Computed Tomography
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Computed tomography (CT), also called computerized axial tomography (CAT), is a medical imaging technique that reconstructs cross-sectional images from X-ray projections taken at multiple angles.
Physical Principle
- An X-ray source rotates around the patient
- X-rays are attenuated according to the Beer-Lambert law as they pass through tissue
- Detectors measure the transmitted intensity
- Each measurement represents a line integral of the attenuation coefficient (a Radon transform)
Mathematical Foundation
The reconstruction relies on the projection-slice theorem:
- Each projection at angle $\theta$ gives one radial slice of the 2D Fourier transform
- Collecting projections at many angles fills the Fourier plane
- Inverse Fourier transform recovers the image
Reconstruction Algorithms
Filtered Back-Projection
The standard clinical algorithm:
- Apply a ramp filter $|\nu|$ to each projection (compensates for non-uniform radial sampling)
- “Smear” each filtered projection back across the image at its original angle
- Sum contributions from all angles
Iterative Methods
Modern approaches that can handle:
- Sparse data (fewer projections)
- Noise reduction
- Prior knowledge constraints
Hounsfield Units
CT images are displayed in Hounsfield units (HU):
$$ HU = 1000 \times \frac{\mu - \mu_{water}}{\mu_{water} - \mu_{air}} $$| Material | HU |
|---|---|
| Air | -1000 |
| Water | 0 |
| Soft tissue | +40 to +80 |
| Bone | +400 to +1000 |
Applications
- Diagnostic imaging (trauma, cancer, cardiovascular)
- Radiation therapy planning
- Industrial inspection (non-destructive testing)