Radon transform, Buffon's needle and Integral geometry

Question

In all the literature that I have seen it is mentioned that these two are "branches" of integral geometry, but no where I can see the exact connection since one is connected with probability and the other is an integral.

Can somebody explain the connection in a clear way. Thanks

Comments

I have seen this, but it is not clear.

also, this material is related to this 

"Conventional Quantum Mechanics Without
Wave Function and Density Matrix"

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-1. This doesn't seem to be related to physics even remotely. Could be my ignorance though, I'd be happy to retract -1 if some one can point out a connection.

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@QSA, ah, ok, I see and +1. I would suggest you put the physics background in the main post.

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@Jia Yiyang, This is related to the relatively new exciting field(for me) of Quantum tomography.

Since it was my first post my browser was doing strange things, so I added the related material in the comments.

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@JiaYiang com'on let him ;-), PhysicsOverflow will never be flooded by mathematics, as the mathematicians have their own nice MathOverflow. And even if there are some questions that are on-topic on both sites (MathOverflow supports some theoretical physics too), so what? PhysicsOverflow should certainly not copy the anti-math attitude represented on some other physics Q&A sites, on the contrary we should adopt a much more reasonable point of view. 

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Probability and integration are the same thing since Kolmogorov. The connections between these things is explained on Wikipedia. The question is very broad, it is not clear what answer is expected.

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@Ron Maimon,Thanks for the reply. Yes, I am aware of the general idea. However, this situation is a bit more complicated. from the reference I gave,it says(referening to Buffon's needle as adjoint):

" As explained in [a3], Chapt. 5, the solution leads to the consideration of a measure  on the space of all lines in the plane and of  invariance under all rigid motions. This measure induces a functional  on the family of compact sets  by....

I need the clarification for that. Any reference is appreciated.

Answer 1

See slide 7 of this talk: http://www.math.utah.edu/~treiberg/IntGeomSlides.pdf you get an explanation of the rigid motions and the invariant measure. It was one of the top hits when I googled "integral geometry". The invariance just means that you need a probability density on lines which is unbiased. Perhaps after reading the background material the question can be made less general, at the moment I can't figure out how to answer it (although it did get me to read about this fascinating topic, so thanks)