Compressive Sampling Jan Pei Wu

Formalism The observation y is linearly related with signal x: y=Ax Generally we need to have the number of observation no less than the number of signal. But we can make less observation if we know some property of signal.

Sparsity A signal is called S-sparse if the cardinality of non-zero element is no more than S. In reality, most signal is sparse by selecting proper basis(Fourier basis, wavelet, etc)

Sparsity in image The difference with the original picture is hardly noticeable after removing most all the coefficients in the wavelet expansion but the 25,000 largest

Compressive Sampling We can have the number of observation much less than the number of signal

Reconstructing Signal can be recovered by minimizing L 1 - norm:

Example

Intuitive explanation: why L1 works(1)

Why L1 works(2) In this case, L1 failed to recover correct signal(point A) This would only happened iff |x|+|y|<|z|((x,y,z) is a tangent vector of the line)

Why L1-works(3) However this will happened in low probability with big m and S<<m<<n. We can have a dominating probability of having correct solution if:

What is φ is the orthonormal basis of signal ψ is the orthonormal basis of observation Definition:

What is The number shows how much these two orthonormal basis is related. Example: – φ i =[0,…,1,…0] – ψ is the Fourier basis: – =1 – This two orthonormal basis is highly unrelated We wish the is as small as possible

Thank You!!