Andrés Iglesias

Particle swarm optimization for non-uniform rational B-spline surface reconstruction from clouds of 3D data points

This work investigates the use of particle swarm optimization (PSO) to recover the shape of a surface from clouds of (either organized or scattered) noisy 3D data points, a challenging problem that appears recurrently in a wide range of applications such as CAD design, data visualization, virtual reality, medical imaging and movie industries. In this paper, we apply a PSO approach in order to reconstruct a non-uniform rational B-spline (NURBS) surface of a certain order from a given set of 3D data points. In this case, surface reconstruction consists of two main tasks: (1) surface parameterization and (2) surface fitting. Both tasks are critical but also troublesome, leading to a high-dimensional non-linear optimization problem. Our method allows us to obtain all relevant surface data (i.e., parametric values of data points, knot vectors, control points and their weights) in a shot and no pre-/post-processing is required. Furthermore, it yields very good results even in presence of problematic features, such as multi-branches, high-genus or self-intersections. Seven examples including open, semiclosed, closed, zero-genus, high-genus surfaces and real-world scanned objects, described in free-form, parametric and implicit forms illustrate the good performance of our approach and its superiority over previous approaches in terms of accuracy and generality.

Efficient particle swarm optimization approach for data fitting with free knot B-splines

Data fitting through B-splines improves dramatically if the knots are treated as free variables. However, in that case the approximation problem becomes a very difficult continuous multimodal and multivariate nonlinear optimization problem. In a previous paper, Yoshimoto et al. (2003) [18] solved this problem for explicit curves by using a real-code genetic algorithm. However, the method does not really deal with true multiple knots, so the cases of data with underlying functions having discontinuities and cusps are not fully addressed. In this paper, we present a new method to overcome such a limitation. The method applies the particle swarm optimization (PSO) paradigm to compute an appropriate location of knots automatically. Our scheme yields very accurate results even for curves with singularities and/or cusps. Several experiments show that our proposal is very efficient and improves previous results (including those by Yoshimoto et al. (2003) in [18]) significantly in terms of data points error, AIC and BIC criteria. Furthermore, the important case of true multiple knots is now satisfactorily solved.

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.

A new iterative mutually coupled hybrid GA-PSO approach for curve fitting in manufacturing

Fitting data points to curves (usually referred to as curve reconstruction) is a major issue in computer-aided design/manufacturing (CAD/CAM). This problem appears recurrently in reverse engineering, where a set of (possibly massive and noisy) data points obtained by 3D laser scanning have to be fitted to a free-form parametric curve (typically a B-spline). Despite the large number of methods available to tackle this issue, the problem is still challenging and elusive. In fact, no satisfactory solution to the general problem has been achieved so far. In this paper we present a novel hybrid evolutionary approach (called IMCH-GAPSO) for B-spline curve reconstruction comprised of two classical bio-inspired techniques: genetic algorithms (GA) and particle swarm optimization (PSO), accounting for data parameterization and knot placement, respectively. In our setting, GA and PSO are mutually coupled in the sense that the output of one system is used as the input of the other and vice versa. This coupling is then repeated iteratively until a termination criterion (such as a prescribed error threshold or a fixed number of iterations) is attained. To evaluate the performance of our approach, it has been applied to several illustrative examples of data points from real-world applications in manufacturing. Our experimental results show that our approach performs very well, being able to reconstruct with very high accuracy extremely complicated shapes, unfeasible for reconstruction with current methods.

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.

Computer graphics for water modeling and rendering: a survey

A key topic in computer graphics is the realistic representation of natural phenomena. Among the natural objects, one of the most interesting (and most difficult to deal with) is water. Its inherent complexity, far beyond that of most artificial objects, represents an irresistible challenge for the computer graphics world. Thus, during the last two decades we have witnessed an increasing number of papers addressing this problem from several points of view. However, the computer graphics community still lacks a survey classifying the vast literature on this topic, which is certainly unorganized and dispersed and hence, difficult to follow. This paper aims to fill this gap by offering a historical survey on the most relevant computer graphics techniques developed during the 1980s and 1990s for realistic modeling, rendering and animation of water.

Extending Neural Networks for B-Spline Surface Reconstruction

Recently, a new extension of the standard neural networks, the so-called functional networks, has been described [5]. This approach has been successfully applied to the reconstruction of a surface from a given set of 3D data points assumed to lie on unknown Bezier [17] and B-spline tensor-product surfaces [18]. In both cases the sets of data were fitted using Bezier surfaces. However, in general, the Bezier scheme is no longer used for practical applications. In this paper, the use of B-spline surfaces (by far, the most common family of surfaces in surface modeling and industry) for the surface reconstruction problem is proposed instead. The performance of this method is discussed by means of several illustrative examples. A careful analysis of the errors makes it possible to determine the number of B-spline surface fitting control points that best fit the data points. This analysis also includes the use of two sets of data (the training and the testing data) to check for overfitting, which does not occur here.

A New Artificial Intelligence Paradigm for Computer-Aided Geometric Design

Functional networks is a powerful and recently introduced Artificial Intelligence paradigm which generalizes the standard neural networks. In this paper functional networks are used to fit a given set of data from a tensor product parametric surface. The performance of this method is illustrated for the case of BEzier surfaces. Firstly, we build the simplest functional network representing such a surface, and then we use it to determine the degree and the coefficients of the bivariate polynomial surface that fits the given data better. To this aim, we calculate the mean and the root mean squared errors for different degrees of the approximating polynomial surface, which are used as our criterion of a good fitting. In addition, functional networks provide a procedure to describe parametric tensor product surfaces in terms of families of chosen basis functions. We remark that this new approach is very general and can be applied not only to BEzier but also to any other interesting family of tensor product surfaces.

Some characterizations of families of surfaces using functional equations

In this article functional equations are used to characterize some families of surfaces. First, the most general surfaces in implicit form <italic>f</italic>(<italic>x, y, z</italic>) = 0, such that any arbitrary intersection with the planes <italic>z</italic> = <italic>z</italic><subscrpt>0</subscrpt>, <italic>y</italic> = <italic>y</italic><subscrpt>0</subscrpt>, and <italic>x</italic> = <italic>x</italic><subscrpt>0</subscrpt> are linear combinations of sets of functions of the other two variables, are characterized. It is shown that only linear combinations of tensor products of univariate functions are possible for <italic>f</italic><italic>x</italic>, <italic>y</italic>, <italic>z</italic>). Second, we obtain the most general families of surfaces in explicit form such that their intersections with planes parallel to the planes <italic>y</italic> = 0 and <italic>x</italic> = 0 belong to two, not necessarily equal, parametric families of curves. Finally, functional equations are used to analyze the uniqueness of representation of Gordon-Coon surfaces. Some practical examples are used to illustrate the theoretical results.

Firefly Algorithm for Explicit B-Spline Curve Fitting to Data Points

This paper introduces a new method to compute the approximating explicit B-spline curve to a given set of noisy data points. The proposed method computes all parameters of the B-spline fitting curve of a given order. This requires to solve a difficult continuous, multimodal, and multivariate nonlinear least-squares optimization problem. In our approach, this optimization problem is solved by applying the firefly algorithm, a powerful metaheuristic nature-inspired algorithm well suited for optimization. The method has been applied to three illustrative real-world engineering examples from different fields. Our experimental results show that the presented method performs very well, being able to fit the data points with a high degree of accuracy. Furthermore, our scheme outperforms some popular previous approaches in terms of different fitting error criteria.