Optimal sampling patterns for Zernike polynomials

TL;DR

This paper identifies optimal sampling patterns on the disk for Zernike polynomial interpolation, ensuring unisolvency and numerical stability, which improves surface reconstruction accuracy from wavefront data.

Contribution

It introduces a specific node pattern that guarantees unisolvency and stability for Zernike polynomial interpolation, enhancing optical surface reconstruction.

Findings

Nodes provide numerically stable surface reconstruction

Sampling improves accuracy of Zernike coefficient recovery

Pattern ensures unisolvency of the interpolation problem

Abstract

A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numerically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface.