---
title: "Stellar Interferometry and Sparse Imaging Interactive Lab | Sandbox Physics"
description: "How can a few telescope pairs make an image of the sky? Each telescope pair measures one spatial pattern in the sky."
canonical_url: "https://sandboxphysics.com/labs/stellar-interferometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/stellar-interferometer/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

## The same signals permit different pictures

The reconstructions and diagnostics use the saved synthetic signals. The known source is shown as a separate reference. Each image has its own stated intensity normalization.

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.

Physics tutorial

## Which parts of the image did the array measure?

**Background** Two telescopes compare arriving radio waves. Their projected separation selects a striped pattern on the sky. Several lengths and directions measure different spatial frequencies.

**Why it matters** A widely separated array can resolve tiny angular detail, yet most spatial frequencies remain unmeasured. Comparing two reconstructions shows where source assumptions influence the picture.

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 misconception** A small signal residual means every reconstructed feature is measured.

**Better mental model** Unsampled 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

1. 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.

2. 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.

3. 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.

Physical conclusion

New baseline directions add shape information. Closure phases cancel station phase errors, and alternative priors reveal which image features need more measurements.

## Related experiments and articles

Continue exploring

### Measuring the sky

What do positions, colors, interference and radio delays reveal about the sky?

1. [Lab **Observing Cadence and Aliases** Move exposures and compare the rhythms that survive the sampling.](https://sandboxphysics.com/labs/observing-cadence-aliases/)
2. [Lab **Supernova Distances and the Distance Ladder** Move a geometric distance anchor and follow its shared calibration error.](https://sandboxphysics.com/labs/supernova-distance-expansion/)
3. [Lab **Stellar Parallax and Proper Motion** Change the stellar distance and follow its annual motion.](https://sandboxphysics.com/labs/stellar-parallax-proper-motion/)
4. [Lab **Crowded-star Photometry** Move the aperture and compare the light admitted from neighbouring stars.](https://sandboxphysics.com/labs/crowded-star-photometry/)
5. [Lab **Stellar Absorption Lines** Change the gas temperature and velocity, then compare the absorption lines.](https://sandboxphysics.com/labs/stellar-absorption-lines/)
6. [Lab **Galaxy Colors and Photometric Redshift** Compare redshift, dust and stellar-population effects in the same five bands.](https://sandboxphysics.com/labs/galaxy-colors-photometric-redshift/)
7. [You are here **Stellar Interferometry and Sparse Imaging** Move a telescope and compare images fitted to the same sparse signals.](https://sandboxphysics.com/labs/stellar-interferometer/)
8. [Lab **Interstellar Radio Propagation** Change the electron column and field, then follow the arriving radio pulse.](https://sandboxphysics.com/labs/interstellar-radio-propagation/)

From telescope pairs to sky images · Reading path

### From telescope pairs to sky images

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How scattered telescopes make an image**](https://sandboxphysics.com/stories/interferometric-imaging/how-scattered-telescopes-make-an-image/)
2. [Article **From wave delay to complex visibility**](https://sandboxphysics.com/stories/interferometric-imaging/from-wave-delay-to-complex-visibility/)
3. [Article **Earth rotation and closure phase**](https://sandboxphysics.com/stories/interferometric-imaging/earth-rotation-and-closure-phase/)
4. [Article **Dirty beams and nonunique images**](https://sandboxphysics.com/stories/interferometric-imaging/dirty-beams-and-nonunique-images/)
5. [Article **Fit a real EHT closure triangle**](https://sandboxphysics.com/stories/interferometric-imaging/fit-a-real-eht-closure-triangle/)
6. [Article **Read EHT imaging and dynamic priors**](https://sandboxphysics.com/stories/interferometric-imaging/read-eht-imaging-and-dynamic-priors/)

Astronomy and gravity · Reading path

### Learning to read radio signals

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How Ryle built a large radio telescope from smaller ones**](https://sandboxphysics.com/stories/astronomy-gravity/ryle-aperture-synthesis/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)