Hichem Barki

Contributing vertices-based Minkowski sum computation of convex polyhedra

Minkowski sum is an important operation. It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. We propose a novel algorithm, named the Contributing Vertices-based Minkowski Sum (CVMS) algorithm for the computation of the Minkowski sum of convex polyhedra. The CVMS algorithm allows to easily obtain all the facets of the Minkowski sum polyhedron only by examining the contributing vertices-a concept we introduce in this work, for each input facet. We exploit the concept of contributing vertices to propose the Enhanced and Simplified Slope Diagram-based Minkowski Sum (ESSDMS) algorithm, a slope diagram-based Minkowski sum algorithm sharing some common points with the approach proposed by Wu et al. [Wu Y, Shah J, Davidson J. Improvements to algorithms for computing the Minkowski sum of 3-polytopes. Comput Aided Des. 2003; 35(13): 1181-92]. The ESSDMS algorithm does not embed input polyhedra on the unit sphere and does not need to perform stereographic projections. Moreover, the use of contributing vertices brings up more simplifications and improves the overall performance. The implementations for the mentioned algorithms are straightforward, use exact number types, produce exact results, and are based on CGAL, the Computational Geometry Algorithms Library. More examples and results of the CVMS algorithm for several convex polyhedra can be found at http://liris.cnrs.fr/hichem.barki/mksum/CVMS-convex.

Contributing vertices-based Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron

We present an original approach for the computation of the Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron, without decomposition and union steps—that constitute the bottleneck of convex decomposition-based algorithms. A non-convex polyhedron without fold is a polyhedron whose boundary is completely recoverable from three orthographic projections defined by three orthogonal basis vectors in ℝ(su3). First, we generate a superset of the Minkowski sum facets using the concept of contributing vertices we accommodate for a non-convex-convex pair of polyhedra. The generated superset guarantees that its envelope is the boundary of the Minkowski sum polyhedron. Secondly, we extract the Minkowski sum facets and handle the intersections among the superset facets by using 3D envelope computation. Our approach is limited to non-convex polyhedra without fold because of the use of 3D envelope computation to recover the Minkowski sum boundary. Models with holes are not handled by our method. The implementation of our algorithm uses exact number types, produces exact results, and is based on CGAL, the Computational Geometry Algorithms Library.

Contributing vertices-based Minkowski sum of a nonconvex--convex pair of polyhedra

The exact Minkowski sum of polyhedra is of particular interest in many applications, ranging from image analysis and processing to computer-aided design and robotics. Its computation and implementation is a difficult and complicated task when nonconvex polyhedra are involved. We present the NCC-CVMS algorithm, an exact and efficient contributing vertices-based Minkowski sum algorithm for the computation of the Minkowski sum of a nonconvex--convex pair of polyhedra, which handles nonmanifold situations and extracts eventual polyhedral holes inside the Minkowski sum outer boundary. Our algorithm does not output boundaries that degenerate into a polyline or a single point. First, we generate a superset of the Minkowski sum facets through the use of the contributing vertices concept and by summing only the features (facets, edges, and vertices) of the input polyhedra which have coincident orientations. Secondly, we compute the 2D arrangements induced by the superset triangles intersections. Finally, we obtain the Minkowski sum through the use of two simple properties of the input polyhedra and the Minkowski sum polyhedron itself, that is, the closeness and the two-manifoldness properties. The NCC-CVMS algorithm is efficient because of the simplifications induced by the use of the contributing vertices concept, the use of 2D arrangements instead of 3D arrangements which are difficult to maintain, and the use of simple properties to recover the Minkowski sum mesh. We implemented our NCC-CVMS algorithm on the base of CGAL and used exact number types. More examples and results of the NCC-CVMS algorithm can be found at: http://liris.cnrs.fr/hichem.barki/mksum/NCC-CVMS

Emergency Response in Complex Buildings: Automated Selection of Safest and Balanced Routes

The extreme importance of emergency response in complex buildings during natural and human-induced disasters has been widely acknowledged. In particular, there is a need for efficient algorithms for finding safest evacuation routes, which would take into account the 3-D structure of buildings, their relevant semantics, and the nature and shape of hazards. In this article, we propose algorithms for safest routes and balanced routes in buildings, where an extreme event with many epicenters is occurring. In a balanced route, a trade-off between route length and hazard proximity is made. The algorithms are based on a novel approach that integrates a multiattribute decision-making technique, Dijkstras classical algorithm and the introduced hazard proximity numbers, hazard propagation coefficient and proximity index for a route.

Exact, robust, and efficient regularized Booleans on general 3D meshes

Computing Boolean operations (Booleans) of 3D polyhedra/meshes is a basic and essential task in many domains, such as computational geometry, computer-aided design, and constructive solid geometry. Besides their utility and importance, Booleans are challenging to compute when dealing with meshes, because of topological changes, geometric degeneracies, etc. Most prior art techniques either suffer from robustness issues, deal with a restricted class of input/output meshes, or provide only approximate results. We overcome these limitations and present an exact and robust approach performing on general meshes, required to be only closed and orientable. Our method is based on a few geometric and topological predicates that allow to handle all input/output cases considered as degenerate in existing solutions, such as voids, non-manifold, disconnected, and unbounded meshes, and to robustly deal with special input configurations. Our experimentation showed that our more general approach is also more robust and more efficient than Maya’s implementation (×3), CGAL’s robust Nef polyhedra (×5), and recent plane-based approaches. Finally, we also present a complete benchmark intended to validate Boolean algorithms under relevant and challenging scenarios, and we successfully ascertain both our algorithm and implementation with it.

Fast simplification with sharp feature preserving for 3D point clouds

This paper presents a fast point cloud simplification method that allows to preserve sharp edge points. The method is based on the combination of both clustering and coarse-to-fine simplification approaches. It consists to firstly create a coarse cloud using a clustering algorithm. Then each point of the resulting coarse cloud is assigned a weight that quantifies its importance, and allows to classify it into a sharp point or a simple point. Finally, both kinds of points are used to refine the coarse cloud and thus create a new simplified cloud characterized by high density of points in sharp regions and low density in flat regions. Experiments show that our algorithm is much faster than the last proposed simplification algorithm [1] which deals with sharp edge points preserving, and still produces similar results.

Re-parameterization reduces irreducible geometric constraint systems

You recklessly told your boss that solving a non-linear system of size n ( n unknowns and n equations) requires a time proportional to n , as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton-Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k ? n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton-Raphson, homotopy, and also p -adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k ? n key unknowns, the cost of a Newton iteration becomes O ( k n 2 ) instead of O ( n 3 ) . Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems. A new re-parameterization for reducing and unlocking irreducible geometric systems.No need for the values of the key unknowns and no limit on their number.Enabling the usage of decomposition methods on irreducible re-parameterized systems.Usage at the lowest linear Algebra level and significant performance improvement.Benefits for numerous solvers (Newton-Raphson, homotopy, p -adic methods, etc.)

Dupin cyclide blends between non-natural quadrics of revolution and concrete shape modeling applications

Abstract In this work, we focus on the blending of two quadrics of revolution by two patches of Dupin cyclides. We propose an algorithm for the blending of non-natural quadrics of revolution by decomposing the blending operation into two complementary sub-blendings, each of which is a Dupin cyclide-based blending between one of the two quadrics and a circular cylinder, thus enabling the direct computation of the two Dupin cyclide patches and offering better flexibility for shape composition. Our approach uses rational quadric Bezier curves to model the relevant arcs of the principal circles of Dupin cyclides. It is quite general and we have successfully used it for the blending of several non-natural surfaces of revolution, such as paraboloids, hyperboloids, tori, catenaries, and pseudospheres. Two complete examples of 3D shape modeling, representing a satellite antenna and a hippocampus are presented to show how quadrics and Dupin cyclide patches can be combined to model concrete objects.

New Geometric Constraint Solving Formulation: Application to the 3D Pentahedron

Geometric Constraint Solving Problems (GCSP) are nowadays routinely investigated in geometric modeling. The 3D Pentahedron problem is a GCSP defined by the lengths of its edges and the planarity of its quadrilateral faces, yielding to an under-constrained system of twelve equations in eighteen unknowns. In this work, we focus on solving the 3D Pentahedron problem in a more robust and efficient way, through a new formulation that reduces the underlying algebraic formulation to a well-constrained system of three equations in three unknowns, and avoids at the same time the use of placement rules that resolve the under-constrained original formulation. We show that geometric constraints can be specified in many ways and that some formulations are much better than others, because they are much smaller and they avoid spurious degenerate solutions. Several experimentations showing a considerable performance enhancement (×42) are reported in this paper to consolidate our theoretical findings.

3D Capture Techniques for BIM Enabled LCM

As a special kind of Product Life cyle Management (PLM), Building Life cycle Management (BLM) is a centric activity for facility owners and managers. This fact motivates the adoption of Building Information Modeling (BIM) approaches as a way to achieve smart BLM strategies for cost reduction, facility knowledge management, and project synchronization among the different stakeholders. Unfortunately, the current BIM state of the art is tailored towards the management of new projects, while ongoing and completed AEC projects could hugely benefit from BIM integration for better BLM strategies. In this regards, it is absolutely necessary to acquire knowledge about the dynamic facility aspects (crowd movement, as-is updates, etc.). Up-to-date, 3D capture appears to be the only reliable way to cope with such situation. In this paper, we analyze 3D capture techniques, ranging from photogrammetry to 3D scanning, with an emphasis on helping 3D capture practitioners to make critical decisions about the choice of adequate acquisition technologies for a particular application. We discuss 3D capture techniques by exposing their pros and cons, according to several relevant criteria, and synthesize our analysis by developing a set of recommendations to enhance the life expectancy of buildings via the integration of BIM into Life Cycle Management (LCM) of the built environment and its buildings.