3.1 Analytic Reconstruction (FBP/FFT)

Analytic reconstruction provides fast, formula-based (or transform-based) solutions under assumptions. Typical examples:

• CT: FBP / FDK
• MRI: FFT-based reconstruction (Cartesian sampling)

This page focuses on CT: geometry → Radon transform → FBP → FDK.

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Core idea (CT)

CT measures line integrals of attenuation. Under ideal sampling, we can recover the image via the inverse Radon transform, implemented efficiently with filtering + backprojection.

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FBP (Filtered Backprojection)

FBP uses:

1. Filtering each projection (ramp / Hann / etc.)
2. Backprojection over angles

Compact form:

$$f(x,y)={\int }_0^{\pi }[Rf(\theta ,s)\ast h(s)]{|}_{s=x\cos \theta +y\sin \theta }d\theta$$

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FDK (Cone-beam CT)

FDK extends FBP to cone-beam geometry using:

• geometric weighting
• 1D filtering (often FFT-based)
• 3D backprojection

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Next

• Iterative reconstruction: /en/guide/ch03/02-iterative-recon
• Deep learning reconstruction: /en/guide/ch03/03-dl-recon