Indexing the archive…
Your Universe of Digital Possibilities
A body is hidden in the bore. A beam crosses it from every angle, and each crossing returns not a picture but a sum — the line integral of attenuation along every ray. Stack the sums and you get the sinogram (Radon, 1917); filter each row by |ω| and smear it back and you get the slice (Cormack 1963, Hounsfield 1971). The rebuild sees only the sums. You may turn around once it is done — and then see what came back, and what no angle could ever have returned.
A beam at angle θ and offset t never sees the body — it returns one number, the integral of attenuation along its line. Stack those numbers over every θ and t and you have the sinogram; a single point of the body traces one sinusoid through it.
The 1-D Fourier transform of one projection is a radial line through the 2-D Fourier transform of the body. Every angle samples one spoke of Fourier space; the spokes you never measured — a missing wedge, the gaps past the Crowther radius — are the null space. The same theorem runs a CT scanner and an interferometer.
Smear each projection back along its rays and sum: plain back-projection blurs every point into a 1/r halo, because polar Fourier samples crowd the centre. Filtering each row by |ω| first undoes that exactly (Bracewell & Riddle 1967; Ramachandran & Lakshminarayanan 1971). Shepp–Logan and Hann windows trade sharpness for noise.
Treat every ray as one linear equation and correct the estimate onto each in turn. Renamed the algebraic reconstruction technique by Gordon, Bender & Herman in 1970; with positivity and a support as priors, it fills more of a missing wedge than FBP can — but never the frequencies the evidence did not sample.
This is the Orfeo Edition’s first shadow, and its whole method in one bore: veil the object, show only the evidence, rebuild from the evidence alone, and turn around last. The misfit is the only number knowable from behind the veil; the true error and the residual come after — and the residual of a short scan lands in exactly the frequencies no angle sampled, the null space drawn as its own inset. The projection-slice theorem at the heart of it was found in 1956 by Ronald Bracewell, strip-scanning the Sun with a radio telescope; the edition’s last instrument, The Dish, points the same theorem back at the sky. What a single time series can rebuild is The Shadow’s question; what a spectrum hides is The Spectrum’s.