Optimization methods for the design of progressive lenses
- Autores: Glòria Casanellas Peñalver
- Directores de la Tesis: Jordi Castro Pérez (dir. tes.)
- Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
- Idioma: español
- Tribunal Calificador de la Tesis: Francisco Javier Prieto Fernández (presid.), Francisco Javier Heredia Cervera (secret.), Manuel Espínola Estepa (voc.)
- Programa de doctorado: Programa de Doctorado en Estadística e Investigación Operativa por la Universidad Politécnica de Catalunya
- Materias:
- Texto completo no disponible (Saber más ...)
- Resumen
This work is the result of an Industrial Doctorate developed through a partnership agreement between the Universitat Politècnica de Catalunya and the company Horizons Optical. This thesis solves the complex design of progressive lenses for eyeglasses, which is a real problem in the industry. The lens is the transparent part of the eye behind the pupil that helps humans to see clearly by focusing light onto the retina. Over time, the lens loses some of its elasticity and therefore can no longer accommodate clearly for near vision. This phenomenon is called presbyopia and explains why people need reading glasses as they become older. Progressive lenses correct presbyopia and have a complex design: they have an upper region for far vision, a low region for near vision (reading), and a corridor that connects these areas which allows clearly vision at an intermediate distance, for example, when looking at a computer screen. The surface of the progressive lens designed in this thesis is the surface that is farthest from the eye, thus the power in the near region is bigger than the power in the far region.
In geometrical terms, power and astigmatism are calculated using the principal curvatures of the lens surface. When the power changes vertically, unwanted lateral astigmatism (aberrations) appear as a result of the Minkwitz theorem. The focus of this thesis is the use of optimization methods in order to design progressive lenses minimizing the lateral aberrations (astigmatism) and providing the power required in each zone.
This thesis presents two different models for computing progressive lens. Both models are highly nonlinear, nonconvex and continuous and were solved using the AMPL modeling language and the interior point solvers IPOPT, LOQO and KNITRO. Both models have approximately 900 variables (the coefficients of a third-degree B-spline basis). The first model has about 7000 constraints, while the second model has about 15000 constraints. Each constraint corresponds to a property of power or astigmatism at a point on the grid that defines the lens surface.
The first model uses Cartesian coordinates and is an improved version of a previous model by the same author, published in a master's thesis. The CPU time in the master thesis was between 10 and 33 minutes, and in this thesis it has been reduced to less than 3 minutes using the same machine and the LOQO solver. In this thesis, all of the proposed instances converged using the LOQO solver and the Cartesian coordinate model, which was not the case in the master's thesis. However, with other solvers some of the instances did not converge using the Cartesian coordinate model of this thesis.
The second model uses spherical coordinates and exhibits better convexity properties than the previous one based on Cartesian coordinates. All of the problem instances converged using all the proposed solvers, and the quality of the solution was improved. CPU time for spherical coordinates increased in relation to the Cartesian coordinate model, due to large calculations involved, but the number of iterations needed to converge decreased considerably (for example, from a maximum of 192 iterations using the Cartesian coordinate model to a maximum of 84 iterations using the spherical coordinate model and the same LOQO solver).
These models resulted in two publications. The first one is a patent for an invention that uses the Cartesian coordinate model and orients the astigmatism gradient, which is useful when personalizing progressive lenses for real users. The second publication is a scientific article published in Optimization and Engineering that proposes the spherical coordinate model.
