Compressed Sensing for the multiplexing of PET detectors

Following up on yesterday's paper and one of the author that had already shown up my radar screen earlier this summer,  Craig Levin forwarded me their use of compressive sensing that quite simply could replace Anger Logic. This is big news.

Here is the paper: Compressed Sensing for the multiplexing of PET detectors by Peter Olcott , Garry Chinn and Craig Levin . The abstract reads:

Compressed sensing can be used to multiplex a large number of individual readout sensors to significantly reduce the number of readout channels in a large area PET block detector. The compressed sensing framework can be used to treat PET data acquisition as a sparse readout problem and achieve sub-Nyquist rate sampling, where the Nyquist rate is determined by the pixel pitch of the individual SiPM sensors. The sensing matrix is fabricated by using discrete elements or wires that uniquely connect pixels to readout channels. By analyzing the recorded magnitude on several ADC channels, the original pixel values can be recovered even though they have been scrambled through a sensing matrix. In a PET block detector design comprising 128 SiPM pixels arranged in a 16 x 8 array, compressed sensing can provide higher multiplexing ratios (128:16) than Anger logic (128:32) or Cross-strip readout (128:24) schemes while resolving multiple simultaneous hits. Unlike Anger and cross-strip multiplexing, compressed sensing can recover the positions and magnitudes of simultaneous, multiple pixel hits. Decoding multiple pixel hits can be used to improve the positioning of events in light-sharing designs, inter-crystal scatter events, or events that pile up in the detector. A Monte-carlo simulation of a compressed sensing multiplexed circuit design for a 16 x 8 array of SiPM pixels was done. Noise sources from the SiPM pixel (dark counts) and from the readout channel (thermal noise) were included in the simulation. Also, two different crystal designs were simulated, a 1x1 coupled design with 128 scintillation crystals, and a 3:2 light-sharing design with 196 crystals. With an input SNR of 37dB (experimentally measured from a single SiPM pixel), all crystals were clearly decoded by the compressed sensing multiplexing with a decoded SNR of the sum signal a 30:6 0:1 dB SNR for the one-to-one coupling, and 26:1 0:1 dB three-to-two coupling. For a 10% energy resolution, and SNR of greater than 20 dB is needed to accurately recover the energy.

This is a very nice paper. Let us note a few things. Their measurement matrix is very sparse and different from the ones we have heard about so far. I note also that in the extreme noiseless case, using the Donoho-Tanner phase transition, a 128x16 measurement matrix like the one implemented here , we therefore have N = 128, m = 16 or delta = 0.125 and using Jared Tanner's app, we can estimate the optimal number of simultaneous detectable events to be about 0.26*16 (Weak Simplex) or about 4. In the noisy case, we should expect a number lower than four. 

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Through the Non Looking-Glass, and What Alice Found There

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

[3] CS: Compressive Coded Apertures for High-Resolution Imaging

[4] CS: A Short Discussion with Gerry Skinner, a Specialist in Coded Aperture Imaging.

[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

[8] CS: What would a Compressive Neutron Detector Look Like ?

[9] Random Lens Imaging

Lensfree optical tomography

View more presentations from sisikman

CS: These Technologies Do Not Exist: Time Modulated Photo-Multiplier Tubes

As an addendum to yesterday's entry on Compressive MultiChannel Analyzers, one couldd also think of a way to do some type of time multiplexing of the Photo-Multiplier Tubes (PMT). The multiplexing could be performed by oscillating resistances between the dynodes or by modulating the power supply of the PMT..

Why would you want to have a technology like this one ? At the very least it could provide some rapid detection of abnormal events from a steady state background. This modulation could also be undertaken in concert with a modulation of the Random Anger Coding Scheme or the Random X-ray Collector mentioned earlier.

This entry will be added to the "These Technologies Do Not Exist" page.