Radon Transform
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The Radon transform is an integral transform that computes projections of a function along straight lines. It is the mathematical foundation of computed tomography (CT) imaging.
Definition
For a 2D function $f(x, y)$, the Radon transform $\mathcal{R}f$ gives the line integral along a line $L$ at angle $\theta$ and distance $s$ from the origin:
$$ \mathcal{R}f(s, \theta) = \int_{-\infty}^{\infty} f(s \cos\theta - t \sin\theta, \, s \sin\theta + t \cos\theta) \, dt $$For the special case of projection along the $x$-axis (θ = 0):
$$ p(y) = \int_{-\infty}^{\infty} f(x, y) \, dx $$Properties
- Linearity: $\mathcal{R}\{af + bg\} = a\mathcal{R}f + b\mathcal{R}g$
- Shifting: A shift in $f$ causes a corresponding shift in its projections
- Rotation: Rotating $f$ rotates its sinogram (the collection of all projections)
Connection to Fourier Transform
The projection-slice theorem states that the 1D Fourier transform of a projection equals a slice through the 2D Fourier transform of the original function:
$$ \mathcal{F}_1\{\mathcal{R}f(s, \theta)\} = F(\nu \cos\theta, \nu \sin\theta) $$Applications
- Medical imaging: CT scanners measure Radon transforms (X-ray projections)
- Electron microscopy: 3D reconstruction from 2D projections
- Radio astronomy: Aperture synthesis imaging