A non-iterative method for robustly computing the intersections between a line and a curve or surface

The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a non-iterative method for computing intersections by solving a matrix singular value decomposition (SVD) and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one-shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton-Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree $q$ of the shape functions and is $2q \times 3q$ for curves and $6 q^2 \times 8 q^2$ for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high-order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/

Total Degree Formula for the Generic Offset to a Parametric Surface

We provide a resultant-based formula for the total degree w.r.t. the spatial variables of the generic offset to a parametric surface. The parametrization of the surface is not assumed to be proper.Comment: Preprint of an article to be published at the International Journal of Algebra and Computation, World Scientific Publishing, DOI:10.1142/S021819671100680

Reparametrizing Swung Surfaces over the Reals

Let K⊆R be a computable subfield of the real numbers (for instance, Q). We present an algorithm to decide whether a given parametrization of a rational swung surface, with coefficients in K(i), can be reparametrized over a real (i.e., embedded in R) finite field extension of K. Swung surfaces include, in particular, surfaces of revolution

Three-Dimensional Object Registration Using Wavelet Features

Recent developments in shape-based modeling and data acquisition have brought three-dimensional models to the forefront of computer graphics and visualization research. New data acquisition methods are producing large numbers of models in a variety of fields. Three-dimensional registration (alignment) is key to the useful application of such models in areas from automated surface inspection to cancer detection and surgery. The algorithms developed in this research accomplish automatic registration of three-dimensional voxelized models. We employ features in a wavelet transform domain to accomplish registration. The features are extracted in a multi-resolutional format, thus delineating features at various scales for robust and rapid matching. Registration is achieved by using a voting scheme to select peaks in sets of rotation quaternions, then separately identifying translation. The method is robust to occlusion, clutter, and noise. The efficacy of the algorithm is demonstrated through examples from solid modeling and medical imaging applications

Memetic electromagnetism algorithm for surface reconstruction with rational bivariate Bernstein basis functions

Surface reconstruction is a very important issue with outstanding applications in fields such as medical imaging (computer tomography, magnetic resonance), biomedical engineering (customized prosthesis and medical implants), computer-aided design and manufacturing (reverse engineering for the automotive, aerospace and shipbuilding industries), rapid prototyping (scale models of physical parts from CAD data), computer animation and film industry (motion capture, character modeling), archaeology (digital representation and storage of archaeological sites and assets), virtual/augmented reality, and many others. In this paper we address the surface reconstruction problem by using rational Bézier surfaces. This problem is by far more complex than the case for curves we solved in a previous paper. In addition, we deal with data points subjected to measurement noise and irregular sampling, replicating the usual conditions of real-world applications. Our method is based on a memetic approach combining a powerful metaheuristic method for global optimization (the electromagnetism algorithm) with a local search method. This method is applied to a benchmark of five illustrative examples exhibiting challenging features. Our experimental results show that the method performs very well, and it can recover the underlying shape of surfaces with very good accuracy.This research is kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project #TIN2012-30768, Toho University, and the University of Cantabria. The authors are particularly grateful to the Department of Information Science of Toho University for all the facilities given to carry out this work. We also thank the Editor and the two anonymous reviewers who helped us to improve our paper with several constructive comments and suggestions