Convergence analysis and validation of low cost distance metrics for computational cost reduction of the Iterative Closest Point algorithm

The Iterative Closest Point algorithm (ICP) is commonly used in engineering applications to solve the rigid registration problem of partially overlapped point sets which are pre-aligned with a coarse estimate of their relative positions. This iterative algorithm is applied in many areas such as the medicine for volumetric reconstruction of tomography data, in robotics to reconstruct surfaces or scenes using range sensor information, in industrial systems for quality control of manufactured objects or even in biology to study the structure and folding of proteins. One of the algorithm’s main problems is its high computational complexity (quadratic in the number of points with the non-optimized original variant) in a context where high density point sets, acquired by high resolution scanners, are processed. Many variants have been proposed in the literature whose goal is the performance improvement either by reducing the number of points or the required iterations or even enhancing the complexity of the most expensive phase: the closest neighbor search. In spite of decreasing its complexity, some of the variants tend to have a negative impact on the final registration precision or the convergence domain thus limiting the possible application scenarios. The goal of this work is the improvement of the algorithm’s computational cost so that a wider range of computationally demanding problems from among the ones described before can be addressed. For that purpose, an experimental and mathematical convergence analysis and validation of point-to-point distance metrics has been performed taking into account those distances with lower computational cost than the Euclidean one, which is used as the de facto standard for the algorithm’s implementations in the literature. In that analysis, the functioning of the algorithm in diverse topological spaces, characterized by different metrics, has been studied to check the convergence, efficacy and cost of the method in order to determine the one which offers the best results. Given that the distance calculation represents a significant part of the whole set of computations performed by the algorithm, it is expected that any reduction of that operation affects significantly and positively the overall performance of the method. As a result, a performance improvement has been achieved by the application of those reduced cost metrics whose quality in terms of convergence and error has been analyzed and validated experimentally as comparable with respect to the Euclidean distance using a heterogeneous set of objects, scenarios and initial situations.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

A methodology for evacuation route planning inside buildings using geospatial technology

Evacuation route planning is a fundamental task for building engineering projects. Safety regulations are established so that all occupants are driven on time out of a building to a secure place when faced with an emergency situation. As an example, Spanish building code requires the planning of evacuation routes on large and, usually, public buildings. Engineers often plan these routes on single building projects, repeatedly assigning clusters of rooms to each emergency exit in a trial-and-error process. But problems may arise for a building complex where distribution and use changes make visual analysis cumbersome and sometimes unfeasible. This problem could be solved by using well-known spatial analysis techniques, implemented as a specialized software able to partially emulate engineer reasoning. In this paper we propose and test an easily reproducible methodology that makes use of free and open source software components for solving a case study. We ran a complete test on a building floor at the University of Alicante (Spain). This institution offers a web service (WFS) that allows retrieval of 2D geometries from any building within its campus. We demonstrate how geospatial technologies and computational geometry algorithms can be used for automating the creation and optimization of evacuation routes. In our case study, the engineers’ task is to verify that the load capacity of each emergency exit does not exceed the standards specified by Spain’s current regulations. Using Dijkstra’s algorithm, we obtain the shortest paths from every room to the most appropriate emergency exit. Once these paths are calculated, engineers can run simulations and validate, based on path statistics, different cluster configurations. Techniques and tools applied in this research would be helpful in the design and risk management phases of any complex building project.