Jaime Puig-Pey

Iterative two-step genetic-algorithm-based method for efficient polynomial B-spline surface reconstruction

Surface reconstruction is a very challenging problem arising in a wide variety of applications such as CAD design, data visualization, virtual reality, medical imaging, computer animation, reverse engineering and so on. Given partial information about an unknown surface, its goal is to construct, to the extent possible, a compact representation of the surface model. In most cases, available information about the surface consists of a dense set of (either organized or scattered) 3D data points obtained by using scanner devices, a todays prevalent technology in many reverse engineering applications. In such a case, surface reconstruction consists of two main stages: (1) surface parameterization and (2) surface fitting. Both tasks are critical in order to recover surface geometry and topology and to obtain a proper fitting to data points. They are also pretty troublesome, leading to a high-dimensional nonlinear optimization problem. In this context, present paper introduces a new method for surface reconstruction from clouds of noisy 3D data points. Our method applies the genetic algorithm paradigm iteratively to fit a given cloud of data points by using strictly polynomial B-spline surfaces. Genetic algorithms are applied in two steps: the first one determines the parametric values of data points; the later computes surface knot vectors. Then, the fitting surface is calculated by least-squares through either SVD (singular value decomposition) or LU methods. The method yields very accurate results even for surfaces with singularities, concavities, complicated shapes or nonzero genus. Six examples including open, semi-closed and closed surfaces with singular points illustrate the good performance of our approach. Our experiments show that our proposal outperforms all previous approaches in terms of accuracy and flexibility.

Bézier curve and surface fitting of 3D point clouds through genetic algorithms, functional networks and least-squares approximation

This work concerns the problem of curve and surface fitting. In particular, we focus on the case of 3D point clouds fitted with Bezier curves and surfaces. Because these curves and surfaces are parametric, we are confronted with the problem of obtaining an appropriate parameterization of the data points. On the other hand, the addition of functional constraints introduces new elements that classical fitting methods do not account for. To tackle these issues, two Artificial Intelligence (AI) techniques are considered in this paper: (1) for the curve/surface parameterization, the use of genetic algorithms is proposed; (2) for the functional constraints problem, the functional networks scheme is applied. Both approaches are combined with the least-squares approximation method in order to yield suitable methods for Bezier curve and surface fitting. To illustrate the performance of those methods, some examples of their application on 3D point clouds are given.

Evaluation of the Run-Length Probability Distribution for CUSUM Charts: Assessing Chart Performance

This article provides a fast and accurate algorithm to compute the run-length probability distribution for cumulative sum charts to control process mean. This algorithm uses a fast and numerically stable recursive formula based on accurate Gaussian quadrature rules throughout the whole range of the computed run-length distribution and, therefore, improves the numerical efficiency and accuracy of existing methods. The algorithm may detect whether or not the geometric approximation is adequate and, when it is possible, it allows switching to the geometric recursion. The procedure may be applied not only to the normal distribution but also to nonsymmetric and long-tailed continuous distributions, some examples of which are provided. Methods to assess chart performance according to the run-length distribution, as well as some multivariate issues in statistical process control, are considered.

Particle Swarm Optimization for Bézier Surface Reconstruction

This work concerns the issue of surface reconstruction, that is, the generation of a surface from a given cloud of data points. Our approach is based on a metaheuristic algorithm, the so-called Particle Swarm Optimization. The paper describes its application to the case of Bezier surface reconstruction, for which the problem of obtaining a suitable parameterization of the data points has to be properly addressed. A simple but illustrative example is used to discuss the performance of the proposed method. An empirical discussion about the choice of the social and cognitive parameters for the PSO algorithm is also given.

Helical Curves on Surfaces for Computer-Aided Geometric Design and Manufacturing

This paper introduces a new method for generating the helical tool-paths for both implicit and parametric surfaces. The basic idea is to describe the helical curves as the solutions of an initial-value problem of ordinary differential equations. This system can be obtained from the fact that the helical curve exhibits a constant angle φ with an arbitrary given vector D, which is assumed to be the axis of the helical curve. The resulting system of differential equations is then integrated by applying standard numerical techniques. The performance of the proposed method is discussed by means of some illustrative examples of helical curves on parametric and implicit surfaces.

Computing the Run Length Probability Distribution for CUSUM Charts

It is widely recognized that the run length probability distribution of a cumulative sum (CUSUM) chart may be rather different from a geometric distribution. This is generally true for the left tail of the distribution, but, when the decision interval is large, it can apply to the whole distribution whether the process is in control or out of control. This paper provides a computer implementation of a fast and accurate algorithm to compute the run length probability distribution for CUSUM charts to monitor the process mean. The program may be used not only under the usual normality assumption but also for nonsymmetric and long-tailed continuous distributions.

Computing optimal adjustment schemes for the general tool-wear problem

An important problem in process adjustment using feedback is how often to sample the process and when and by how much to apply an adjustment. Minimum cost feedback schemes based on simple, but practically interesting, models for disturbances and dynamics have been discussed in several particular cases. The more general situation in which there may be measurement and adjustment errors, deterministic process drift, and costs of taking an observation, of making an adjustment, and of being off target, is considered in this article. Assuming all these costs to be known, a numerical method to minimize the overall expected cost is presented. This numerical method provides the optimal sampling interval, action limits, and amount of adjustment; and the resulting average adjustment interval, mean squared deviation from target, and minimum overall expected cost. When the costs of taking an observation, of making an adjustment, and of being off target are not known, the method can be used to choose a particular schem...

Some applications of scalar and vector fields to geometric processing of surfaces

In this paper, two geometric processing problems are considered: (1) point on a surface nearest to an external point, and (2) silhouette curve of a surface when observed from a given point. Problem (1) is solved by constructing gradient curves on the surface associated with a distance scalar field. Problem (2) appears as the intersection of surfaces (implicit case), or as tracing a plane curve (parametric case). Formulations are geometric-differential, and lead to explicit, first-order systems of ordinary differential equations (ODEs), with initial conditions that can be efficiently integrated by standard numerical methods. The methodology allows us to deal with both implicit and parametric representations, these having any functional structure for which the differential statements are meaningful.

A Differential Method for Parametric Surface Intersection

In this paper, a new method for computing the intersection of parametric surfaces is proposed. In our approach, this issue is formulated in terms of an initial value problem of first-order ordinary differential equations (ODEs), which are to be numerically integrated. In order to determine the initial value for this system, a simple procedure based on the vector field associated with the gradient of the distance function between points lying on each of the parametric surfaces is described. Such a procedure yields a starting point on the nearest branch of the intersection curve. The performance of the presented method is analyzed by means of some illustrative examples that contain many of the most common features found in parametric surface intersection problems.

An accurate algorithm to compute the run length probability distribution, and its convolutions, for a Cusum chart to control normal mean

A fast and accurate algorithm to compute the run length probability distribution for a Cusum chart to control process mean is provided. The algorithm is based on an integral equation for the characteristic function of the run length and uses efficient and accurate formulas based on Gaussian quadrature rules. The algorithm also provides the probability distribution for the sum of r independent and identically distributed run lengths, where r is a positive integer. Examples are provided to demonstrate the excellent performance of the algorithm, even for very extreme combinations of the design parameters, and to compare this performance with that of previous methods.