###### tags: `BioI-maging`

# Radon Transform
Radon Transform for 3-D cases:
When we want to observe a 3-D object, we can think this object as a function $f: R^3 \to R$. The function f has unknown values at any given coordinate $(x, y , z) \in R^3$. To reconstruct the 3-D model of the object is to know the value of the function everywhere.
We can take the Radon transform of $f$ with respect to an angle $\theta$ to obtain a function: $\pmb p_\theta :A \to R$ is a function of integrals on a plane where $A \subset R^3$ such that
$\pmb p _\theta(t, \phi) = \int_{L(t, \theta, \phi)} f(x, y, z) ds$ ,
Here, $L$ is a function (using polar coordinate!) that determines a line in the $R^3$. In real practice, this line will be the x-ray beam. Each $\pmb p_\theta$ of a particular angle $\theta$ is represented by a 2-D image of the 3-D object taken at that angle. The set of all $\pmb p_\theta$ from different angle $\theta$ is the Radon transform of $f$. Each $\pmb p_\theta$, which represents a projection from a certain agnle, receives two input: $t$ and $\phi$. All the possible $t$ and $\phi$ determine a plane (this plane will be where detectors are located in the real practice). Plugging any point of that plane in $\pmb p_\theta$ will have a value, and that value is the line integral of $f$ on the line $L$.

Here is a demostration for the 2-D cases:

In 3-D case, function L has an additional parameter $\phi$, and the domain of $\pmb p_\theta$ will be a plane.

The purpose of the 3-D reconstruction is to obtain the function $f$ from the Radon transform of $f$ i.e. the set of all $\pmb p_\theta$'s obtained from different angles. This process is called the inverse Radon transform.

1. Is this way of understanding Radon Transform correct?

Inverse Radon Transform is inverting the linear projection map.

$\to \pmb {Further-question}$: What does it mean to "invert a linear projection map"? When you talk about a "linear projection map", do you mean $\pmb p_\theta$ in my notation?

2. Is back projection a way of implementing the Radon Transform?

Back projection is for the Inverse Radon transform
>The purpose of the 3-D reconstruction is to obtain the function $f$ from the Radon transform of $f$ i.e. the set of all $\pmb p_\theta$'s obtained from different angles. This process is called the inverse Radon transform.

$\to$ Maybe I will reread back projection.

* What is their relationship? In particular, this figure shows Radon transform and backprojection separately. However, I feel Radon transform is an abstract thing, and backprojection is implementing the Radon transform. (I am little confused by back projection)

Ill explain tomorrow.