Exact computation of the medial axis of a polyhedron

doi:10.1016/j.cagd.2003.07.008

Computer Aided Geometric Design

Volume 21, Issue 1, January 2004, Pages 65-98

https://doi.org/10.1016/j.cagd.2003.07.008 Get rights and content

Abstract

We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding neighboring junctions along the seam curves. To speed up the computation, we have designed specialized algorithms for fast computation with algebraic curves and surfaces. These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

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morphological thinning-based methods that was first applied for discrete binary images [17], and improved by Huang et al. [22] and other related notions on the development of 3D thinning algorithm [23,48]. geometry-based methods using medial axis transformation for planar shape [18] including Voronoi diagrams [19] and Delaunay triangulation [47]. distance-based functions such as skeleton generation and centerline by the distance transform [40], skeletonization by discrete Euclidean distance maps [41], and Euclidean skeleton based on connectivity criterion [42].
Automatic building extraction and delineation from airborne LiDAR point cloud data of urban environments is still a challenging task due to the variety and complexity at which buildings appear. The Medial Axis Transform (MAT) is able to describe the geometric shape and topology of an object, but has never been applied for building roof outline extraction. It represents the shape of an object by its centerline, or skeleton structure instead of its boundary. Notably, end points of the MAT in principle coincide with corner points of building outlines. However, the MAT is sensitive to small boundary irregularities, which makes shape detection in airborne point clouds challenging. We propose a robust MAT-based method for detecting building corner points, which are then connected to form a building boundary polygon. First, we approximate the 2D MAT of a set of building edge points acquired by the alpha-shape algorithm to derive a so-called building roof skeleton. We then propose a hierarchical corner-aware segmentation to cluster skeleton points based on their properties which are the so-called separation angle, radius of the maximally inscribe circle, and defining edge point indices. From each segment, a corner point is then estimated by extrapolating the position of the zero radius inscribed circle based on the skeleton point positions within the segment. Our experiment uses point cloud datasets of Makassar, Indonesia and EYE-Amsterdam, The Netherlands. The average positional accuracy of the building outline results for Makassar and EYE-Amsterdam is 65 cm and 70 cm, respectively, which meet one-meter base map accuracy criteria. The results imply that skeletonization is a promising tool to extract relevant geometric information on e.g. building outlines even from far from perfect geographical point cloud data.
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Indoor navigation networks are the foundation of indoor localization-based services. To build navigation graph networks of buildings, the medial axis is an effective way to represent the paths of indoor spaces, as medial axis-based navigation networks are consistent with common human cognition of paths and satisfy the requirements of spatial queries in most indoor map-matching algorithms. The existing methods for generating medial axis-based indoor navigation networks compute the modified medial axis of indoor spaces to represent indoor paths. However, these methods cannot deal with rooms with irregular shapes or generate detours. Especially, these methods fail to generate path lines to represent floor-to-floor paths accurately. To address these issues, this study develops a novel approach called an improved generation of straight skeletons-based navigation networks (I-GSSNN) to produce straight skeletons based indoor navigation graph networks. The straight skeleton is adopted in I-GSSNN to catch the medial axis of indoor spaces. The Industry Foundation Classes (IFC) model, a widely-used format of Building Information Modeling (BIM), is used as input in the I-GSSNN. The I-GSSNN is a workflow that includes constructing path lines of corridors, connecting doors to corridors, generating floor-to-floor path lines, and removing redundant lines. Compared with existing methods, the I-GSSNN can handle rooms with any polygon shape and produces fewer detours. Novel algorithms are developed in the I-GSSNN to generate path lines for accurately representing stairs and elevators. An educational building is used as a case study to validate the result of I-GSSNN. The case study indicates that the I-GSSNN can generate a complete three-dimensional (3D) navigation network of the educational building. In addition, the accuracy test shows that the route length based on the generated navigation networks is very close to the real length. Significantly, the results show that the I-GSSNN can accurately generate full 3D navigation graph networks of buildings that can serve as indoor maps for indoor route planning and navigation and can also be used in code checking and pedestrian circulation analysis during building design.
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This is achieved by using algorithms such as distance transform, potential functions, thinning erosion and Voronoï diagrams. Even if effective, the medial axis is highly sensitive to small perturbations on boundaries of the shape, difficult and expensive to extract in 3D [25], and hard to store and manipulate [24]. Adding to this the fact that, in this work, we focus on beam-like structures, our attention is attracted by curve skeletons.
This paper presents a fully-automated reconstruction of beam-like CAD solid structures from 3D topology optimization (TO) results. Raw TO results are first processed to generate a triangulation that represents boundaries of the optimal shape derived. This triangulation is then smoothed and a curve skeletonization procedure is carried out to recover meaningful characteristics of this smoothed triangulation. The resulting skeleton, made with curvilinear geometry, is transformed into straight lines through a normalization process. These straight lines are used to generate a 3D beam structure. Thus, following these steps, a 3D beam structure is automatically derived from TO results. This 3D beam structure is meshed with beam finite elements and since TO non-design material is represented by 3D solid geometry, which is meshed using tetrahedron, the FEA beam structure needs to be rigidly connected with these tetrahedrons. Rigid connections between beam elements and 3D solid elements are ensured using specific FEA beam elements referred to as mini-beams. This results in a mixed-dimensional FEA model with beam and solid finite elements. Results obtained with this mixed-dimensional FEA model allow validating the beam structure obtained from TO results. Performance of the approach is demonstrated on several TO examples.

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^☆: Supported in part by ARO Contract DAAH04-96-1-0257, NSF award 9876914, ONR Young Investigator Award (N00014-97-1-0631), Intel and DOE ASCI grant.

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Exact computation of the medial axis of a polyhedron☆

Abstract

Computer-Aided Design

Computer-Aided Design

Computer Aided Geometric Design

Computer-Aided Design

Computing envelopes in four dimensions with applications

SIAM J. Comput.

A transformation for extracting new descriptors of shape

On Euclid's algorithm and the theory of subresultant

J. ACM

The Complexity of Robot Motion Planning

Accurate computation of the medial axis of a polyhedron

A hybrid approach for determinant signs of moderate-sized matrices

Internat. J. Comput. Geom. Appl.

Computer Algebra: Systems and Algorithms for Algebraic Computation

LAPACK: A portable linear algebra library for supercomputers

The eliminant of the equations of four quadric surfaces

Proc. London Math. Soc.

Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts

A sweepline algorithm for Voronoi diagrams

Algorithmica

Matrix Computations