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This paper describes a method based on a bi-objective evolutionary algorithm to obtain the profile of a convex aspherical surface, which is defined by a set of synthetic points placed on an xyz coordinate system. The set of points to be analyzed is constructed considering the sources of measurement error in a coordinate measuring machine (CMM), such as machine, probe, and positioning errors. The proposed method is applied to solve a bi-objective optimization problem by minimizing two objective functions. By minimizing the first objective function the positioning error is removed from the coordinates of each affected point. Once the first goal is achieved, the second objective function is minimized to determine from the resulting data all parameters related to the test surface, such as paraxial radius of curvature, the conic constant and the deformation constants. Hence, this method can obtain the correct surface profile even when the positioning error tends to increase the CMM measurement error in the set of analyzed points. The bi-objective evolutionary algorithm (BEA) was tested against a single-objective evolutionary algorithm, and illustrative numerical examples demonstrate that the BEA performs better.

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Este artículo presenta un estudio numérico comparativo que demuestra la factibilidad de la optimización estocástica (OE) para reconstruir la diferencia de caminoóptico (DCO) a partir de un interferograma real degradado por ruido, ya sea por la maximización del coeficiente de correlación o por la minimización de la distancia euclidiana donde la optimización conseguida para cada función objetivo corresponde a la solución más cercana alóptimo global, sin ser dominada por unóptimo local. Con la finalidad de mostrar la eficacia de diferentes algoritmos de OE, inspirados en cómputo evolutivo, nosotros proponemos una solución al problema de maximización mediante el uso de un algoritmo genético con las aberraciones primarias descritas por Kingslake, mientras que para la solución del problema de minimización se propone una estrategia evolutiva con los polinomios de Zernike. Los resultados numéricos muestran la sencillez, robustez y precisión de ambos algoritmos de optimización para calcular sus correspondientes coeficientes de aberración. De esta forma, este trabajo ofrece una oportunidad idónea para integrar las habilidades adquiridas por los estudiantes universitarios de ciencias e ingeniería en materias como metrologíaóptica interferométrica, métodos numéricos y programación con el propósito de realizar análisis de interferogramas.Descriptores: Reconstrucción numérica; problemas inversos; optimización estocástica; experimentos de demostración para estudiantes no graduados; estudiantes no graduados de ciencias e ingeniería. This paper presents a comparative numerical study that demonstrates the feasibility of stochastic optimization (SO) for reconstructing the optical path difference (OPD) from a real interferogram degraded by noise, either by the maximization of the correlation coefficient or by the minimization of the Euclidean distance where the optimization achieved for each objective function corresponds to the near-optimal solution without being dominated by a local optimum. In order to show the efficacy of different SO algorithms based on evolutionary computation, we propose a solution to the maximization problem by using a genetic algorithm with the primary aberrations described by Kingslake, while for the solution of the minimization problem an evolutionary strategy with Zernike polynomials is proposed. The numerical results show the simplicity, robustness, and accuracy of both SO algorithms to calculate their corresponding aberration coefficients. Thus, this work offers an ideal opportunity to integrate the skills acquired by university students of science and engineering in subjects such as interferometric optical metrology, numerical methods, and programming with the purpose of performing interferogram analysis.

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