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Friday, February 03, 2012

Calibrating a Compressive Sensing System

As I was reading this presentation from Andrew Waters, I was reminded of the calibration issue I keep on mentioning every once in a while. Dynamic range is a known issue in compressive sensing because every measurement is the linear combination of many several "normal" measurements. In other words, compressive measurements are larger while requiring a similar resolution as "normal" measurements. All the quantization work generated around compressive sensing stems from needing to understand in the interplay between quantization and dynamic range issues. As it turns out, the example used here is the single pixel camera (see how it works here) which is used as a spectral imager. The concern is that the pixel may not be deep enough or have enough resolution in its A/D system.

What's important is really to note that saturation is in fact an issue and one wonders how one can recover from it or be able to live with it.

and it looks like in order to not pollute your system, one rather want to reject saturated measurements.

As Mark Davenport mentioned in Google + thread, the blue curve is the recovery if one uses the conventional approach a rescale the signals so that they fit into the depth of the dynamic range (at the expense of the quantization of the sample) whereas the red one is the one shows  a system where saturated measurements are discarded so that the rest of the sample that fit can have better discretization. In the attendant paper [1], another method enforces consistency:whereby the saturated measurements are included in the optimization steps and reconstructed with SC-CoSAMP, From [1]

1) the conventional approach, scaling the signal so that the saturation rate is zero and reconstructing with the program (3);
2) the rejection approach, discarding saturated measurements before reconstruction with (4); and
3) the consistent approach, incorporating saturated measurements as a constraint in the program (6).

 As one can see, the consistent approach does not help much besides increasing the complexity of the solver. However, I wonder if something could not be done using the fact that most calibrating images could come from a low dimensional manifold or use the recent matrix completion or low rank solvers in some way. There is more information on this Google+ thread.

[1] Democracy in Action: Quantization, Saturation, and Compressive Sensing, Jason N. Laska, Petros T. Boufounos, Mark A. Davenport, and Richard G. Baraniuk

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