Wednesday, November 09, 2011

Why Stop There ?

There is a trend here. Out of many solutions to an underdetermined system of linear equations, we were initially looking at one aspect of a solution: For the past two hundred years, it used to be that the solution had to have minimum error. Then recently, it had to be an exactly sparse signal in compressive sensing or it had to be positive when performing NMF. As in any application, the issue of noise surfaces and one needs a second regularizer: the reconstruction error which leads to LASSO type of algorithms.

Why stop there ?

Recently I mentioned the possibility of maybe using two regularizers in the noiseless case (three in the noisy case) to find low complexity solutions. Toying around is one thing but are there any mathematical work that take a stab at two or more properties like positivity and sparsity ?

The only instance I know where several items are being enforced is the following: A variant on the compressed sensing of Emmanuel Candes by Basarab Matei and Yves Meyer where it is proven that using sampling set similar to the sets found in quasicrystral tilings provide positivity, compactness, and to a certain extent compressibility. 

As a side note I was reminded of this paper recently by going through the Structurally Random Matrices page of Thong Do where he actually used this deterministic subsampler to perform compressed sensing on Lena with results similar to a randomized subsampler approach.

I'd be curious what these figures would look like if instead of PSNR for the quality reconstruction, one were to use the Structural Similarity Index. The SSI being a physiological measure, this may bring some light some similarity between the quasicrystal tilings and the cones location in the fovea[2]

[1] Duncan,J.L., Zhang,Y., Gandhi,J., Nakanishi,C., Othman,M., Branham,K.H., Swaroop,A., Austin Roorda High resolution imaging of foveal cones in patients with inherited retinal degenerations using adaptive optics, Invest.Ophthalmol.Vis.Sci. 48: 3283-3291 (2007)

[2] Austin Roorda, A., Williams, D.R., The Arrangement of the Three Cone Classes in the Living Human Eye Nature 397, 520-522 (1999).

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