Compressive Sampling

Jan 25.2013Pei 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 L1-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:– φi=[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!!