A new method for aspherical surface fitting with large-volume datasets

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. a b s t r a c tIn the framework of form characterization of aspherical surfaces, European National Metrology Institutes (NMIs) have been developing ultra-high precision machines having the ability to measure aspherical lenses with an uncertainty of few tens of nanometers. The fitting of the acquired aspherical datasets onto their corresponding theoretical model should be achieved at the same level of precision. In this article, three fitting algorithms are investigated: the Limited memory-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), the Levenberg-Marquardt (LM) and one variant of the Iterative Closest Point (ICP). They are assessed based on their capacities to converge relatively fast to achieve a nanometric level of accuracy, to manage a large volume of data and to be robust to the position of the data with respect to the model. Nevertheless, the algorithms are first evaluated on simulated datasets and their performances are studied. The comparison of these algorithms is extended on measured datasets of an aspherical lens. The results validate the newly used method for the fitting of aspherical surfaces and reveal that it is well adapted, faster and less complex than the LM or ICP methods.

Modelling and traceability for computationally-intensive precision engineering and metrology

“…Moore-Penrose Quasi-Newton methods attempt to build an approximation of the Hessian matrix (or its inverse) that incorporates second order information by incorporating first order information as the optimisation proceeds. The Broyden-Fletcher-Goldfarb-Shanno Algorithm (BFGS) [49,126,136,144] is one of the most famous quasi-Newton algorithms for unconstrained optimization. Moving away from deterministic algorithms, intelligence-oriented algorithms (Genetic algorithms, Swarm algorithms) [97,173,179,180] with their simplicity are another way to search the solution of extreme problems with many local minima.…”

“…It automatically selects the most efficient algorithm for the computed mathematical problem. Matlab uses several algorithms depending on the type of problem to be solved: interior reflective Newton [17][18] [ 49,126,136,144] method, trust-region-dogleg, trust-region-reflective, Levenberg-Marquardt, simplex, BFGS, MiniMax, and so on.…”

Novel automated methods for coarse and fine registrations of point clouds in high precision metrology

“…Moreover, Levenberg Marquardt and L-BFGS methods were also respectively studied and discussed by NIST and LNE [36] for an optimal solution of the problem (Eq.17).…”

Robust and precise algorithm for aspheric surfaces characterization by the conic section

“…The description of the aspheric surface by the conic section with a polynomial correction is common in ray tracing software and in producer specifications of aspheric lens. The conic section surface fitting with a polynomial correction was addressed by several authors [1][2][3][4][5]. Also alternative descriptions of aspheric surfaces were introduced e.g.…”