Annie Hui

Data structures for simplicial complexes: an analysis and a comparison

In this paper, we review, analyze and compare representations for simplicial complexes. We classify such representations, based on the dimension of the complexes they can encode, into dimension-independent structures, and data structures for three- and for two-dimensional simplicial complexes. We further classify the data structures in each group according to the basic kinds of the topological entities they represent. We present a description of each data structure in terms of the entities and topological relations encoded, and we evaluate it based on its expressive power, on its storage cost and on the efficiency in supporting navigation inside the complex, i.e., in retrieving topological relations not explicitly encoded. We compare the various data structures inside each category based on the above features.

A scalable data structure for three-dimensional non-manifold objects

In this paper, we address the problem of representing and manipulating non-manifold, mixed-dimensional objects described by three-dimensional simplicial complexes embedded in the 3D Euclidean space. We describe the design and the implementation of a new data structure, that we call the non-manifold indexed data structure with adjacencies (NMIA), which can represent any three-dimensional Euclidean simplicial complex compactly, since it encodes only the vertices and the top simplexes of the complex plus a restricted subset of topological relations among simplexes. The NMIA structure supports efficient traversal algorithms which retrieve topological relations in optimal time, and it scales very well to the manifold case. Here, we sketch traversal algorithms, and we compare the NMIA structure with data structures for manifold and regular 3D simplicial complexes.

A Dimension-Independent Data Structure for Simplicial Complexes

We consider here the problem of representing non-manifold shapes discretized as d-dimensional simplicial Euclidean complexes. To this aim, we propose a dimension-independent data structure for simplicial complexes, called the Incidence Simplicial (IS) data structure, which is scalable to manifold complexes, and supports efficient navigation and topological modifications. The IS data structure has the same expressive power and exibits performances in query and update operations as the incidence graph, a widely-used representation for general cell complexes, but it is much more compact. Here, we describe the IS data structure and we evaluate its storage cost. Moreover, we present efficient algorithms for navigating and for generating a simplicial complex described as an IS data structure. We compare the IS data structure with the incidence graph and with dimension-specific representations for simplicial complexes.

A semantic web environment for digital shapes understanding

In the last few years, the volume of multimedia content available on the Web significantly increased. This led to the need for techniques to handle such data. In this context, we see a growing interest in considering the Semantic Web in action and in the definition of tools capable of analyzing and organizing digital shape models. In this paper, we present a Semantic Web environment, be-SMART, for inspecting 3D shapes and for structuring and annotating such shapes according to ontology-driven metadata. Specifically, we describe in details the first module of be-SMART, the Geometry and Topology Analyzer, and the algorithms we have developed for extracting geometrical and topological information from 3D shapes. We also describe the second module, the Topological Decomposer, which produces a graph-based representation of the decomposition of the shape into manifold components. This is successively modified by the third and the fourth modules, which perform the automatic and manual segmentation of the manifold parts.

A data structure for non-manifold simplicial d -complexes

We propose a data structure for d-dimensional simplicial complexes, that we call the Simplified Incidence Graph (SIG). The simplified incidence graph encodes all simplices of a simplicial complex together with a set of boundary and partial co-boundary topological relations. It is a dimension-independent data structure in the sense that it can represent objects of arbitrary dimensions. It scales well to the manifold case, i.e. it exhibits a small overhead when applied to simplicial complexes with a manifold domain, Here, we present efficient navigation algorithms for retrieving all topological relations from a SIG, and an algorithm for generating a SIG from a representation of the complex as an incidence graph. Finally, we compare the simplified incidence graph with the incidence graph, with a widely-used data structure for d-dimensional pseudo-manifold simplicial complexes, and with two data structures specific for two-and three-dimensional simplicial complexes.

A two-level topological decomposition for non-manifold simplicial shapes

Modeling and understanding complex non-manifold shapes is a key issue in shape analysis. Geometric shapes are commonly discretized as two- or three-dimensional simplicial complexes embedded in the 3D Euclidean space. The topological structure of a nonmanifold simplicial shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we present a topological decomposition of a shape at two different levels, with different degrees of granularity. We discuss the topological properties of the components at each level, and we present algorithms for computing such decompositions. We investigate the relations among the components, and propose a graph-based representation for such relations.

A decomposition-based representation for 3D simplicial complexes

We define a new representation for non-manifold 3D shapes described by three-dimensional simplicial complexes, that we call the Double-Level Decomposition (DLD) data structure. The DLD data structure is based on a unique decomposition of the simplicial complex into nearly manifold parts, and encodes the decomposition in an efficient and powerful two-level representation. It is compact, and it supports efficient topological navigation through adjacencies. It also provides a suitable basis for geometric reasoning on non-manifold shapes. We describe an algorithm to decompose a 3D simplicial complex into nearly manifold parts. We discuss how to build the DLD data structure from a description of a 3D complex as a collection of tetrahedra, dangling triangles and wire edges, and we present algorithms for topological navigation. We present a thorough comparison with existing representations for 3D simplicial complexes.

Computing and Visualizing a Graph-Based Decomposition for Non-manifold Shapes

Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a decomposition of a non-manifold shape into components which are almost manifolds, and we present a novel graph representation which highlights the non-manifold singularities shared by the components as well as their connectivity relations. We describe an algorithm for computing the decomposition and its associated graph representation. We present a new tool for visualizing the shape decomposition and its graph as an effective support to modeling, analyzing and understanding non-manifold shapes.

A Dimension-Independent Representation for Multiresolution Nonmanifold Meshes

We consider the problem of representing and manipulating nonmanifold objects of any dimension and at multiple resolutions. We present a modeling scheme based on (1) a multiresolution representation, called the vertex-based nonmanifold multitessellation, (2) a compact and dimension-independent data structure, called the Simplified Incidence Graph (SIG), and (3) an atomic mesh update operator, called vertex-pair contraction/ vertex expansion. We propose efficient algorithms for performing the vertex-pair contraction on a simplicial mesh encoded as a SIG, and an effective representation for encoding this multiresolution model based on a compact encoding of vertex-pair contractions and vertex expansions.

Identification of Form Features in Non-Manifold Shapes Through a Decomposition Approach

In Computer-Aided Design (CAD), the idealization process reduces the complexity of the model of a solid object, thus resulting in a simplified representation which captures only the essential elements of its shape. Form features extraction is a relevant issue in order to recover semantic information from an idealized object model, since such information is typically lost during the idealization process. An idealized model is usually composed of non-manifold parts, whose topology carries significant structural information about the object shape. To this aim, we define form features for non-manifold object by extending the taxonomy of form features provided by STEP [19]. We describe an approach for the identification of features, which interact with non-manifold singularities in the object, based on a decomposition of a non-manifold object into nearly manifold components and on the properties of the graph representing such decomposition.Copyright