Radon transform, Buffon's needle and Integral geometry

[ftr]

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.

I have seen this, but it is not clear.

http://www.encyclopediaofmath.org/index.php/X-ray_transform


Can somebody explain the connection in a clear way. Thanks

 

[Stephen Tashi]

While we're waiting for someone who really knows the answer, I'll make some comments.

From the abstract point of vew, a probability distribution is a "measure" on a space of things. When you do integration from the abstract point of view, you integrate functions on a space of things with respect of a "measure" on the space. Much of probability theory (such as finding the expected value of a random variable, the variance of a random variable etc.) involves doing integrals. If $p(x)$ is a probability density function on the real line, you can regard the expected value of $E(f(x)) = \int f(x) p(x) dx$ as the integral of the product $f(x)p(x)$ with respect to the ordinary way of measuring length on the real line (denoted by $dx$) or you can regard it as an integral of $f(x)$ with respect to another way of assigning a "measure" to an interval on the real line given by $p(x) dx$.

The high class way to think about $E(f(x))$ is to think about it as an integral of $f(x)$ with respect to the measure $p(x) dx$ because this view generalizes to cases where integration with respect to the ordinary notion of length doesn't work. For example, suppose $X$ is a random variable realized as follows. Flip a fair coin. If the coin lands heads then $X = 1/3$. If the coin lands tails then pick the value of $X$ from a uniform distribution on $[0,1]$. To find the expected value of $X$ you can't do a simple Riemann integral since it would assign zero length to the point $1/3$ and the correct calculation of the expected value of $X$ somehow has to justify adding the term $(1/2)(1/3)$ to the result. If you think about a kind of measure on the real line where the point $1/3$ has measure $1/2$ then you can justify doing that.

So there is an intimate connection between integration and measures. A probability disribution defines a special kind of measure.

If a transform is defined conceptually as an integration "over all possible lines" that satisfy a certain condition, you may be able to parameterize such a line by an n-tuple of real numbers and do an n-variable Riemann integral in the ordinary way, thinking of the measure as the ordinary measure of n-dimensional volume. But if parameterizing the integral of $f(x,y,z...)$ that way introduces other functions as factors in the integral, the high class way of thinking about it may be to think of those factors as defining a new sort of measure on the space of lines. Someone who really knows integral geometry will have to comment on whether that's the way to look at it.

 

[ftr]

Thanks for the reply. I am still searching for answers, it is getting a bit complicated.

 

[ftr]

I have found an answer for it in here. The connection is in the second solution.

 

[blue_raver22]

I can provide a response to your inquiry about the connection between Radon transform, Buffon's needle, and integral geometry.

Firstly, it is important to understand what integral geometry is. Integral geometry is a branch of mathematics that deals with the integration of geometric quantities over a space or manifold. It is a powerful tool for solving problems in various fields such as physics, engineering, and computer science.

Now, let's look at the Radon transform and Buffon's needle. The Radon transform is a mathematical operation that maps a function or distribution in one space onto a function or distribution in another space. It is commonly used in medical imaging to reconstruct images from X-ray projections. On the other hand, Buffon's needle is a mathematical problem that involves randomly dropping a needle onto a lined surface and calculating the probability of the needle crossing a line.

So, how are these two seemingly different concepts related to integral geometry? The key connection lies in the concept of integration. Both the Radon transform and Buffon's needle involve integrating a function or probability over a space. In the case of the Radon transform, it involves integrating a function over a space to reconstruct an image. In Buffon's needle, it involves integrating the probability of the needle crossing a line over a space.

Moreover, integral geometry provides the mathematical framework for understanding and solving problems related to the Radon transform and Buffon's needle. It allows for the manipulation and analysis of the integrals involved in these problems, leading to solutions and insights. In fact, the Radon transform is often referred to as the X-ray transform in integral geometry literature.

In conclusion, the connection between Radon transform, Buffon's needle, and integral geometry lies in the concept of integration. Integral geometry provides the mathematical tools and framework for understanding and solving problems related to these concepts. I hope this explanation helps clarify the connection between these branches of integral geometry.