Download - Compressive Sampling
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
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