Understanding the Fourier Slice Theorem
The Fourier Slice Theorem (also known as the Projection-Slice Theorem) is a fundamental principle underpinning medical imaging modalities like CT scans. In simple terms, it states that the 1D Fourier transform of a parallel projection of a 2D object is exactly equal to a 1D slice of the 2D Fourier transform of that object through its origin.
Mathematically, let's represent an object's density as a 2D function $ f(x, y) $. If we take a projection of this object along the y-axis, we integrate over $ y $ to get a 1D function $ p(x) $:
$$ p(x) = \int_{-\infty}^{\infty} f(x, y) \, dy $$
Next, we take the 1D Fourier transform of this projection:
$$ P(u) = \int_{-\infty}^{\infty} p(x) e^{-i 2\pi u x} \, dx $$
Now, consider the 2D Fourier transform of the original object $ f(x, y) $:
$$ F(u, v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) e^{-i 2\pi (ux + vy)} \, dx \, dy $$
If we examine the slice of this 2D transform exactly along the x-axis (where $ v = 0 $), the equation simplifies perfectly to match our 1D transform of the projection:
$$ F(u, 0) = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(x, y) \, dy \right) e^{-i 2\pi u x} \, dx = P(u) $$
Because of this relationship, we can take physical projections of an object from multiple angles, compute their 1D Fourier transforms to build out a 2D frequency domain plane, and then apply an inverse 2D Fourier transform to reconstruct the original object's internal structure.