Reconstruction of a conic-section surface from autocollimator-based deflectometric profilometry

TL;DR

This paper introduces a method to reconstruct a conic-section surface from multiple autocollimator-based line-scans taken in different, unknown coordinate frames by using a complex optimization approach.

Contribution

The paper presents a novel optimization-based technique to accurately reconstruct conic surfaces from deflectometric data with unknown orientations.

Findings

Successful reconstruction of conic surfaces from multiple scans

Robustness of the method against measurement noise

Effective convergence using gradient-based optimization

Abstract

We present a description of our method to process a set of autocollimator-based deflectometer 1-dimensional line-scans taken over a large optical surface and reconstruct them to a best-fit conic-section surface. The challenge with our task is that each line-scan is in a different (unknown) coordinate reference frame with respect to the other line-scans in the set. This problem arises due to the limited angular measurement range of the autocollimator used in the deflectometer and the need to measure over a greater range; this results in the optic under measurement being rotated (in pitch and roll) between each scan to bring the autocollimator back into measurement range and therefore each scan is taken in a different coordinate frame. We describe an approach using a 6N+2 dimension optimisation (where N is the number of scan lines taken across the mirror) that uses a gradient-based non-linear least squares fitting combined with a multi-start global search strategy to find the best-fit surface. Careful formulation of the problem is required to reduce numerical noise and allow the routine to converge on a solution of the required accuracy.