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Compressive Sampling

Compressive Sampling. Jan 25.2013 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.

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Compressive Sampling

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  1. Compressive Sampling Jan 25.2013 Pei Wu

  2. 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.

  3. 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)

  4. 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

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

  6. Reconstructing • Signal can be recovered by minimizing L1-norm:

  7. Example

  8. Intuitive explanation: why L1 works(1)

  9. 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)

  10. 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:

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

  12. 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

  13. Thank You!!

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