Morse Decomposition of Geometric Meshes with Applications

9th Eurographics Italian Chapter Conference 2011 (EG-IT 2011), 2011

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 representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on nonmanifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.

We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multiresolution representations.

Visualization and Computer …, 2004

With improvements in sensor technology and simulation methods, datasets are growing in size, calling for the investigation of efficient and scalable tools for their analysis. Topological methods, able to extract essential features from data, are a prime candidate for the development of such tools. Here, we examine an approach based on discrete Morse theory and compare it to the well-known watershed approach as a means of obtaining Morse decompositions of tessellated manifolds endowed with scalar fields, such as triangulated terrains or tetrahedralized volume data. We examine the theoretical aspects as well as present empirical results based on synthetic and real-world data describing terrains and 3D scalar fields. We will show that the approach based on discrete Morse theory generates segmentations comparable to the watershed approach while being theoretically sound, more efficient with regard to time and space complexity, easily parallelizable, and allowing for the computation of all descending and ascending i-manifolds and the topological structure of the two Morse complexes.

Parole chiave: Shape comparison, size function, natural pseudo-distance, persistent homology module,Čech homology, shape occlusion.

1995

In computer graphics and geometric modeling, shapes are often represented by triangular meshes. With the advent of laser scanning systems, meshes of extreme complexity are rapidly becoming commonplace. Such meshes are notoriously expensive to store, transmit, render, and are awkward to edit. Multiresolution analysis offers a simple, unified, and theoretically sound approach to dealing with these problems. Lounsbery et al. have recently developed a technique for creating multiresolution representations for a restricted class of meshes with subdivision connectivity. Unfortunately, meshes encountered in practice typically do not meet this requirement. In this paper we present a method for overcoming the subdivision connectivity restriction, meaning that completely arbitrary meshes can now be converted to multiresolution form. The method is based on the approximation of an arbitrary initial mesh M by a mesh M J that has subdivision connectivity and is guaranteed to be within a specified tolerance.

IEEE Transactions on Image Processing, 2004

This paper introduces the concept of digital planar surfaces and corresponding Morse operators. These operators offer a novel and powerful method for construction and de-construction of such surfaces in a way that global topological control of the resulting object is always maintained. In that respect, this paper offers a complete pixel characterization tool. Image handling is a natural application for such approach. We present a novel fast algorithm for image segmentation using Morse operators for digital planar surfaces. It classifies as a region growing technique with added topological control and is extremely useful for applications that need proper object description. Results from real data are stimulating, and show that the segmentation algorithm compares very well with other methods. The topological approach also forms a base for future expansion to applications such as volume segmentation.

2004

We propose a new topological method for shape description that is suitable for any multi-dimensional data set that can be modelled as a manifold. The description is obtained for all pairs (M, f), where M is a closed smooth manifold and f a Morse function defined on M. More precisely, we characterize the topology of all pairs of lower level sets (My, Mx) of f, where Ma = f-1((-∞,a]), for all a ∈ R. Classical Morse theory is used to establish a link between the topology of a pair of lower level sets of f and its critical points lying between the two levels.

From Geometric Modeling to Shape Modeling, 2002

This paper investigates the possible role of the new field of computational topology for incorporating abstraction mechanisms in shape modelling. The effectiveness of computational topology techniques is exemplified with an application of discrete differential topology. In particular, a method is proposed for the extraction of a critical point configuration graph from a triangulated surface. Starting from the definition of the Reeb graph in the smooth domain, the concept of critical point is extended to critical areas, which may represent isolated as well as degenerated critical points in the discrete domain. The resulting graph effectively represents the surface shape and has been successfully used as a basis for model compression and restoring purposes.

2012

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.