I have mentioned exotic sampling schemes before in photography and other areas. These exotic sampling techniques are generally devised to obtain newer sorts of information (hyperspectral, 3-d,....). One of these exotic means of collecting images is coded aperture. How are Compressive Sensing and Coded Aperture related in a simple way ? They differ in the reconstruction scheme: traditional coded aperture rely on a linear technique and no assumption of sparsity where CS uses a nonlinear technique and assume sparsity of the signal. But then, why has traditional coded aperture been successful in some niche areas ? Gerry Skinner, a specialist in coded aperture used for X-ray observatories in orbiting satellites, provides the beginning of an answer in Coded Mask Imagers: when to use them, and when not ? The following slide is extraordinarily expressive: (CM4 in this presentation)
In the context of Compressive Sensing, projecting a delta function onto a widespread function is natural whereas it is totally counter-intuitive in a normal setting. Yet it works! When reading about the "worst" Point Spread Function imaginable, one cannot but be reminded of George Constanza's "doing the opposite" in an unforgetable Seinfeld episode:
To use the wording used in Optics, when imaging stars, it looks as though designing a Compressive Sensing system is equivalent to designing a point spread function that is incoherent with the target of interest. The rest of the paper reads as if it were a CS paper except for the reconstruction aspect of it. In Compressed Sensing, the assumption is that we are looking at a sparse signal whereas in a normal setting, no such assumption is made The reason coded aperture has had continued success in Astronomy stems from the fact that the scene of interest is sparse in point-like sources (no extended objects) and these sources can be considered at infinity. In other words, the signal is sparse in the physical world already. It is embedded in the problem statement. Therefore, it seems likely that most algorithms developed for theses cases do in fact parallel those in compressed sensing (the one mentioned here seems to implement some sort of matching pursuit algorithm). For show, it interesting to see how rich the imaging systems have been:
Let us note also, that the arrows in the graph above show the movement of certain detectors. They are in effect using the movements of the sensor/spacecraft as a way to perform time multiplexing. When put in the context of compressed sensing, we should see this presentation as a 40-year jump start in our understanding of what to expect from these systems.
This lessons learned can be mapped to Compressive Sensing within the context of:
- low light flux
- far field approximation
- point like sources (diracs)
Obviously we need to find the same sets of rules for close field of view situations (nuclear medicine) and/or high fluxes and extended sources. In turn, we also need to understand how applying other type of mask properties (such as the Restricted Isometry Property) improve X-ray astronomy. We have already seen that Compressed Sensing is an enabling technology in traditional astronomy by providing a simple and efficient time coding of star field data (example of Herschel compression scheme using CS). Can CS provide a similar jump in improvement of Coded Aperture processing by a modified use of the mask or the time encoding?
Curiously, the reason coded aperture has not been more extensively used may have to do with the slow realization that most of our surrounding world is sparse, albeit not sparse in point-like features. Until we figured that out in the mid-1990's, we could only conceive coded aperture to be a viable option when we had not other way to go about it. This is pretty clear in the case of nuclear medicine, where particle/rays cannot be focused.
Finally, it is interesting to note that because star shining is a low flux problem and that spreading light is not efficient because you are drowning your compressed signal in the noise (recall that only the star-point-like-diracs is incoherent with the mask, not the background noise), recent development goes into avoiding Coded Aperture altogether by developing focusing lenses for gamma rays using Fresnel like lenses for gamma rays:
And so the question is: Could CS reconstruction techniques improve current coded aperture imaging to the point where this newer focusing technology is made "obsolete" by an improvement in the algorithm development of current coded masks ? What about raising the ability to provide data from current orbiting spacecrafts that have low ranking in NASA's latest review ? or provide better time modulating schemes [1] in nuclear medicine ?
Acknowledgment: Thanks to Ben Kowash (via Glenn Knoll) for pointing to some of the references above.
References:
[1] Digital Tomographic Imaging with Time-Modulated Pseudorandom Coded Aperture and Anger Camera by Kenneth F. Koral, W. Leslie Rogers, and Glenn F. Knoll, Journal of Nuclear Medicine, Volume 16, Number 5, 1974.
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