Sunday, March 25, 2012

You explore the world with the sensor you have, not the sensor you wished you had.

Within the realm of Compressive Sensing, the answer to the following question seems obvious, initially at least, but then, it is not:
I have a problem with the reconstruction algorithm based on wavelet sampling. For example y=Ex, where E is partial wavelet transform but can select significant wavelet coefficients. Is there reconstruction algorithm for that or any papers to discuss this problem?
The canned answer stating that you have to have an incoherent basis for sampling is not optimal for several reasons:

  • Who says the problem is for a signal that is sparse in some wavelet basis ?
  • Who says that curvelets are not the real basis in which you want to decompose your signal, I mean you need many wavelets to get the equivalent of a curvelet.
  • Who says the undersampling needs to be optimal ? Who are you to think that the practitionner needs to have an optimal set-up ? there are many couples of bases for which incoherence is not optimal but then the requirement for the number of samples is not optimal either. You explore the world with the sensor you have, not the sensor you wished you had.
  • Who says that using any one of these solvers, the undersampling will not get better ? I mean most results are in O notation and most problems are finite.

Thanks Akshay and Zheng for this insightful discussion.

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