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What is a Phantom in Tomography? 📂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.

head_radon.png

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

head_iradon.png

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.

SL_phantom.png

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