Concept

Image reconstruction — where it appears

The construction of a sky image from incomplete measurements of its Fourier transform. The data determine a family of images rather than one, and the algorithm chooses among them by a criterion the data do not contain.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

$20 closure phases, unmoved by an atmosphere that ruins every baseline. Closure phase measured against closure phase true, for all 20 triangles of a 6-antenna array observing a binary 3.2 mas apart with a flux ratio of 0.35. Each antenna has been given an independent atmospheric phase of 65° rms, which corrupts the individual baseline phases by 102° rms — several times the 31.2° the source itself produces, so no single visibility phase in this simulation carries usable information. Every point here nonetheless sits exactly on the diagonal: the per-antenna terms cancel identically round any triangle, and the largest departure over all 20 is 2.5e-14 degrees, which is round-off. The price is in the counting. 6 antennas give 15 baseline phases of which 5 are consumed by the unknowns, so of the 20 triangles only 10 closures are independent — a fraction (N−2)/N = 0.667 of the phase information. For two antennas that fraction is zero and there is no closure at all; the Event Horizon Telescope's image rests on quantities of this kind and on no absolute phase whatever.$

A phase that survives what corrupts it

An atmosphere over each antenna adds an unknown to the phase of every baseline that antenna takes part in. Sum the phases round a triangle and every one of those unknowns cancels identically — which is the reason an image can be made across ten thousand kilometres, and the reason it has no position on the sky.

starlight · Interferometry

A hole's shadow at 17° to its spin axis. The critical curve — the outline of a black hole's shadow on a distant observer's sky — for spins from 0 to 0.998, seen 17 degrees from the spin axis, in gravitational radii. Every curve is the projection of the unstable spherical photon orbits, and nothing about an accretion flow enters it. The sizes are 5.20, 5.12, 4.92, 4.84: a spread of 6.9 per cent across the entire range of spin, which is what makes an image of one of these a mass measurement with almost no model in it. What spin does instead is push the curve sideways and flatten one edge — the fast hole's outline sits 0.70 gravitational radii off centre, because photons co-rotating with the hole are dragged round and escape from closer in than counter-rotating ones can. That displacement is the spin signal, and it is measured against a centre nothing else marks.

A ring whose size is almost only a mass

The dark patch at the centre of an image of a black hole is the projection of the orbits light cannot escape from, and its size changes by seven per cent across the whole range of spin and inclination. That insensitivity is what makes an image a mass measurement with almost no model in it — and what leaves the spin in the part of the picture that barely moves.

gravitation · Black hole spin

Three pictures, one set of measurements. A one-dimensional sky, the transform of it sampled at 10 spatial frequencies out of 128, and three images reconstructed from those samples. Every one of the three reproduces every measured visibility exactly — checked to a part in a million — so the data cannot choose between them. The dirty image is the samples transformed back and nothing else, complete with the negative sidelobes that incomplete sampling produces. The other two are the members of the same family that minimise a penalty: one the total curvature, one the total absolute brightness. They differ from each other by 7 per cent of the peak, and where they differ is exactly where the array did not look. The smooth reconstruction blurs the two compact sources into one another and keeps the broad component; the sparse one splits them cleanly and loses the broad one. Neither is a lie and neither is the sky. A published image is the data plus a criterion, and the criterion is a choice made by whoever ran the reduction.

How much of the picture is the prior

An array measures the sky's transform at a few dozen spatial frequencies and at no others, so the data do not determine an image. They determine a family of them, and the reconstruction picks one member by a criterion that is not in the data — smoothness, or sparsity, or entropy. Three reconstructions of one measurement fit it exactly and disagree by a tenth of the peak.

starlight · Interferometry

Named alongside it

The objects these essays reach for when they reach for this one.

Closure phase Antenna gain Aperture synthesis Astrometric reference Atmospheric coherence time Black hole spin Closure amplitude Critical curve Deconvolution Degeneracy Degrees of freedom Dirty beam

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