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Filtered back projection, sinogram & voxel/pixel

A rotating source and detector acquire projections (ray-sums) at many angles; a point object traces a sinusoid in the sinogram. Reconstruction has two routes. Simple unfiltered back-projection smears each ray back and produces a star or streak artefact with 1 over r blurring. Filtered back-projection first convolves each projection with a ramp high-pass reconstruction kernel (equivalent to multiplication in the frequency domain, a Fourier step), which cancels the blur and recovers a sharp point. Iterative reconstruction is an alternative that models noise and physics for lower noise and dose but is computationally demanding, while filtered back-projection remains the common default.

CT image reconstruction: from projections and the sinogram to filtered back-projection A rotating source and detector acquire projections (ray-sums) at many angles; a point object traces a sinusoid in the sinogram. Reconstruction has two routes. Simple unfiltered back-projection smears each ray back and produces a star or streak artefact with 1 over r blurring. Filtered back-projection first convolves each projection with a ramp high-pass reconstruction kernel (equivalent to multiplication in the frequency domain, a Fourier step), which cancels the blur and recovers a sharp point. Iterative reconstruction is an alternative that models noise and physics for lower noise and dose but is computationally demanding, while filtered back-projection remains the common default. CT image reconstruction rotating source–detector projections at many angles sinogram detector position vs angle θ a point object → a sinusoid two reconstruction routes Simple back-projection (no filter) Star / streak artefact each ray smeared back along its path; finite angles → streaks; blurs as 1/r Convolution filter ramp kernel (high-pass) Back-projection Sharp point recovered (correct) Filtered back-projection (FBP): the ramp filter is applied before back-projection, cancelling the 1/r blur. Convolution in space = multiplication in frequency (Fourier). Kernel selectable: smooth (low noise) ↔ sharp (detail). Iterative reconstruction — alternative to FBP Starts from an estimate, repeatedly forward-projects and compares with the measured data, modelling noise / physics. Lower noise and dose, but more computationally demanding — FBP remains the common default.
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