Modal Data Processing for High Resolution Deflectometry

In this paper, we present a modal data processing methodology, for reconstructing high resolution surfaces from measured slope data, over rectangular apertures. One of the primary goals is the ability to effectively reconstruct deflectometry measurement data for high resolution and freeform surfaces, such as telescope mirrors. We start by developing a gradient polynomial basis set which can quickly generate a very high number of polynomial terms. This vector basis set, called the G polynomials set, is based on gradients of the Chebyshev polynomials of the first kind. The proposed polynomials represent vector fields that are defined as the gradients of scalar functions. This method yields reconstructions that fit the measured data more closely than those obtained using conventional methods, especially in the presence of defects in the mirror surface and physical blockers/markers such as fiducials used during deflectometry measurements. We demonstrate the strengths of our method using simulations and real metrology data from the Daniel K. Inouye Solar Telescope (DKIST) primary mirror.