A Mathematician Becomes a Pirate.

If you recall the islands of knowledge and the people that discover them, you might get a kick out of the following.

Last week, Emmanuel Candes was at Cambridge University giving math focused talks for the whole week. I'll feature all the videos in a latter post but the one that got my interest is this one entitled "Some Applications and Hardware Implementations". It is available here and in different formats (wow! way to go Cambridge)

           Format               Quality    Bitrate       Size

• MPEG-4 Video *640x360   1.84 Mbits/sec1.11 GB View Download
• Flash Video484x272   568.8 kbits/sec340.02 MB View Download
• iPod Video480x270   506.2 kbits/sec302.66 MB View Download
• MP344100 Hz125.0 kbits/sec74.54 MB Listen Download

The host points to the fact that the upcoming talk is not the usual mathematical fare which brings a giggle to the crowd. Anyway after the math interlude and 30 minutes into the talk, Emmanuel mentions hardware calibration issues that take a very long time to iron out. It echoes some of the concern I mentioned earlier on Nuit Blanche and is the reason you are all entertained in this exquisite series on the Multiplicative Noise Effect on the Donoho-Tanner phase transition.

I note that the current dearth of papers on actual implementation of a high frequency A/D converters (besides the Modulated Wideband Converter of Yonina Eldar and Moshiko Mishali) seems to come from some elements in the set-up that produces noise. In the presentation, Emmanuel points to several culprits including thermal noise. However, the most important annoyance seems to be the amplification stage that is right before the ramdomly modulated pipeline. It is a nonlinear process that creates harmonics. That process thereby makes the initial signal non sparse. He also mentions the fact that the clock jitter can be modeled as additive noise and that he is getting close the Heisenberg limit. wow.

I have some probably non-useful ideas in this respect. First, while the signal is somehow getting harmonics, can we not get some sort of structured sparsity element in the reconstruction that would make this amplified signal sparse ? What about using this amplification process as a way to 'spread' the signal ? With regards to the clock jitter and the Heisenberg limit, it cannot be as bad as what Rick Trebino goes through when he has to determine the time width of the shortest time scale possible ( other entries on the subject include this one and this Q&A with Rick ). Also what about using a nonlinear compressive sensing solver (Sparsify by Thomas Blumensath) ?

Then Emmanuel goes on to present a confocal microscopy project where random patterns are used to avoid the raster scanning. It so happens that the example of the structured random pattern projected on the biological sample  is none other than a Simpson movie featuring Homer Simpson.

Uh, a Homer Simpson movie, that's not some well known structured Hadamard transform...that's what a pirate would do. Go watch the video.