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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