EDIT (after comments and subject matter review)
CS is based on a choice of a sensing basis $\Phi$ relative to a representation basis $\Psi$. Using an "Incoherence Property" $\mu$ that measures the correlation between vectors from each basis it has been shown that the CS paradigm can achieve good reconstructions if the incoherence between $\Phi$ and $\Psi$ is low. Examples of low $\mu$ come from: The spike basis ($\Phi$) and the Fourier basis ($\Psi$), the noiselet basis ($\Phi$) and the wavelet basis ($\Psi$), a random matrix ($\Phi$) and a fixed basis ($\Psi$).
updated question is:
Is there a way to calculate the $\mu$ when the user is not able to chose $\Phi$ ahead of time to sample the signal, i.e. when the samples are already provided through some arbitrary sampling scheme that is sub-optimal vis-vis CS. My aim is to show/quantify, via $\mu$, that the arbitrary sampling scheme I have been given is sub-optimal if one is expecting CS to work.
With respect to Matrix Completion and Compressive Sampling (CS) I'm trying to understand how to calculate an incoherence property μ between two bases Φ and Ψ. Getting this incoherence is important because if Φ and Ψ are highly correlated there is little chance of succesfully reconstructing a signal from sparse samples. It is stated here (page 3) that μ is given by:
where n represents the number of elements in a matrix M - say an image signal.
I understand Φ to be a sensing basis and Ψ to be a sparse representation basis. I am using uniform random sampling to get a set of a samples from M. But I would also like to try arbitrary sampling patterns. (I'm not using wavelets or noiselets or Fourier coefficients - although I would like to try this eventually)
My question is:
How do I actually obtain Φ and Ψ?