FWHM minimization

From a series of images, the FWHM of a single star is determined. The focus is then the position of minimal FWHM. Propbaly the easiest method to calculate the FWHM of a star (or a measure close enough to it), is by assuming a Gaussian point-spread-function (PSF):

psf(r)=I₀·exp(-r²/2σ²), summing psf(r) as Σ and psf²(r) as ΣΣ over the entire range and simply writing

FWHM=√(2ln(2)/π)·Σ²/ΣΣ

Outliers should be rejected to stabilize the search, an initial focal position can be derived from a temperture-depending model.

Measuring the FWHM is strongly affected by varying seeing conditions, thus a parabola is fitted through the points. Outlier removal is advisable (in the example the measurement with FWHM≈14px, f≈43.25 has to be discarded) . The minimum  of the fitting parabola is called the optimal focus.