Geometri Uydurma için En İyi Eşlenik Gradyan Yönteminin Tespit Edilmesi

Abstract

In this study, it is aimed to determine the optimal conjugate gradient (CG) method for the geometry fitting of 2D measured profiles. To this end, the three well-known CG methods such as the Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel were employed. For testing those methods performances, the five primitive geometries accommodating circle, square, triangle, ellipse and rectangle were first built with a 3D printer, and then they were scanned with a coordinate measuring machine (CMM) to achieve their 2D profiles. The nonlinear least squares procedure was implemented to minimize the error between those measured data and modeled ones. An iterative line search was utilized for this task. The search direction was calculated using the above-mentioned CG methods. During the geometry fitting process, the number of function evaluations at each iteration were computed and the total number of function evaluations were set to be a performance measure of the CG method in question when it converged. By using these performance measures, the performance and data profiles were created to efficiently determine the optimal CG method. Based on performance profiles, it can be stated that the Fletcher-Reeves and Polak-Ribiere methods are the fastest ones on three test geometries out of five. In addition to that, all the CG methods were able to complete the geometry fitting of 80% of test geometries. On the other hand, by examining the data profiles, it was determined that the Polak-Ribiere and Hestenes-Stiefel methods achieve their maximum capabilities of the completing geometry fitting (i.e., 80%) with much lower number of function evaluations than the Fletcher-Reeves method. Besides, in most geometries, the Polak-Ribiere method outperformed the others, thereby it was determined to be the optimal one for the geometry fitting. As a conclusion, the reported results in this work might help the end-users who study on the CMM data processing to conduct an efficient geometry fitting.