Decomposing non-manifold objects in arbitrary dimensions
Introduction
A manifold object is a subset of the Euclidean space for which the neighborhood of each internal point is locally equivalent to an open ball. Objects that do not fulfill this property at one or more points are called non-manifold objects. A geometric modeling system must deal with non-manifold objects, since they are important to provide a faithful description of real-world situations [7], [12], [34], [35], [36], [38], [39], [40]. On the other hand, most objects encountered in the applications satisfy the manifold condition at most points, in the sense that they contain a relatively small number of non-manifold geometric singularities. Non-manifold singularities in modeled objects may sometimes occur as a side-effect of feature extraction from images, of 3D reconstruction or as a byproduct of severe discretization. However, singularities are essential when, for instance, we choose to model the semantic content of an image [20] with an object of mixed dimensionality.
A possible approach for representing non-manifold objects consists of decomposing them into simpler parts, splitting an object at those elements (vertices, edges, faces, etc.) where singularities occur [7], [12], [14], [15], [35], [36]. The result of such a decomposition should be a collection of singularity-free components. Different components should be linked together at geometric elements where singularities occur. Fig. 1 depicts an example of a non-manifold object and of one of its possible decompositions. In order to be effective, the decomposition process should remove as many singularities as possible, while guaranteeing that the resulting components still represent meaningful parts of the object. A decomposition should not introduce artificial or arbitrary “cuts” through manifold parts. Under these assumptions, a decomposition into manifold components is possible, in general, only for two-dimensional complexes.
In three or higher dimensions a decomposition into manifold components may need to introduce artificial cuts through the object. For instance, Fig. 2a shows an example of a pinched pie which is described by a 3D complex of tetrahedra forming a fan around point p. This object is non-manifold and point p is a singularity. In order to eliminate the singularity, we necessarily have to cut the object along a manifold face, like triangle pqr (see Fig. 2b).
In dimension six or higher a decomposition into manifold components is not feasible in general, since the class of d-manifolds is not decidable for d⩾6 [19], [32]. Thus, it is not always convenient, and even not always possible, to remove all singularities from components. Here, we define a sound decomposition of d-dimensional non-manifold objects. Our decomposition is unique since it does not make any arbitrary choice in deciding where the object has to be decomposed, and it is natural, since it removes all singularities which can be removed by splitting the object only at non-manifold elements, i.e., without introducing artificial cuts.
We consider a description of non-manifold objects by using abstract simplicial complexes as basic modeling tools. In this way it is possible to study singularities from a purely combinatorial point of view. Moreover, combinatorial models are the necessary basis for designing effective data structures in solid modeling.
In order to define a sound decomposition, we consider first all the possible decompositions of a simplicial complex and we show that the resulting set is both a partially ordered set (poset) and a lattice. Then, we focus on a subset of all decompositions, that we call the subset of essential decompositions: essential decompositions do not cut the complex at manifold faces. We are interested in finding an essential decomposition, which is not arbitrary and removes as many singularities as possible. To this aim, we show that there exists a unique essential decomposition, which is the least upper-bound of the subset of essential decompositions in the lattice. We call this least upper-bound the standard decomposition of the original complex.
We show that the components of a standard decomposition, that we call initial quasi-manifolds, admit a local characterization in terms of combinatorial properties around each vertex. Thus, they are decidable in any dimension (in contrast with the class of manifolds which are not decidable in dimension ⩾6). We have compared initial quasi-manifolds with the well-known classes of regular, manifold, pseudomanifold, and quasi-manifold [24] complexes. We show that, up to dimension two, the class of initial quasi-manifold coincides with that of manifolds. In general, in dimension 3 and higher, an initial quasi-manifold is not always a manifold and not even a pseudomanifold. However, in dimension d⩾3, if an initial quasi-manifold is embeddable in , then must be a pseudomanifold complex. In fact, in the hyperplane supporting a (d−1)-face γ divides into two disjoint open half-spaces. In each half-space there can be only one d-simplex incident at γ from the given simplicial complex . The previous results are relevant for defining data structures for representing three-dimensional objects described either through their boundary or through a decomposition of their interior with tetrahedral cells.
The remainder of this paper is organized as follows. In Section 2, we review related work. In Section 3, we present some background notions in combinatorial topology. In Section 4, we analyze the set of all possible decompositions and define the poset structure for such a set. In Section 5, we introduce essential decompositions, the standard decomposition, and its components (i.e., initial quasi-manifolds). Next, we analyze initial quasi-manifolds and compare them with known classes of combinatorial objects (regular complexes, manifolds, pseudomanifolds, and quasi-manifolds). In Section 6, we discuss optimality issues for standard decomposition. In Section 7, we present an algorithm for computing the standard decomposition, illustrate it on some examples and show that worst-case complexity of the algorithm is slightly superlinear. Section 8 contains some concluding remarks, and discusses future work. Appendix A shows how to construct a three-dimensional complex which is an initial quasi-manifold, but not a pseudomanifold.
Section snippets
Related work
Motivations for developing effective representations for non-manifold objects have been pointed out by several authors [4], [17], [27], [36], [39], [40] in solid modeling. A considerable amount of work has been done on models and data structures for boundary representation of three-dimensional manifold and non-manifold objects. Some work also exists on dimension-independent descriptions for d-dimensional subdivided manifolds (see for instance [3], [33]). Since CAD and computer graphics have
Background
In this section, we summarize some background notions from algebraic and combinatorial topology. We refer to [13], [18], [37] for a more thorough treatment.
Abstract simplicial complexes. Let V be a finite set of elements, that we call vertices. An abstract simplicial complex with vertices in V is a subset of the set of the (non-empty) parts of V such that, for every vertex v∈V, we have that , and, if γ⊂V is an element of , then every subset of γ is also an element of . Each element of
Decompositions of a complex
A complex is a decomposition of another complex whenever can be obtained from by cutting along some faces. If is a decomposition of , then any other decomposition of will also be a decomposition for .
This fact induces a partial order over the set of all possible decompositions of a complex, in which is the minimum and the complex obtained by decomposing into the collection of its top simplices is the maximum. We will call this latter complex the totally exploded
The standard decomposition and initial quasi-manifolds
Usually one perceives non-manifold simplices as “joints” between manifold parts, and it looks reasonable to build a decomposition by splitting the complex at such joints. On the other hand, it does not seem reasonable to decompose some non-manifold situations, such as the one of the pinched pie in Fig. 2. In this section, we formally define this concept of “reasonable” decomposition and show, with Theorem 3, that the set of such decompositions admits a least upper-bound. We study the structure
Optimality of the standard decomposition
In this section we discuss optimality issues for the standard decomposition. Note that we came to the definition of standard decomposition by considering three requirements: (i) the decomposition must be essential, i.e., it must not contain cuts along manifold faces; (ii) the decomposition must eliminate all singularities, which can be eliminated without violating the previous constraint; and (iii) the decomposition must be unique, i.e., not arbitrary. Constraint (ii) above seems to be
An algorithm for computing the standard decomposition
In this section, we present a decomposition algorithm that builds the standard decomposition by splitting at vertices that violate the condition for initial quasi-manifold. The algorithm works iteratively on the vertices of the input complex and recursively on its dimension. The decomposition of either a 0-complex, or of an empty complex trivially coincides with the complex itself. The algorithm for computing is given by the pseudo-code in the Algorithm 1. This algorithm defines a
Concluding remarks
We have defined a mathematically sound decomposition of non-manifold objects in arbitrary dimensions into components that belong to a well-understood class, that we call initial quasi-manifolds. This class is a decidable superset of d-manifolds for d⩾3, and coincides with the class of d-manifolds for d⩽2. We have shown that our decomposition, that we have termed the standard decomposition, is unique, since it does not make any arbitrary choice in deciding where the object should be decomposed.
Acknowledgements
This work has been performed while Leila De Floriani has been visiting the Computer Science Department of the University of Maryland, College Park (USA). This work has been partially supported by the European Research Training Network “MINGLE—Multiresolution in Geometric Modelling,” contract number HPRN-CT-1999-00117. The support of the Italian Space Agency (ASI) under project “Augmented Reality for Teleoperation of Free Flying Robots” and of the Italian Ministry of Education University and
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