Angel Cobo

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.

Working with differential, functional and difference equations using functional networks

Abstract In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multiargument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.

Orthogonal sets and polar methods in linear algebra : applications to matrix calculations, systems of equations, inequalities, and linear programming

LINEAR SPACES AND SYSTEMS OF EQUATIONS. Basic Concepts. Orthogonal Sets. Matrix Calculations Using Orthogonal Sets. More Applications of Orthogonal Sets. Orthogonal Sets and Systems of Linear Equations. CONES AND SYSTEMS OF INEQUALITIES. Polyhedral Convex Cones. Polytopes and Polyhedra. Cones and Systems of Inequalities. LINEAR PROGRAMMING. An Introduction to Linear Programming. The Exterior Point Method. APPLICATIONS. Applications. Appendices. References. Index.

An Orthogonally Based Pivoting Transformation of Matrices and Some Applications

In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, and Pruneda [Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming, John Wiley, New York, 1999] and its multiple applications. The meaning of each sequential tableau appearing during the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors and its complement. This transformation, which is based on the orthogonality concept, allows us to solve many problems of linear algebra, such as calculating the inverse and the determinant of a matrix, updating the inverse or the determinant of a matrix after changing a row (column), determining the rank of a matrix, determining whether or not a set of vectors is linearly independent, obtaining the intersection of two linear subspaces, solving systems of linear equations, etc. When the process is applied to inverting a matrix and calculating its determinant, not only is the inverse of the final matrix obtained, but also the inverses and the determinants of all its block main diagonal matrices, all without extra computations.

A general framework for functional networks

In this paper, we introduce functional networks as a generalization and extension of the standard neural networks in the sense that every problem that can be solved by a neural network can also be formulated by a functional network. But, more importantly, we give examples of problems that cannot be solved using neural networks but can be naturally formulated using functional networks. Functional networks are defined as a collection of connected functional units on a set of nodes. A functional unit or neuron connects input nodes to output nodes. The values of the output nodes are calculated from the values of the input nodes by given functions of one or several arguments. The main differences with neural networks are that (a) the neural functions can be multivariate and can be different from neuron to neuron (in which case, no weights are necessary, because they subsume by the different functions) and (b) the neuron outputs can be coupled, that is, coincident. This mathematical model of functional networks parallels printed circuit boards with electronic components, thus giving an intuitive interpretation to functional networks and an interesting and natural additional application. The existence of functional units with common outputs leads to functional equations whose solution can lead to substantial simplification of the initial topology of the network and the neural functions involved. Two types of functional networks (the one-layer and serial functional networks) are discussed in detail. For the one-layer functional networks, a very simple simplification algorithm is given. For the serial functional networks, systems of functional equations are obtained. The methods are illustrated by several examples of applications.

A Minimax Method for Learning Functional Networks

In this paper, a minimax method for learning functional networks is presented. The idea of the method is to minimize themaximum absolute error between predicted and observed values. In addition, the invertible functions appearing in the modelare assumed to be linear convex combinations of invertible functions. This guarantees the invertibilityof the resulting approximations. The learning method leads to a linear programming problem and then: (a) the solution isobtained in a finite number of iterations, and (b) the global optimum is attained. The method is illustrated withseveral examples of applications, including the Hénon and Lozi series. The results show that the method outperforms standard least squares direct methods.

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.

Functional Networks: A New Network‐Based Methodology

In this article we give a general methodology to build and work with functional networks, a network-based alternative to the neural networks paradigm. In functional networks, neural functions are allowed to be not only multivariate but also truly multiargument and different for all neurons. Thus neural functions instead of weights are learned. In addition, outputs coming from different neurons can be connected, that is, forced to output the same values. The topology and neuron functions of functional networks can be selected based on data, domain knowledge, or a combination of the two. Functional equations play an important role in functional networks, since the preceding types of connections lead to functional equations that impose a substantial reduction in the degrees of freedom of the initial neural functions. Some methods are given to obtain equivalent functional and differential equations, and they are applied to approximating the solutions of differential equations problems. The examples of an associative operator, a cantilever beam, and a mass supported by two springs and a viscous damper are given to illustrate the methods and show their power.

SOME LEARNING METHODS IN FUNCTIONAL NETWORKS

This article discusses some methods for learning functional networks. After a short introduction and motivation of functional networks using a CAD problem, 4 steps used in learning functional networks are described: 1) selection of the initial topology of the network, which is derived from the physical properties of the problem being modeled; 2) simplification of this topology, using functional equations; 3) estimation of the parameters or weights, using least squares and minimax methods; and 4) selection of the subset of basic functions leading to the best fit to available data, using the minimum-description-length principle. Several examples are presented illustrating the learning procedure, including the use of a separable functional network to recover the missing data of the significant wave height records in 2 different locations, based on a complete record from a third location where the record is complete.

Evaluation of the interactivity of students in virtual learning environments using a multicriteria approach and data mining

This work seeks to provide a new multicriteria approach to evaluate and classify the level of interactivity of students in learning management systems (LMS). We describe, step by step, the complete methodological development process of the evaluation model as well as detailing the results obtained when applying it to a higher education teaching experience. This research demonstrates that the combined use of multicriteria decision methodologies and data mining prove to be particularly suitable for identifying behavioural patterns of the users through the analysis of records generated in LMS. The results reveal that the behavioural patterns in LMS offer certain indicators as to students’ academic performance, although the study does not permit to state that those students who adopt passive attitudes in these spheres may necessarily produce low academic performance.