📂Tomography

What is a Phantom in Tomography?

Definition

A hypothetical image used for the numerical simulation of tomography is called a phantom.

Description

The problem dealt with in tomography is, given a function $f$ and an operator $A$, finding $f$ when $Af$ is given.

For example, in the case of CT imaging, $Af$ is the Radon transform $\mathcal{R}f$, which refers to the data obtained by a CT scanner passing radiation through our body. Specifically, the brain CT data is as follows in the picture.

Suppose we derived some inverse transform formula or developed an algorithm and applied it to the above data, and obtained the following result.

However, we cannot know whether the inverse transform formula or algorithm worked well. This is because we do not have the correct answer. It’s a formula created to see inside the body that we cannot see, but to check if it actually calculates properly, we need to see inside the body.

Hence, when creating data $\mathcal{R}f$ for numerical simulation, it must be made from a $f$ that we already know accurately (one that we have artificially created). This is so we can check how close $\mathcal{R}^{-1}\mathcal{R}f$ and $f$ are to each other. This $f$ is called a phantom. A commonly used phantom is the Shepp-Logan phantom¹ proposed by Shepp and Logan, which is an image that describes features appearing in a cross-section of the brain, as shown below.

If we calculate $\mathcal{R}f$ and $\mathcal{R}^{-1} \mathcal{R} f$ using this as $f$, the results are as follows.