David Canino

ABSTRACT - Decomposing a non-manifold shape into its almost manifold components is a powerful tool for analyzing its complex structure. Many techniques for decomposing a non-manifold shape are available in the current literature, and provide a structural model, which exposes its non-manifold singularities, as well as the connectivity of its relevant subcomponents, connected through the singularities. However, the majority of the decompositions are static, and are not automatically updated, if the corresponding non-manifold shape is modified by an editing operator. In many cases, the resulting decomposition is recomputed from scratch without reusing the unchanged portions of the existing decomposition. In this paper, we describe how updating automatically a specific decomposition of a non-manifold shape. Here, we show that our approach may be useful for adapting many geometry processing techniques also to non-manifold shapes, where several problems may arise. One of the most promising applications consists of defining a multiresolution version for the specific structural model of interest, due to its good topological properties.

In this document, I summarize briefly results I have obtained in my academic career, in particular during my PhD. and Postdoc Studies in Computer Science at the Department of Computer Science, University of Genova (GE), Italy, under the supervision of Professor Leila De Floriani.

ABSTRACT - Simplicial complexes are extensively used for discretizing digital shapes in two, three, and higher dimensions within a variety of application domains. There have been many proposals of topological data structures, which represent the connectivity information among simplices. We introduce the Mangrove Topological Data Structure (Mangrove TDS) framework, a tool which supports the efficient implementation of data structures for simplicial complexes of any dimension under the same application interface. Our framework is based on a graph-based representation of connectivity relations, that we call the mangrove. It can be customized in order to simulate the content of any topological data structure with a negligible overhead. Thus, the Mangrove TDS framework is extensible, and supports the most diverse modeling needs. We also provide implicit representations of those simplices, which are not directly encoded in a specific topological data structure. Our tests show that these representations, that we call ghost simplices, improve the expressive power and the efficiency of topological queries. In order to prove the validity of our approach, we design two topological data structures, specific for non-manifold complexes, within our framework. We perform comparisons with some widely-used representations in the literature as well as with libraries available in the public domain.

We introduce the Mangrove Topological Data Structure (Mangrove TDS) framework for modeling simplicial and cell complexes. It is based on a graph-based representation of the data structures, called mangroves, which ensures an extensible description of any data structure without restrictions under a common application interface. Mangroves can be easily customized for any modeling need, including the efficient representation of non-manifold shapes, and of those cells, not directly encoded in a mangrove, that we call ghost entities. We discuss here the properties of this framework, and current and future developments.

Simplicial complexes are extensively used for discretizing digital shapes in several applications. A structural description of a non-manifold shape can be obtained by decomposing the input shape into a collection of meaningful components with a simpler topology. Here, we consider a unique and dimension-independent decomposition of a non-manifold shape into nearly manifold components, known as the Manifold-Connected (MC-) decomposition. We present the Compact Manifold-Connected (MC-) graph, an efficient graph-based representation for the MC-decomposition, which can be combined with any topological data structure for encoding the underlying components. We present the main properties of this representation as well as algorithms for its generation. We also show that this representation is more compact than several topological data
structures, which do not explicitly describe the non-manifold structure of a shape.

We introduce the Mangrove Topological Data Structure (Mangrove TDS) framework for modeling simplicial complexes. It is based on a graph-based representation of the data structures, called mangroves, which ensures an extensible representation of a data structure for simplicial complexes. Mangroves can be easily customized for any modeling need, including the efficient representation of non-manifold shapes, and of those simplices, not directly encoded in a mangrove, that we call ghost simplices. We discuss here the properties of this framework, and current and future developments.

In this thesis, we address the effective representation of arbitrary shapes, called non-manifold shapes, discretized through simplicial complexes, and we introduce a set of tools for their modeling and analysis.

Specifically, we propose two dimension-independent data structures for simplicial complexes in arbitrary dimensions. The first contribution is the Incidence Simplicial (IS) data structure, based on the incidence relations for simplices of consecutive dimensions. The second contribution is the Generalized Indexed Data Structure with Adjacencies (IA∗), based on the adjacency relations for top simplices. The IS and IA∗ data structures are compact, support efficient navigation, and exhibit a small overhead, if restricted to manifolds. In the literature, there are several topological data structures for cell and simplicial complexes, thus a framework targeted to their fast prototyping is a valuable tool. Here, we introduce the dimension-independent and extensible Mangrove Topological Data Structure (Mangrove TDS) framework. This framework describes any data structure through a graph-based representation, which we call a mangrove. In this thesis, we provide extensive experimental comparisons for several data structures implemented in the Mangrove TDS framework, including the IS and IA∗ data structures. At the same time, we complete the definition of several data structures, previously proposed in the literature.

In the second part of the thesis, we decompose any non-manifold shape into almost manifold parts in order to deal with its intrinsic complexity. We consider a dimension-independent decomposition of a non-manifold shape, called Manifold-Connected Decomposition (MC-Decomposition), previously investigated only for two- and three-dimensional complexes. Here, we propose several graph-based representations of such a decomposition, which can be combined with any topological data structure. We provide experimental comparisons about building times and storage costs of these data structures.

Recently, the computation of topological invariants, like the simplicial homology, has drawn much attention in several applications. Here, we design and implement the dimension-independent and modular Mayer-Vietoris (MV) algorithm, which exploits the MC-Decomposition for computing the simplicial homology of a non-manifold simplicial shape in arbitrary dimensions. The MV algorithm offers an elegant way for computing the homology of any simplicial complex from the homology of its MC-components and of their intersections.

We propose a compact, dimension-independent data structure for manifold, non-manifold and non-regular simplicial complexes, that we call the Generalized Indexed Data structure with Adjacencies (IA∗ data structure). It encodes only top simplices, i.e., the ones that are not on the boundary of any other simplex, plus a suitable subset of the adjacency relations. We describe the IA∗ data structure in arbitrary dimensions, and compare the storage requirements of its two-dimensional and three-dimensional instances with both dimension-specific and dimension-independent representations. We show that the IA∗ data structure is more cost effective than other dimension-independent representations and is even slightly more compact than the existing dimension-specific ones. We present efficient algorithms for navigating a simplicial complex described as an IA∗ data structure. This shows that the IA∗ data structure allows retrieving all topological relations of a given simplex by considering only its local neighborhood and thus it is a more efficient alternative to incidence-based representations when information does not need to be encoded for boundary simplices.

Nowadays, gigantic models can be easily produced in many applications and their dimension often exceeds the RAM size in a common workstation. Thus, using an external memory technique is mandatory in this case. In this paper, we define a dimension-independent and extensible framework, called Objects Management in Secondary Memory (OMSM), for managing huge models. The OMSM framework can be easily adapted to the users needs through dynamic plugins, providing many techniques to be integrated in a storing architecture.

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 exhibits performances in the 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.

Nowadays, some gigantic models can be easily produced in many applications and their storage cost often exceeds the RAM size in a common workstation. The model simplification and the multi-resolution techniques are a rather mature technologies that in many cases can efficiently manage complex data: however, the RAM size is often a severe bottleneck, even for high–performance graphics workstations. Thus, using an external memory technique is mandatory in this case: it is important to encode huge models in the most efficient way as possible, maintaining the opportunity to analyze, to manage and to display them according to the user requests and choices.

In this paper we define an extensible framework, called OMSM (short for Objects Management in Secondary Memory), for managing huge models: it supports external memory management of complex models, loading dynamically in RAM only the selected sections. All the functionalities implemented on this framework can be applied to a generic geometric model, independently from its dimension, on low–cost PC platforms."

This document is an extented abstract of our Master’s Thesis defended at the D.I.S.I., Universitá degli Studi di Genova: you can refer the original version of our Master’s Thesis, written in Italian. In our Master’s Thesis, we have analyzed the state of art about the out–of–core simplification techniques and we have designed a prototype of an extensible framework for managing huge geometric models, called OMSM (short for Objects Management in Secondary Memory).