Part 1: What is a Fourier Transform?

Imagine you are listening to a chord played on a piano. Your ear hears one, rich, complex sound. But you know that this single sound is actually made up of several individual, pure notes being played at the same time.

The Fourier Transform is a mathematical genius that can do exactly this: it takes any complex signal—like a sound, a radio wave, or even an image—and breaks it down into its simple, basic ingredients. These ingredients are pure waves (called sine waves) that have different:

Frequencies: How "close together" the waves are (low pitch vs. high pitch for sound, large smooth areas vs. fine details for an image).

Amplitudes: How "strong" or "loud" each wave is.

Phases: The timing of where each wave starts.

Think of it like a magical recipe decoder:

The complex signal is your finished, baked cake.

The Fourier Transform gives you the recipe, listing all the exact amounts of flour, sugar, eggs, etc. (the frequencies and amplitudes).

The reverse process, the Inverse Fourier Transform, takes the recipe and puts all those ingredients back together to bake the cake. It reconstructs the original signal from its basic parts.

Part 2: How This Applies to Images (2D Fourier Transform)

An image isn't a sound wave over time; it's a 2D object with height and width. So, we use the 2D version of the Fourier Transform.

Instead of breaking down a sound into simple 1D waves, it breaks down an image into simple 2D wave patterns.

In this case:

Low frequencies correspond to the big, smooth areas, the overall shapes, and the slow changes in color—the "big picture" of the image.

High frequencies correspond to the sharp edges, fine details, textures, and rapid changes—all the "little details" that make an image crisp.

The result of this 2D Fourier Transform is called k-space. K-space is not an image you can recognize. It's the image's recipe written in a secret code. It's a map that tells you exactly which wave patterns, with which strength, are needed to build your image.

Part 3: Applying this to MRI Scans

Here's the crucial part: An MRI machine doesn't take a picture like a camera. It directly measures this k-space recipe.

The Setup: The patient is in a strong magnet. Our bodies are mostly water, and water molecules act like tiny magnets. The MRI machine sends a radio wave pulse that makes these tiny magnets "sing."

Listening to the Song: The machine then listens to the signal these molecules send back as they relax. However, it uses magnetic gradients (like clever magnets that change strength across space) to tag the signals based on where they came from.

Filling the Recipe Book (k-space): Each measurement the scanner takes doesn't correspond to a single pixel. Instead, each measurement fills in a whole line of the k-space recipe. It does this over and over, from different directions, until the entire k-space grid is filled with data.

Baking the Image (Inverse Fourier Transform): The computer now has the complete recipe (k-space). It then performs the Inverse 2D Fourier Transform. This is the mathematical magic that follows the recipe, adds all the wave patterns together, and reconstructs them into the final, detailed anatomical image that doctors see.

In short: MRI scanner measures k-space (the recipe) → Computer uses Inverse Fourier Transform to create the image (the cake).

Part 4: Applying this to CT Scans

A CT scanner works a bit differently, but it still relies on the same mathematical magic.

Taking Shadows (Projections): A CT scanner rotates around you. At each angle, an X-ray tube shoots rays through your body, and a detector on the other side measures how much the X-rays are weakened. This measurement is like taking a single, flattened "shadow" or silhouette of your body from that angle.

The Magic Theorem (Fourier Slice Theorem): There is a brilliant mathematical rule called the Fourier Slice Theorem. It states that if you take one of these "shadow" projections and perform a 1D Fourier Transform on it, the result is exactly equal to one single line, slicing right through the center of the k-space of your final image.

Building the Recipe: So, as the CT scanner rotates and takes hundreds of these shadow projections from all different angles, it is effectively gathering hundreds of lines that, together, fill up the entire k-space recipe of your body's slice.

Reconstructing the Image: Once it has enough lines in k-space, the computer can use the Inverse Fourier Transform to turn that data into an image. (The most common method, called Filtered Back Projection, is a smart and efficient way to do this Inverse Fourier Transform using all the individual projections).

In short: CT scanner measures projections (shadows) → Math theorem says these shadows are slices of k-space (the recipe) → Computer uses Inverse Fourier Transform to create the image (the cake).

HNID KARIM MD