Inside the core SDSS pipelines and inside the Astrometry.net source-detection code simplexy, centroiding—measurement of star positions in the image—is performed by fitting two-dimensional second-order polynomials to the central 3×3 pixel patch centered on the brightest pixel of each star. This is known to work far better than taking first moments of the light distribution (integrals of x and y times the brightness above background) for the (possibly obvious) reason that it is a quasi-justified (in terms of likelihood) fit.
Of course not all of the information about a star's position is contained in that central 3×3 pixel patch (and the method doesn't make use of any point-spread function information to boot). For this reason, Jo Bovy (IAS) and I did some work a few years ago to test it. Things intervened and we never finished, but our preliminary results were really surprising: For well-behaved point-spread functions, the two-dimensional quadratic fit in the 3×3 patch performed almost indistinguishably from fits that made use of the true point-spread function and larger patches. That is, it appeared in our early tests that the 3×3 patch does contain most of the centroiding information! A good research project would see how the 3×3 patch inference degrades relative to the point-spread-function inference, as a function of PSF properties and the signal-to-noise, with an eye to analyzing when we need to be thinking about doing better. I will call out Adrian Price-Whelan (Columbia) here, because he is all set up with the machinery to do this!