8,542 research outputs found
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
Fast B-spline Curve Fitting by L-BFGS
We propose a novel method for fitting planar B-spline curves to unorganized
data points. In traditional methods, optimization of control points and foot
points are performed in two very time-consuming steps in each iteration: 1)
control points are updated by setting up and solving a linear system of
equations; and 2) foot points are computed by projecting each data point onto a
B-spline curve. Our method uses the L-BFGS optimization method to optimize
control points and foot points simultaneously and therefore it does not need to
perform either matrix computation or foot point projection in every iteration.
As a result, our method is much faster than existing methods
Optimization in Differentiable Manifolds in Order to Determine the Method of Construction of Prehistoric Wall-Paintings
In this paper a general methodology is introduced for the determination of
potential prototype curves used for the drawing of prehistoric wall-paintings.
The approach includes a) preprocessing of the wall-paintings contours to
properly partition them, according to their curvature, b) choice of prototype
curves families, c) analysis and optimization in 4-manifold for a first
estimation of the form of these prototypes, d) clustering of the contour parts
and the prototypes, to determine a minimal number of potential guides, e)
further optimization in 4-manifold, applied to each cluster separately, in
order to determine the exact functional form of the potential guides, together
with the corresponding drawn contour parts. The introduced methodology
simultaneously deals with two problems: a) the arbitrariness in data-points
orientation and b) the determination of one proper form for a prototype curve
that optimally fits the corresponding contour data. Arbitrariness in
orientation has been dealt with a novel curvature based error, while the proper
forms of curve prototypes have been exhaustively determined by embedding
curvature deformations of the prototypes into 4-manifolds. Application of this
methodology to celebrated wall-paintings excavated at Tyrins, Greece and the
Greek island of Thera, manifests it is highly probable that these
wall-paintings had been drawn by means of geometric guides that correspond to
linear spirals and hyperbolae. These geometric forms fit the drawings' lines
with an exceptionally low average error, less than 0.39mm. Hence, the approach
suggests the existence of accurate realizations of complicated geometric
entities, more than 1000 years before their axiomatic formulation in Classical
Ages
Fast and numerically stable circle fit
We develop a new algorithm for fitting circles that does not have drawbacks
commonly found in existing circle fits. Our fit achieves ultimate accuracy (to
machine precision), avoids divergence, and is numerically stable even when
fitting circles get arbitrary large. Lastly, our algorithm takes less than 10
iterations to converge, on average.Comment: 16 page
Use of lp norms in fitting curves and surfaces to data
Given a family of curves or surfaces in R s , an important problem is that of finding a member of the family which gives a "best" fit to m given data points. A criterion which is relevant to many application areas is orthogonal distance regression, where the sum of squares of the orthogonal distances from the data points to the surface is minimized. For example, this is important in metrology, where measured data from a manufactured part may have to be modelled. The least squares norm is not always suitable (for example, there may be wild points in the data, accept/reject decisions may be required, etc). So we use this to justify looking at the use of other l p norms. There are different ways to formulate the problem, and we examine methods which generalize in a natural way those available for least squares. The emphasis is on the efficient numerical treatment of the resulting problems
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