Modal Data Processing for High Resolution Deflectometry

Abstract

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.

Freeform surface selection based on parametric fitness function using modal wavefront fitting

“…Aftab et al derive a vector polynomial basis set, called ì G polynomials [21]. This set is obtained from gradients of the two-dimensional Chebyshev polynomials of the first kind, named F polynomials.…”

“…They are well-suited for applications where a very large number of polynomials are needed for fitting data such as freeform optics or the reconstruction of surfaces from slope data, where mid-to-high spatial frequencies must be correctly represented. Some features of these polynomial sets that make this possible and efficient [21] are: (a) development of recursive relations for both F and ì G polynomial sets (b) a one-to-one correspondence between the coefficients of the two polynomial sets (c) ease and accuracy of generating the gradient ( ì G) polynomials which, unlike most other gradient polynomial sets, do not need an orthogonalization process. Shown in Eq.…”

“…where n and m are double index variables describing the polynomials' order [21] whileî and j are unit vectors indicating a Cartesian coordinate system's axes. Quiver plots of the first 5 non-trivial ì G polynomials are given in Fig.…”

“…2. F and ì G polynomials based data processing has also shown to be adaptable and efficient in various practical situations [21,24] such as the presence of NaNs (not a number) in data (representing missing data, apertures, blockers etc. ), uneven sampling or a different sampling in two directions (e.g.…”

Rectangular domain curl polynomial set for optical vector data processing and analysis

“…Previous works highlight the importance of fitting vector slope data in the domain of measurement as well as using orthonormal polynomials for the fitting. [1][2][3] To summarize, the fit to a nonorthogonal basis set can require many more terms than are necessary, and expansion coefficients themselves are not meaningful, because the value for any particular coefficient changes as the number of expansion terms changes. Also, when fitting to real data, the noise propagation is increased with the use of nonorthogonal basis functions.…”