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

In 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. Nev-ertheless, 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.EMR

Reconstruction of freeform surfaces for metrology

The application of freeform surfaces has increased since their complex shapes closely express a product's functional specifications and their machining is obtained with higher accuracy. In particular, optical surfaces exhibit enhanced performance especially when they take aspheric forms or more complex forms with multi-undulations. This study is mainly focused on the reconstruction of complex shapes such as freeform optical surfaces, and on the characterization of their form. The computer graphics community has proposed various algorithms for constructing a mesh based on the cloud of sample points. The mesh is a piecewise linear approximation of the surface and an interpolation of the point set. The mesh can further be processed for fitting parametric surfaces (Polyworks® or Geomagic®). The metrology community investigates direct fitting approaches. If the surface mathematical model is given, fitting is a straight forward task. Nonetheless, if the surface model is unknown, fitting is only possible through the association of polynomial Spline parametric surfaces. In this paper, a comparative study carried out on methods proposed by the computer graphics community will be presented to elucidate the advantages of these approaches. We stress the importance of the pre-processing phase as well as the significance of initial conditions. We further emphasize the importance of the meshing phase by stating that a proper mesh has two major advantages. First, it organizes the initially unstructured point set and it provides an insight of orientation, neighbourhood and curvature, and infers information on both its geometry and topology. Second, it conveys a better segmentation of the space, leading to a correct patching and association of parametric surfaces.EMR

Fast B-Spline 2D Curve Fitting for unorganized Noisy Datasets

In the context of coordinate metrology and reverse engineering, freeform curve reconstruction from unorganized data points still offers ways for improvement. Geometric convection is the process of fitting a closed shape, generally represented in the form of a periodic B-Spline model, to data points [WPL06]. This process should be robust to freeform shapes and convergence should be assured even in the presence of noise. The convection's starting point is a periodic B-Spline polygon defined by a finite number of control points that are distributed around the data points. The minimization of the sum of the squared distances separating the B-Spline curve and the points is done and translates into an adaptation of the shape of the curve, meaning that the control points are either inserted, removed or delocalized automatically depending on the accuracy of the fit. Computing distances is a computationally expensive step in which finding the projection of each of the data points requires the determination of location parameters along the curve. Zheng et al [ZBLW12] propose a minimization process in which location parameters and control points are calculated simultaneously. We propose a method in which we do not need to estimate location parameters, but rather compute topological distances that can be assimilated to the Hausdorff distances using a two-step association procedure. Instead of using the continuous representation of the B-Spline curve and having to solve for footpoints, we set the problem in discrete form by applying subdivision of the control polygon. This generates a discretization of the curve and establishes the link between the discrete point-to-curve distances and the position of the control points. The first step of the association process associates BSpline discrete points to data points and a segmentation of the cloud of points is done. The second step uses this segmentation to associate to each data point the nearest discrete BSpline segment. Results are presented for the fitting of turbine blades profiles and a thorough comparison between our approach and the existing methods is given [ZBLW12, WPL06, SKH98]

Comparison of tactile and chromatic confocal measurements of aspherical lenses for form metrology

Both contact and non-contact probes are often used in dimensional metrology applications, especially for roughness, form and surface profile measurements. To perform such kind of measurements with a nanometer level of accuracy, LNE (French National Metrology Institute (NMI)) has developed a high precision profilometer traceable to the SI meter definition. The architecture of the machine contains a short and stable metrology frame dissociated from the supporting frame. It perfectly respects Abbe principle. The metrology loop incorporates three Renishaw laser interferometers and is equipped either with a chromatic confocal probe or a tactile probe to achieve measurements at the nanometric level of uncertainty. The machine allows the in-situ calibration of the probes by means of a differential laser interferometer considered as a reference. In this paper, both the architecture and the operation of the LNE’s high precision profilometer are detailed. A brief comparison of the behavior of the chromatic confocal and tactile probes is presented. Optical and tactile scans of an aspherical surface are performed and the large number of data are processed using the L-BFGS (Limited memory-Broyden-Fletcher-Goldfarb-Shanno) algorithm. Fitting results are compared with respect to the evaluated residual errors which reflect the form defects of the surface.EMR

Dil davasında kati karara doğru:Türkçe'nin fesahatçılarla uzun cengi

Taha Toros Arşivi, Dosya No: 154-İsmail Habib Sevü

Contribution à la reconstruction de surfaces complexes à partir d'un grand flot de données non organisées pour la métrologie 3D.

Complex surfaces exhibit real challenges in regard to their design specification, their manufacturing, their measurement and the evaluation of their manufacturing defects. They are classified according to their geometric/shape complexity as well as to their required tolerance. Thus, the manufacturing and measurement processes used are selected accordingly. In order to transcribe significant information from the measured data, a data processing scheme is essential. Here, processing involves surface reconstruction in the aim of reconstituting the underlying geometry and topology to the points and extracting the necessary metrological information (form and/or dimensional errors). For the category of aspherical surfaces, where a mathematical model is available, the processing of the data, which are not necessarily organized, is done by fitting/associating the aspherical model to the data. The sought precision in optics is typically nanometric. In this context, we propose the L-BFGS optimization algorithm, first time used in metrological applications and which allows solving unconstrained, non-linear optimization problems precisely, automatically and fast. The L-BFGS method remains efficient and performs well even in the presence of very large amounts of data.In the category of general freeform surfaces and particularly turbine blades, the manufacturing, measurement and data processing are all at a different scale and require sub-micrometric precision. Freeform surfaces are generally not defined by a mathematical formula but are rather represented using parametric models such as B-Splines and NURBS. We expose a detailed state-of-the-art review of existing reconstruction algorithms in this field and then propose a new active contour deformation of B-Splines approach. The algorithm is independent of problems related to initialization and initial parameterization. Consequently, it is a new algorithm with promising results. We then establish a thorough study and a series of tests to show the advantages and limitations of our approach on examples of closed curves in the plane. We conclude the study with perspectives regarding improvements of the method and its extension to surfaces in 3D.Les surfaces complexes ont des applications dans divers domaines tels que ceux de la photonique, de l'énergie, du biomédical, du transport... Par contre, elles posent de véritables défis quant à leur spécification, fabrication et mesure ainsi que lors de l'évaluation de leur défaut de forme. Les processus de fabrication et de mesure de surfaces complexes sont fortement tributaires des dimensions, des tolérances et des formes spécifiées. Afin de rendre exploitable les informations données par le système de mesure, une étape importante de traitement s'impose. Il s'agit ici de la reconstruction de surfaces afin de reconstituer la géométrie et la topologie de la surface sous-jacente et d'en extraire les informations nécessaires pour des besoins de métrologie dimensionnelle (caractéristiques dimensionnelles et évaluation des défauts de forme). Dans la catégorie des surfaces asphériques pour lesquelles un modèle mathématique est associé, le processus de traitement de données géométriques, non nécessairement organisées, se fait par l'association du modèle aux données. Les résidus d'association recherchés en optique sont typiquement de l'ordre du nanomètre. Dans ce cadre, nous proposons l'utilisation de l'algorithme L-BFGS qui n'a encore jamais été utilisé en métrologie. Ce dernier permet de résoudre des problèmes d'optimisation non-linéaires, sans contraintes et d'une manière robuste, automatique et rapide. La méthode L-BFGS reste efficace pour des données contenant plusieurs millions de points. Dans la catégorie des surfaces gauches et notamment des aubes de turbines, la fabrication, la mesure et le traitement sont à une toute autre échelle, sub-micrométrique. Les surfaces gauches ne sont généralement pas définies par un modèle mathématique mais sont représentées par des modèles paramétriques de type B-Spline et/ou NURBS. Dans ce cadre, nous exposons un état de l'art détaillé et proposons une nouvelle approche itérative d'association B-Spline. L'algorithme s'affranchit de tous les problèmes liés à l'initialisation et au paramétrage initial. Par conséquent, un tel algorithme constitue une nouveauté dans ce domaine. Nous établissons une étude approfondie en évoquant les avantages et les limites actuelles de cette approche sur des exemples de courbes fermées en 2D. Nous complétons ensuite cette étude par des perspectives d'amélioration et de généralisation aux surfaces en 3D

Contribution to complex surfaces reconstruction from large and unorganized datasets for 3D metrology.

Les surfaces complexes ont des applications dans divers domaines tels que ceux de la photonique, de l'énergie, du biomédical, du transport... Par contre, elles posent de véritables défis quant à leur spécification, fabrication et mesure ainsi que lors de l'évaluation de leur défaut de forme. Les processus de fabrication et de mesure de surfaces complexes sont fortement tributaires des dimensions, des tolérances et des formes spécifiées. Afin de rendre exploitable les informations données par le système de mesure, une étape importante de traitement s'impose. Il s'agit ici de la reconstruction de surfaces afin de reconstituer la géométrie et la topologie de la surface sous-jacente et d'en extraire les informations nécessaires pour des besoins de métrologie dimensionnelle (caractéristiques dimensionnelles et évaluation des défauts de forme). Dans la catégorie des surfaces asphériques pour lesquelles un modèle mathématique est associé, le processus de traitement de données géométriques, non nécessairement organisées, se fait par l'association du modèle aux données. Les résidus d'association recherchés en optique sont typiquement de l'ordre du nanomètre. Dans ce cadre, nous proposons l'utilisation de l'algorithme L-BFGS qui n'a encore jamais été utilisé en métrologie. Ce dernier permet de résoudre des problèmes d'optimisation non-linéaires, sans contraintes et d'une manière robuste, automatique et rapide. La méthode L-BFGS reste efficace pour des données contenant plusieurs millions de points. Dans la catégorie des surfaces gauches et notamment des aubes de turbines, la fabrication, la mesure et le traitement sont à une toute autre échelle, sub-micrométrique. Les surfaces gauches ne sont généralement pas définies par un modèle mathématique mais sont représentées par des modèles paramétriques de type B-Spline et/ou NURBS. Dans ce cadre, nous exposons un état de l'art détaillé et proposons une nouvelle approche itérative d'association B-Spline. L'algorithme s'affranchit de tous les problèmes liés à l'initialisation et au paramétrage initial. Par conséquent, un tel algorithme constitue une nouveauté dans ce domaine. Nous établissons une étude approfondie en évoquant les avantages et les limites actuelles de cette approche sur des exemples de courbes fermées en 2D. Nous complétons ensuite cette étude par des perspectives d'amélioration et de généralisation aux surfaces en 3D.Complex surfaces exhibit real challenges in regard to their design specification, their manufacturing, their measurement and the evaluation of their manufacturing defects. They are classified according to their geometric/shape complexity as well as to their required tolerance. Thus, the manufacturing and measurement processes used are selected accordingly. In order to transcribe significant information from the measured data, a data processing scheme is essential. Here, processing involves surface reconstruction in the aim of reconstituting the underlying geometry and topology to the points and extracting the necessary metrological information (form and/or dimensional errors). For the category of aspherical surfaces, where a mathematical model is available, the processing of the data, which are not necessarily organized, is done by fitting/associating the aspherical model to the data. The sought precision in optics is typically nanometric. In this context, we propose the L-BFGS optimization algorithm, first time used in metrological applications and which allows solving unconstrained, non-linear optimization problems precisely, automatically and fast. The L-BFGS method remains efficient and performs well even in the presence of very large amounts of data.In the category of general freeform surfaces and particularly turbine blades, the manufacturing, measurement and data processing are all at a different scale and require sub-micrometric precision. Freeform surfaces are generally not defined by a mathematical formula but are rather represented using parametric models such as B-Splines and NURBS. We expose a detailed state-of-the-art review of existing reconstruction algorithms in this field and then propose a new active contour deformation of B-Splines approach. The algorithm is independent of problems related to initialization and initial parameterization. Consequently, it is a new algorithm with promising results. We then establish a thorough study and a series of tests to show the advantages and limitations of our approach on examples of closed curves in the plane. We conclude the study with perspectives regarding improvements of the method and its extension to surfaces in 3D

3D Measurement and Characterization of Ultra-precision Aspheric Surfaces

Aspheric surfaces have become widely used in various fields ranging from imaging systems to energy and biomedical applications. Although many researches have been conducted to address their manufacturing and measurement, there are still challenges in form characterization of aspheric surfaces considering a large number of data points. This paper presents a comparative study of 3D measurement and form characterization of an aspheric lens using tactile and optical single scanning probing systems. The design of the LNE high precision profilometer, traceable to standard references is presented. The measured surfaces are obtained from the aforementioned system. They are characterized with large number of data points for which a suitable process chain is deployed. The form characterization of the aspheric surfaces is based on surface fitting techniques by comparing the measured surface with the design surface. A comparative study of registration methods and non-linear Orthogonal Least-Squares fitting Methods is presented. Experimental results are analyzed and discussed to illustrate the effectiveness of the proposed approaches

Fast B-Spline 2D Curve Fitting for unorganized Noisy Datasets

National audienceno abstrac