Indexing the archive…
Your Universe of Digital Possibilities
A detector far from a small object records how brightly its diffraction lands — |F|² — and throws away the phases, which carry where everything is. This is the problem behind Photo 51, the X-ray pattern of DNA Franklin and Gosling took in 1952. Yet with the pattern sampled finely enough (Sayre, 1952), a loop between two constraints — fit the data, stay inside the box — finds the object (Gerchberg & Saxton 1972, Fienup 1982), up to a twin no data can tell apart. The rebuild sees only the intensities. Turn around when it is done.
A detector records how brightly each spatial frequency lands, never its phase — and the phases carry where everything is. The object g(x), its 180° twin g(−x) and every translation share one |F|: those ambiguities are the null space of this evidence.
Impose the measured modulus, then impose the object constraints — real, non-negative, inside the support S — and repeat. Each step cannot raise the error, which is why it so often stops at the wrong place: error-reduction stagnates.
Fienup’s fix: where the new estimate violates the constraints, do not zero it — push back against it with feedback β (≈ 0.9). The iterate is no longer an estimate but a driving signal, and it climbs out of the traps error-reduction falls into; a few ER steps at the end polish.
A helix of radius r and pitch P diffracts onto layer lines spaced 1/P apart; on the l-th line the intensity follows the Bessel function Jl, whose first peak moves outward as l grows — so the brightest spots march away from the axis and draw an X.
The Orfeo Edition’s third shadow keeps only half of every measurement — the modulus — and still rebuilds the object, because the two constraints (the data, the box) intersect at it once the pattern is oversampled. What the evidence cannot settle is drawn, not hidden: the twin and the translations are the null space of |F|, so the instrument reports FLIPPED or UPRIGHT only after you turn around, and the error is measured modulo both. The prior is named on the pane (real, non-negative, inside the box). Two slits and their fringes are The Slit; the phase this loop recovers is exactly what The Hologram records directly, by interfering the object with a reference wave; and the line-integral cousin of the problem is The Scan.