CT Reconstruction: Radon Transform & Filtered Back-Projection

Acquisition Technique

Tube voltage (kVp) 80 kVp

Tube current-time (mAs) 150 mAs

Projection views 180

Angular coverage 180°

Reconstruction Quality

Avg. transmission

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Bone/lung contrast

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Recon RMSE vs truth

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Entrance dose (approx)

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How it works

Each detector reading still obeys the same Beer-Lambert law as plain-film radiography — a beam loses intensity exponentially as it crosses tissue:

I = I₀ · exp(−Σ μᵢ·xᵢ)
p(θ,s) = −ln(I/I₀) = ∫ μ(x,y) dl   (line integral along the ray)

But a CT scanner never sees the cross-section directly — it only measures p(θ,s), the Radon transform of the attenuation map, one straight-line integral at a time, at many angles θ as the gantry rotates. Stacking every angle's readings gives the sinogram shown in the middle panel. Recovering the image from the sinogram needs filtered back-projection (FBP): each projection row is convolved with a Ram-Lak ramp filter (which undoes the 1/r blurring that plain summation would otherwise produce), then every filtered ray is smeared back across the image and all angles are added together:

f(x,y) ≈ Δθ · Σ_θ  Qθ(x·cosθ + y·sinθ)
Qθ = p(θ,·) ⊛ h_RamLak

Two real artifacts appear directly from the acquisition settings, exactly as on a physical scanner:

Photon-starvation streaks — lowering mAs starves each ray of photons, so quantum (shot) noise on the raw counts propagates through the ramp filter into streaky artifacts across the reconstructed slice, worst through the densest (bone) paths.
Angular under-sampling / missing-wedge artifacts — cutting the number of projection views, or restricting angular coverage below a full half-turn, leaves the Radon transform incompletely sampled; the reconstruction develops streak or wedge-shaped smear artifacts even with zero noise.

Bone (high effective Z) still attenuates mostly via the photoelectric effect (∝E⁻³, strongly kVp-dependent), while soft tissue attenuates mostly via nearly energy-independent Compton scattering — so raising kVp washes out bone/soft-tissue contrast in the reconstructed slice much faster than it changes overall brightness, same physics as plain radiography, just recovered through an inverse problem instead of read straight off a detector.

Phantom (ground truth μ) Sinogram p(θ,s) FBP reconstruction