Generation of representation models for complex systems using Lagrangian functions

In this article, a new methodology is presented to obtain representation models for a priori relation z = u(x1, x2, . . . ,xn) (1), with a known an experimental dataset zi; x1i ; x2i ; x3i ; . . . ; xni i=1;2;...;p· In this methodology, a potential energy is initially defined over each possible model for the relationship (1), what allows the application of the Lagrangian mechanics to the derived system. The solution of the Euler–Lagrange in this system allows obtaining the optimal solution according to the minimal action principle. The defined Lagrangian, corresponds to a continuous medium, where a n-dimensional finite elements model has been applied, so it is possible to get a solution for the problem solving a compatible and determined linear symmetric equation system. The computational implementation of the methodology has resulted in an improvement in the process of get representation models obtained and published previously by the authors.

Three-dimensional planar model estimation using multi-constraint knowledge based on k-means and RANSAC

Plane model extraction from three-dimensional point clouds is a necessary step in many different applications such as planar object reconstruction, indoor mapping and indoor localization. Different RANdom SAmple Consensus (RANSAC)-based methods have been proposed for this purpose in recent years. In this study, we propose a novel method-based on RANSAC called Multiplane Model Estimation, which can estimate multiple plane models simultaneously from a noisy point cloud using the knowledge extracted from a scene (or an object) in order to reconstruct it accurately. This method comprises two steps: first, it clusters the data into planar faces that preserve some constraints defined by knowledge related to the object (e.g., the angles between faces); and second, the models of the planes are estimated based on these data using a novel multi-constraint RANSAC. We performed experiments in the clustering and RANSAC stages, which showed that the proposed method performed better than state-of-the-art methods.

A three-dimensional representation method for noisy point clouds based on growing self-organizing maps accelerated on GPUs

The research described in this thesis was motivated by the need of a robust model capable of representing 3D data obtained with 3D sensors, which are inherently noisy. In addition, time constraints have to be considered as these sensors are capable of providing a 3D data stream in real time. This thesis proposed the use of Self-Organizing Maps (SOMs) as a 3D representation model. In particular, we proposed the use of the Growing Neural Gas (GNG) network, which has been successfully used for clustering, pattern recognition and topology representation of multi-dimensional data. Until now, Self-Organizing Maps have been primarily computed offline and their application in 3D data has mainly focused on free noise models, without considering time constraints. It is proposed a hardware implementation leveraging the computing power of modern GPUs, which takes advantage of a new paradigm coined as General-Purpose Computing on Graphics Processing Units (GPGPU). The proposed methods were applied to different problem and applications in the area of computer vision such as the recognition and localization of objects, visual surveillance or 3D reconstruction.