The story goes that Lewis Caroll gave a copy of "Alice in Wonderland" to Queen Victoria who then asked him in return to send her his next book as she fancied the first one. The joke is that the book was "An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear Equations and Algebraic Equations". Sir Lewis Caroll then went on to write a follow-on to Alice in Wonderland called "Through the Looking-Glass, and What Alice Found There" that features a chess board.
And then it all aligned in one fell swoop a little like Alice falling in a different magic world of wonders. That chess board reminded me of the grid used in the compressive coded aperture work mentioned yesterday in Rebecca Willett's work[2][3].
The simultaneous linear equations reminded me of the obvious connection to compressive sensing and underdetermined linear systems of equations. The determinant functional of the second book reminded of the current solution technique using the rank proxy log(det) [1] to recover image from a quadratic compressive sensing approach and finally, the looking glass is nothing short of a lens. What is linking of these elements ? Quite simply: Lens Free imaging [5][6][7].
And so what did Alice found through the Non Looking Glass you ask ? Initially there are some successes but eventually we have to deal with the pesky details, actually those were details when using lenses, they are now your main issues and the way you solve for them is what makes you famous. These pesky details are ?
- Since we are getting non human readable image, we need to perform calibration to figure out if what we see is what we want. The issue is that we have to form a combinatorial amount of trials to since now the transfer function is not symmetric or has no known features [9].
- Noise gets in the way as found by the earlier folks who have done coded aperture for the past fifty years [4] since now the PSF is widespread. You lose resolution with a lens system but this is because you are decreasing noise.
Let us also note that these issues are similar to the ones we face when looking at other radiation than light[8].
[1]Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing by Yoav Shechtman, Yonina C. Eldar, Alexander Szameit, Mordechai Segev
[2] Rebecca Willett's 5/20;lecture 4 Sparsity: Generalized Sparsity Measures and Applications
[5] Holes Better than Microlenses on Vladimir Koifman's blog
[6] Lens-free optical tomographic microscope with a large imaging volume on a chip by Serhan O. Isikmana, Waheb Bisharaa, Sam Mavandadia, Frank W. Yua, Steve Fenga, Randy Laua, and Aydogan Ozcan
[7] Lensfree on-chip holography facilitates novel microscopy applications by Aydogan Ozcan, Serhan Isikman, Onur Mudanyali, Derek Tseng and Ikbal Sencan
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