1. Aishwarya sreenivasan 15 December 2006.
Compressive sensing
Aishwarya sreenivasan
15 December 2006.

2. transform coding *linear process *no information is lost
*number of coefficients = number of pixels transformed
*transform coefficients and their locations
*imaging sensor is forced to acquire and process the entire signal.

3. Basic Idea
Compressive sampling shows us how data compression can be implicitly incorporated into the data acquisition process .

4. Concept
uses nonlinear recovery algorithms the ones based on convex optimization using which highly resolved signals and images can be reconstructed from what appears to be highly incomplete or insufficient data.
data compression –data acquisition

5. concept
Y M x 1 =A M x N x X N x 1
X is K sparse N x 1. Y is the projections.
K<<M<N
M x 1 = M x N N x 1

6. Reconstruction of signal
Lo norm-X (hat) = arg min ||Z|| Lo
z belongs to S
Lo norm is the correct way. But very slow.This is easier for binary images.
L2 norm X (hat) = arg min ||Z|| L2
X(hat)=(AAT)-1 ATy.
This is wrong though fast.

7. L1 norm-X (hat) = arg min ||Z||L1
z belongs to S
M =K log(N) under this constraint L1 norm is same as Lo norm
A sparse vector can be recovered exactly from a small number of Fourier domain observations. More precisely, let f be a length-N discrete signal which has B nonzero components samples at K different frequencies which are randomly selected. Then for K on the order of B log N, we can recover f perfectly (with very high probability) through l1 minimization

8. L1_eq-from L1 magic site.
Modifying the image as a vector and using L1_eq program from L1 magic Site.

12. Using block diagonal matrix
Image in the form of matrix is taken as a vector so use block diagonal matrix to span it instead of a normal random or Fourier or DCT matrix
A block diagonal matrix, also called a diagonal block matrix, is a square diagonal matrix in which the diagonal elements are square matrices of any size (possibly even ), and the off-diagonal elements are 0.
The square matrix here used is a n x n Fourier matrix where n is the length of the image.

13. Using block diagonal matrix
Y = AX
If x is the image in the form of a vector and is not sparse in A
Then let S = FX is sparse
Then Y = BS
Where B = AFinv
We can find S (hat) such that it is more close to X then X(hat)= ATY
Finv is a N2 x N2 block diagonal matrix if X is a N2 x 1 vector.

15. Total Variation ||g|| tv = ∑t1t2(|D1g(t1,t1)|2+|D2g(t1,t2)|2)1/2
||g|| tv- Total variance norm for a 2D object g.
D1g = g(t1,t2)-g(t1-1,t2)
D2g = g(t1,t2)-g(t1,t2-1)

16. Total Variation Images using tveq-example.m(L1 magic)

17. Applications
Imaging Camera Architecture using Optical-Domain Compression.
Imaging for Video Representation and Coding.
Application of Compressed Sensing for Rapid MR Imaging.

18. Thank you and questions