O052 · Build a planet-sized aperture

Stellar Interferometry and Sparse Imaging

How can a few telescope pairs make an image of the sky? Each telescope pair measures one spatial pattern in the sky. Changing pair directions and observing as Earth turns supplies more patterns. Sparse measurements leave several possible images, so reconstructions combine the signals with stated assumptions about the source.

Move a telescope across Earth, extend the observing track and watch the sampled spatial patterns change. Compare the dirty image with two reconstructions of the same saved signals.

Recorded complex samples $54$

Longest projected baseline $7.28\,\mathrm{G}\lambda$

One directed closure phase $0.288^{\circ}$

Smooth-image signal residual $7.04\,\mathrm{mJy}$

Choose a view of the sky

Central hour angle

105^{\circ}

H\;[{}^{\circ}]

Observation span

3.00\,\mathrm h

\Delta t\;[\mathrm h]

Drag a dish to change its real latitude and longitude. Arrow keys move the selected dish; Shift makes a larger step.

What is measured and what is assumed

These are noiseless synthetic target signals. A separate ideal point calibrator measures station phases, and an ideal total-power observation fixes flux. Reconstruction reads these saved records only.

Read the imaging course and EHT tutorial

Model assumptions

The interactive array is a synthetic four-station experiment on a spherical Earth, observing a fixed-declination source at a prescribed 1.3-millimetre wavelength. Stations below ten degrees elevation are omitted. A narrow-field scalar Fourier model uses a finite 24-by-24 flux-per-pixel sky, fixed total flux of 0.6 jansky and noiseless complex signals with a stated reference uncertainty. An ideal simultaneous one-jansky point calibrator measures phase-only station gains; an ideal total-power observation supplies the flux constraint. Both are saved separately from the source signals. Two positivity-and-flux-constrained numerical inversions use different explicit regularizers. Their images depend on the chosen field, grid, priors and finite iteration count. No scattering, bandwidth/time smearing, polarization, gain-amplitude error, relativistic sky model or unique recovery claim is included. Globe size, signal paths and wave motion are schematic. Baselines are geometric chords, not cables. Real EHT data appear only in the separate fixed-record tutorial, where amplitudes and closure phases are fitted; their uncalibrated phases are not treated as a sky image.

Sampled baselines; faint dots are conjugates

Dirty beam · unit-peak response

Phase-calibrated dirty image · Jy per beam

Smooth reconstruction · Jy per pixel

Compare the source, another prior and uncalibrated phases

Synthetic source truth · separate reference

Compactness preference · same saved signals

Uncalibrated dirty image · failure diagnostic

Fourier units and reconstruction choices

V_k=\sum_p x_p e^{-2\pi i(u_kl_p+v_km_p)},\qquad x_p\geq0,\quad\sum_p x_p=0.6\,\mathrm{Jy}

Pixels store flux, so the Fourier sum has no extra pixel-area factor. Maps display east left and north up. Dirty maps retain signed sidelobes; reconstructions enforce positivity and the independently saved total flux. Smoothness penalizes neighboring-pixel differences; compactness adds a radius-weighted flux penalty and a small squared-pixel penalty. Ninety accelerated projected-gradient steps are a finite numerical choice.

Start with the essentials

Focus question: Why can two different images predict almost the same measured signals?
One-sentence intuition: The array measures selected Fourier components; reconstruction adds explicit preferences in directions the sampling leaves uncertain.

Core mathematical model

One complex sample

V_k=\sum_p x_p e^{-2\pi i(u_kl_p+v_km_p)}

Each pixel stores flux in jansky. Sky coordinates are east-positive and north-positive radians; baseline coordinates are in wavelengths. Opposite baselines have conjugate signals.

The directed triangle

\Phi_{ijk}=\arg(V_{ij}V_{jk}V_{ki})

A station phase occurs with both signs when the signals are multiplied around a simultaneous triangle. The first available triangle supplies the readout; station phase errors leave it unchanged.

Fit the signals with a stated preference

\min_{x_p\geq0,\,\sum_px_p=F}\left[\frac1{2N}\sum_k|(Ax)_k-y_k|^2+R(x)\right]

Both images read the same separately calibrated record and independent total flux. One penalizes neighboring-pixel differences; the other favors central flux with a small squared-pixel term. The known synthetic source is a separate comparison.

Common difficulties

A sharp image can still depend on assumptions

Typical misconceptionA small signal residual means every reconstructed feature is measured.

Better mental modelUnsampled spatial frequencies can change the image without changing the saved signals. Compare priors and seek new baselines where their predicted measurements differ.

Run the experiment

01
Move one station
Use Crossed baselines. Predict how dragging one telescope changes the coverage, then move it and compare the dirty beam.
What to observe: One moved station changes every baseline connected to it. The signed beam responds to the new set of measured directions.
02
Let the ground rotate
Compare One narrow direction with Let Earth turn. Open the alternate-prior images and compare their long directions.
What to observe: A longer track supplies additional projected directions and changes which source shapes remain plausible.
03
Cancel station phase errors
Choose Atmospheric phase errors. Compare closure with the default, then inspect the raw and phase-calibrated dirty maps.
What to observe: The closure survives the station phases. A separate ideal point calibrator restores the complex signals used for the two inversions.

Stellar Interferometry and Sparse Imaging

The same signals permit different pictures

Which parts of the image did the array measure?

Core mathematical model

One complex sample

The directed triangle

Fit the signals with a stated preference

Common difficulties

A sharp image can still depend on assumptions

Run the experiment

Move one station

Let the ground rotate

Cancel station phase errors

Related experiments and articles

Measuring the sky

From telescope pairs to sky images

Learning to read radio signals