Sampling and seeing: where the 2–3 pixel rule comes from and why sources disagree

What gets sampled

A star image has no sharp edge, so imagers measure its full width at half maximum (FWHM): the diameter at which the brightness falls to half of the peak above the background. The AAVSO photometry guide ties that width to image scale in one line:

seeing (″) = image scale (″/px) × FWHM (px)

The FWHM in your frames includes more than the atmosphere. Stan Moore treats optics, tracking and seeing as blur terms that add in quadrature:

image FWHM = √( optics² + tracking² + seeing² )

With 3″ seeing, 1″ of optical blur and 1″ of tracking error, the stars come out at about 3.3″ (computed). Moore notes that seeing usually dominates for medium and long focal lengths once it is worse than about 3″.

The Nyquist argument

Diffraction Limited’s Douglas George starts from the sampling theorem: to reproduce a signal you need at least two samples per cycle of its highest spatial frequency. Carried over to stars, that becomes two pixels across the FWHM. He then raises the optimum to 2.5–2.8 px because pixels are square and stars are round. Turning “twice the highest frequency” into “two pixels per FWHM” is an analogy, and the sources part ways at exactly that step.

Why the sources disagree

Source	Recommendation	Written for
AAVSO guide (2022)	2–3 px across the FWHM; under 2 px probably undersampled, over 3 px may be oversampled	photometry
Diffraction Limited (2020)	2 px absolute minimum, 2.5–3 px excellent, up to 4–5 px still good	camera matching
Stan Moore (2004)	3.33 px critical; 3.5 px or more for high-resolution work	resolution, deconvolution
Atik (2018)	1–2″ per pixel	general imaging, sensitivity
Robertson, PASA (2017)	2 px per FWHM is not the Nyquist limit; often an acceptable minimum at moderate S/N	spectra

Moore’s argument runs like this: the factor of 2 belongs to sine waves, while for a Gaussian star the relevant width is its standard deviation, which gives critical sampling at FWHM = 2.355 px along one axis. The diagonal of a square pixel is √2 longer, so the critical value becomes 3.33 px. Robertson’s refereed paper studies spectra rather than images, but its simulations reach a related verdict: most real profiles show some aliasing at 2 px per FWHM, and he recommends simulating the actual profile instead of trusting the maxim. Atik skips pixels per FWHM and states a fixed scale, based on about 2″ seeing on a good, still night.

The goals explain much of the spread. AAVSO optimizes photometric precision. Moore wants headroom for the resampling that registration applies and for deconvolution. Diffraction Limited and Atik also weigh signal per pixel, because spreading a star over more pixels adds more pixels’ worth of noise.

What the calculator assumes

The sampling calculator divides your FWHM by your image scale and compares the result with the AAVSO 2–3 px band by default. You can switch to Diffraction Limited’s 2.5–3 px or to 3.3–4 px after Moore. Turned around, the 2–3 px band asks for an image scale between FWHM ÷ 3 and FWHM ÷ 2: 0.67–1.0″ per pixel for 2″ stars and 1.0–1.5″ for 3″ stars (computed).

How much seeing to expect

The amateur-oriented sources cluster between 2″ and 4″. Moore puts average seeing at most non-professional sites near 3″, sometimes under 2″ and sometimes over 4″. AAVSO gives 3–4″ for most locations. Diffraction Limited gives 2.5–3.5″ at non-alpine sites, with 2″ only on exceptional nights. Frédéric Michaud of the Société Astronomique du Havre uses about 3″ for a normal site.

Professional sites sit in another regime. ESO gives a median of 0.72″ at Paranal for 2016–2023, and an archived Gemini fact sheet gives a median site seeing of 0.43″ at Mauna Kea at 0.55 µm. Sampling Paranal’s median at 2–3 px would take 0.24–0.36″ per pixel (computed). These are site statistics; the FWHM in a finished amateur frame also carries the optics and tracking terms above.

Measuring your own FWHM

AAVSO’s method takes a few minutes. Use well-focused frames taken near the zenith, measure several unsaturated stars with good signal near the center of the field with your capture software’s star tool, and read the FWHM in pixels. Bright and faint stars share roughly the same FWHM even though bright ones look bigger. Multiply by your image scale to get seeing in arcseconds. Diffraction Limited suggests leaning toward the upper end of its pixel range so that rare nights of excellent seeing are not wasted.

Worked example

Five setups with 3.76 µm pixels (IMX571 class), using native focal lengths from the Askar FRA400 and Esprit 100ED pages and the EdgeHD 8 values in Celestron’s reducer article. Scales and pixel counts are computed.

Setup	Image scale	px across 2″	px across 3″	px across 4″
Askar FRA400, 400 mm	1.94″	1.03	1.55	2.06
Esprit 100ED, 550 mm	1.41″	1.42	2.13	2.84
EdgeHD 8 with 0.7× reducer, 1422 mm	0.55″	3.67	5.50	7.34
EdgeHD 8, 2032 mm	0.38″	5.24	7.86	10.48
EdgeHD 8, 2032 mm, binned 2×2	0.76″	2.62	3.93	5.24

On a 3″ night the FRA400 falls below AAVSO’s 2 px floor and the Esprit lands just inside the band. Unbinned at 2032 mm, stars span almost 8 px, past the 6 px point where AAVSO suggests considering a focal reducer. Binning 2×2 brings them to 3.9 px, inside Diffraction Limited’s “still good” range.

What goes wrong at each end

Undersampling throws information away. AAVSO calls it much worse than oversampling, because nothing in post-processing restores what the pixels never recorded, and Diffraction Limited warns of square-looking stars. Oversampling spreads the same light over more pixels, each adding its own noise, which lowers signal-to-noise per pixel. Moore adds that modern low-noise sensors make that penalty smaller than it was for early CCDs.

Remedies

• Oversampled: a focal reducer, or binning. AAVSO points out that binning after readout, as CMOS cameras do, brings no read-noise advantage, but it still matches the scale to the stars and shrinks the files.
• Undersampled: a Barlow or focal extender, or a camera with smaller pixels (AAVSO). The other route is to dither between frames and combine them with drizzle, the method Fruchter and Hook (2002) developed for linear reconstruction of undersampled, dithered images. The drizzle and dithering guide covers the practice, and the reducer and Barlow calculator shows the new scale for a given factor.