Computation of 3D skeletons using a generalized Delaunay triangulation technique

doi:10.1016/0010-4485(94)00025-9

Computer-Aided Design

Volume 27, Issue 9, September 1995, Pages 677-694

https://doi.org/10.1016/0010-4485(94)00025-9 Get rights and content

Abstract

The skeletal representation of 3D solids based on the medial axis transform has many applications in engineering. However, these applications are seldom realized, owing to the lack of viable computational techniques for generating skeletons. Such a computational technique, based on a notion of the generalized Voronoi diagram of a set of mixed-dimensional entities, is presented. It is shown that the generalized Voronoi diagram of a set of specific mixed dimensional set derived from the set of boundary entities of a polyhedron is, in fact, the exact skeleton of the polyhedron. Rather than the generalized Voronoi diagram being directly computed, its dual, an abstract Delaunay triangulation, is computed, from which the skeleton can be derived. An approach based on the Voronoi diagram of a well chosen representative point set on the boundary is also discussed as a special case; it is shown that the limitations of this approach are overcome by the generalization developed. Overall, it is argued that this generalization of the Voronoi diagram and the notion of the abstract generalized Delaunay triangulation are useful, and that they provide a viable approach to the computation of skeletons. Finally, details of the implementation, results, and an evaluation are presented.

References (33)

H Blum
Biological shape and visual science: Part 1
J. Theor. Biol.
(1973)
H Blum et al.
Shape description using weighted symmetric axis features
Pattern Recognition
(1978)
HN Gursoy et al.
Automated interrogation and adaptive subdivision of shape using medial axis transform
Adv. Engng Software and Workstations
(1991)
PV De Souza et al.
Computer location of medial axes
Comput. & Biomed. Res.
(1977)
FL Bookstein
The line skeleton
Comput. Graph & Image Proc.
(1979)
H Blum
Transformation for extracting new descriptors of shape
LR Nackman
Three-dimensional shape description using symmetric axis transform
DH Ballard
Model-directed detection of ribs in chestradiographs
KG Shin et al.
Robot path planning using goedesic and straight line segments with Voronoi diagrams
H Hjalmarsson
Tools for computer-based analysis of injection moulding designs

FB Prinz et al.

Geometric abstractions using medial axis transformation

M Reddy

Skeletons: generation, representation and applications to finite element modeling

V Srinivasan et al.

Automatic mesh generation using the symmetric axis transformation of polygonal domains

TKH Tam et al.

2d finite element mesh generation by medial axis subdivision

G Turkiyyah et al.

Generation and interpretation of finite element models in a knowledge based environment

NM Patrikalakis et al.

Shape interrogation by medial axis transform

Cited by (62)

Building outline extraction from als point clouds using medial axis transform descriptors
2020, Pattern Recognition
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.
Isogeometric generalized n th order perturbation-based stochastic method for exact geometric modeling of (composite) structures: Static and dynamic analysis with random material parameters
2019, Computer Methods in Applied Mechanics and Engineering
Citation Excerpt :
However, in deterministic numerical techniques, such as classical FEM, there is a huge gap between CAD and CAE [18,19] because the CAD and CAE models inherently use different types of geometric model. Some works that attempt to integrate CAD and CAE [20–23] truly improve it to an extent rather than solving it radically. So far as we know, they generally use lines or planes to represent (interpolate) the curve or surface approximately.
The contribution herein proposes an isogeometric generalized n th order perturbation-based stochastic method for exactly modeling/representing composite structures comprising of different materials with particular attention to both static and dynamic analysis of structures with random material characteristics. Herein, we exactly represent the geometric model via isogeometric analysis (IGA), such as exactly modeling the interfaces in composite structures with dissimilar materials and also continuous variable thicknesses that cannot be achieved by existing traditional methods such as FEM. Besides, only a limited or scant work has been conducted thus far with IGA and stochastic methods. We consider the uncertainties of elastic modulus and mass density into account as stochastic inputs in static and dynamic isogeometric stochastic analyses. Moreover, we derive and expand the IGA based random-input parameter and all state functions included in static and dynamic equilibrium equations around their expectations via a generalized n th order Taylor series using a small perturbation parameter $ε$ . In addition, we determine the probabilistic moments of the stochastic solution that satisfy the given accuracy requirement by expanding to n th order. The results obtained by the proposed method, the finite element method (wherever feasible), and Monte Carlo simulations for both benchmark and engineering applications verify the following: (a) the proposed methodology can achieve more accurate deterministic solutions with improved efficiency, thereby strengthening the effectiveness and efficiency brought by the stochastic method. This is in contrast to the FEM based method which weakens them, (b) on the other hand, it can more efficiently acquire reliable and more accurate stochastic results, both for expected values and standard deviations in the static and dynamic (free vibration) analyses. Note that the larger the problem size is, the more efficient the proposed method will be.
Optimal mesh topology generation for CFD
2017, Computer Methods in Applied Mechanics and Engineering
Citation Excerpt :
Therefore, various approaches have been proposed to approximate the medial axis. These include surface sampling approaches [5–8], thinning algorithms [9–12], distance field based algorithms [13–16] and the algebraic methods [17–21]. The Voronoi diagram and the distance field based methods are more widely used for block topology generation.
Multiblock structured mesh generation is amongst the most popular meshing techniques for flow simulations. However, the time consuming and user intensive process of blocking complex domains presents a major bottleneck in structured mesh generation. To this end, we have considered various automatic blocking strategies where some typical aeroengine domains have been showcased. An adjoint based error analysis procedure is then used to assess these methods. It is found that, in general, the medial axis based approaches provide optimal blocking and yields better accuracy in computing the functional of interest. This is because the medial axis based methods produce meshes which have better flow alignment especially in case of internal flows. A new hybrid blocking approach, combining the existing methods with the distance field isosurface is also presented to overcome the shortcomings of the current methods. Such a blocking technique has the potential to be applied to a wide range of flow domains.
Research on 3D medial axis transform via the saddle point programming method
2012, CAD Computer Aided Design
Citation Excerpt :
Dutta and Hoffmann [4] constructed the Voronoi diagrams of CSG objects (obtained by Boolean combinations of primitive objects such as a block, sphere or cylinder) between pairs of shape elements, and the MA can be generated by trimming the Voronoi diagrams. Reddy and Turkiyyah [5] proposed an algorithm for computing an abstract Delaunay triangulation of a polyhedral object, the geometric dual of which is the generalized Voronoi diagram. Some algorithms employed a set of sample points to handle freeform objects, and then approximated the MA by the Voronoi diagrams of these points.
The present paper investigates the 3D medial axis transform of objects bounded by freeform surfaces via the saddle point programming method, a mathematical programming approach used to identify the saddle points of a function. After exploring the local geometry and saddle point property of 3D medial axis transform, the mathematical programming method is employed to construct the saddle point programming models. Based on the optimality conditions that the optimal solutions should satisfy, a generic algorithm for computing various medial axis points is developed. In order to identify the junction points and localize the problem, the boundary and the skeletal curves are divided into skeletal segments, and it is proved to be efficient and accurate by numerical examples.
Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces
2010, CAD Computer Aided Design
Citation Excerpt :
The MAT has been used in pattern analysis and image analysis [4,5], finite element mesh generation [6,7], and path planning [8], to name a few applications. There exists a bijective mapping between the MAT and the object’s boundary [9] which means that for an object, there will be a unique MAT. Moreover, it is possible to reconstruct the object given its MAT [10].
This paper presents an algorithm for generating the Interior Medial Axis Transform (iMAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description of the iMAT. The algorithm generates the iMAT by a tracing technique that marches along the object’s boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiply-connected objects as well. Results of the implementation are provided.
A divide and conquer algorithm for medial surface calculation of planar polyhedra
2007, CAD Computer Aided Design
The Medial Axis Transform surface, (or MAT or MS) is proving to be a useful tool for several applications and geometric reasoning tasks. However, calculation of the MAT is a time-consuming task and the benefits of the mathematical-based tool are offset by the cost of the calculation. This paper presents a method for medial surface calculation which uses subdivision to simplify the problem and hence speed up the calculation, a so-called ‘divide-and-conquer’ approach. The basis for this is a modification of the dual structure of the original object. As the calculation proceeds this structure is broken up into sub-pieces each representing a simpler sub-part of the MAT. Perhaps more importantly, this method creates a correct node decomposition of the dual structure. The paper goes on to demonstrate some applications of the results for geometric tasks, specifically offsetting and model subdivision, which are normally expensive but are simpler based on the MAT calculation results.

View all citing articles on Scopus

View full text

ResearchComputation of 3D skeletons using a generalized Delaunay triangulation technique

Abstract

J. Theor. Biol.

Pattern Recognition

Adv. Engng Software and Workstations

Comput. & Biomed. Res.

Comput. Graph & Image Proc.

Transformation for extracting new descriptors of shape

Three-dimensional shape description using symmetric axis transform

Model-directed detection of ribs in chestradiographs

Robot path planning using goedesic and straight line segments with Voronoi diagrams

Tools for computer-based analysis of injection moulding designs