How we can determine the number of measurements required by a compressive sensing technique. Or what is the number of measurements required by a specific sampling matrix used to sample the signal.
I am also looking for some references that can answer my question,
Thanks,
Yes Professor, this is why I am looking for an answer.
Could you please suggest me a reference in which I can find an example.
Thanks,
The number of measurement depends on the sparsity of to be sampled signal and the sampling matrix that you are using. If your signal is compressible (or sparse in selected domains) then there is no exact required measurement. It is because sparsity (number of non-zero component in sparsify basis) is varied depends on your definition of sparsity.
With gaussian i.i.d matrix, many work assum number of measuremetn should be twice times larger than the signal sparsity.
@Thuong: thank you for yor response, i am using circulant and toeplitz sensing matrices for signal samplimg.
@Fatima then you should check paper on that structural sensing matrix. The authors often provide theoretical proof on how many measurements are required for reconstruction. The question that you ask often related to "adaptive sampling" topic which researcher try to capture proper number of measurement for corresponding signal. You should check it out.
FYI, the book titled "A mathematical introduction to compressive sensing" explained the structural sensing matrix that you are using in quite great detail. You can check the references therein. Beside, there are several more structural sensing matrix that you would check out (rather than circulant and toeplitz). In my field (image processing) those matrix are not perform the best.
http://www.springer.com/la/book/9780817649470
@Thuong:
thank you for sharing the book, i am working on signal processing where these matrices are more efficient for signal sampling. In image processing, I think random matrices perform better, am I right?
In image processing, we use random matrices due to its theoretical proof (it is easier to proof something with Gaussian i.i.d random matrix. However, fully random matrix is not practical. We prefer some sort of structural random matrix like the work of Thong T. Do [1] .
Regards to problem of adaptive sampling, you can check out some of our recent work on this such as: adaptive sampling based on relative sparsity (multi-scale/multi-resolution paper [3]), or based on k-term approximation (Visually weighted paper [2] and Compressive sensing of video. [4]).
Article Fast and Efficient Compressive Sensing Using Structurally Ra...
Article Visually Weighted Compressive Sensing: Measurement and Reconstruction
Conference Paper Multi-scale/multi-resolution Kronecker compressive imaging
Conference Paper Compressive sensing of video with weighted sensing and measu...
@Thuong: thank you so much, this is very intersting work, thank you for sharing t.he papers.
Has you are working on circulant matrices go through the following link below :
http://www.caam.rice.edu/~optimization/L1/RecPC/
this link provides complete details about different types of partial circulant matrices and also shown that some of the partial circulant matrices perform at par with gaussian measurement matrices.
There is one more paper that discusses about circulant matrices
Bajwa, Waheed U., et al. "Toeplitz-structured compressed sensing matrices." 2007 IEEE/SP 14th Workshop on Statistical Signal Processing. IEEE, 2007.
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