Average gradient of Zernike polynomials over polygons.

Wavefront estimation from slope sensor data is often achieved by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures. Here we discuss how the calculation of these average derivatives can be reduced to one-dimensional integrals of the Zernike polynomials, rather than their derivatives, along the perimeter of each subaperture. We then use this result to derive closed-form expressions for the average Zernike polynomial derivatives over polygonal areas, only requiring evaluation of polynomials at the polygon vertices. Finally, these expressions are applied to simulated Shack-Hartmann wavefront sensors with 7 and 23 fully illuminated lenslets across a circular pupil, with their accuracy and calculation time compared against commonly used integration methods.