with y = MX (fewer linear projections)
Y – L*1, M - L*J, X- J*1 matrices.
K<L<J
M random measurements are sufficient for a good recovery of the original signal or image.
The key here is projection onto incoherent basis where data is not compressible/sparse and recover using nonlinear processing.
The reconstruction can use the L0 norm, L1 norm or the L2 norm.L2 norm is fast is wrong.L0 exactly reconstruct the signal but is hard to implement and is slow. So we go in for the L1 norm,which is given by
X (hat) = arg min||X||1
Y= MX
Objective of the project:-
The main objective of this project is to implement an algorithm to perform compressed sensing using on of the basis like DCT or wavelets or DFT and demonstrate the underlying principle behind the technique. Also how the L1 norm is used here and the difficulty in using L2 norm is shown as part of the report.
References
1. Compressed sensing –David.L.Donoho, Department of Statistics Stanford University, 2004
2 2007 IEEE Statistical signal processing workshop.
3. Fast algorithm for compressed sensing-A.C.Gilbert, J.A.Tropp, M.J.Strauss and R.Vershynin.
4. Gonzalez and Woods, Digital Image Processing, Second Edition
5. J.A.Tropp and A.C.Gilbert signal recovery from random measurements via orthogonal matching pursuits- 2005