I just came upon Michael Robinson's Research overview for the Fall 2011 and found this reference to Euler Calculus and compressive sensing in the context of radar sensing. An item you really want to look at is Michael's research page and his demo page on opportunistic imaging where he shows an unusual bend (for a mathematician) to get things working in hardware. Anyway, if you want to know more about Euler Calculus, you may want to watch this video entitled Euler Calculus and Topological Data Management from Robert Ghrist.Go ahead and watch it, I'll wait....Minesweeper gets my vote as an explanatory tool.
What are the use for this type of calculus ? For the time being, I need for it to sink in but if Euler characterisation is a generalization of counting, and since compressed sensing techniques show the connection between the l_1 norm and the l_0 semi-norm which itself is a counting function then those fields are bound to be on sort of collision course.
While the authors above use the sensor network as an example, I can think of other sensors that act only as counters and for which one would wants to know more about the field: SPECT comes to my mind for instance. In another direction, since the main pain of compressive sensing reconstruction solvers is to figure out the support of the elements being sought, then maybe this calculus can provide a way to craft better solvers. In yet another direction, maybe some of the calculus mentioned might somehow explain why in CT computations it sometimes does not really matter whether ray casting is performed by taking into account the attenuation of the material or not. Once some simple application are shown and explained in some detail, I am sure a whole community of engineer would embrace sheaves :-) Anyway, for further reading, you might want to check:
While the authors above use the sensor network as an example, I can think of other sensors that act only as counters and for which one would wants to know more about the field: SPECT comes to my mind for instance. In another direction, since the main pain of compressive sensing reconstruction solvers is to figure out the support of the elements being sought, then maybe this calculus can provide a way to craft better solvers. In yet another direction, maybe some of the calculus mentioned might somehow explain why in CT computations it sometimes does not really matter whether ray casting is performed by taking into account the attenuation of the material or not. Once some simple application are shown and explained in some detail, I am sure a whole community of engineer would embrace sheaves :-) Anyway, for further reading, you might want to check:
- Topological localization via signals of opportunity by Michael Robinson, Robert Ghrist.
- Euler Calculus with Applications to Signals and Sensing by Justin Curry, Robert Ghrist, and Michael Robinson
- Tomogaphy of Constructible Functions by Pierre Shapira
- Inversion of Euler integral transforms with applications to sensor data by Yuliy Baryshnikov, Robert Ghrist, David Lipsky.
- Applied Algebraic Topology and Sensor Networks, Robert Ghrist
- Euler-Bessel and Euler-Fourier Transforms, Robert Ghrist, Michael Robinson
- Target enumeration via integration over planar sensor networks by Yuliy Baryshnikov, Robert Ghrist
- Target enumeration via Euler characteristic integrals by Yuliy Baryshnikov, Robert Ghrist
- Euler integration for definable functions by Yuliy Baryshnikov, Robert Ghrist
- Barcodes: The persistent topology of data by Robert Ghrist
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2 comments:
I wrote some code based on Ghrist's target enumeration. http://blog.sigfpe.com/2010/01/target-enumeration-with-euler.html
Dan,
This is outstanding.
Cheers,
Igor.
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