L. De Floriani

Describing shapes by geometrical-topological properties of real functions

Differential topology, and specifically Morse theory, provide a suitable setting for formalizing and solving several problems related to shape analysis. The fundamental idea behind Morse theory is that of combining the topological exploration of a shape with quantitative measurement of geometrical properties provided by a real function defined on the shape. The added value of approaches based on Morse theory is in the possibility of adopting different functions as shape descriptors according to the properties and invariants that one wishes to analyze. In this sense, Morse theory allows one to construct a general framework for shape characterization, parametrized with respect to the mapping function used, and possibly the space associated with the shape. The mapping function plays the role of a lens through which we look at the properties of the shape, and different functions provide different insights. In the last decade, an increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods proposed range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same mathematical constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner. The term geometrical-topological used in the title is meant to underline that both levels of information content are relevant for the applications of shape descriptions: geometrical, or metrical, properties and attributes are crucial for characterizing specific instances of features, while topological properties are necessary to abstract and classify shapes according to invariant aspects of their geometry. The approaches surveyed will be discussed in detail, with respect to theory, computation, and application. Several properties of the shape descriptors will be analyzed and compared. We believe this is a crucial step to exploit fully the potential of such approaches in many applications, as well as to identify important areas of future research.

Efficient implementation of multi-triangulations

Multi-triangulation (MT) is a general framework for managing the level-of-detail in large triangle meshes, which we have introduced in our previous work. In this paper, we describe an efficient implementation of an MT based on vertex decimation. We present general techniques for querying an MT, which are independent of a specific application, and which can be applied for solving problems, such as selective refinement, windowing, point location, and other spatial interference queries. We describe alternative data structures for encoding an MT, which achieve different trade-offs between space and performance. Experimental results are discussed.

Selective refinement queries for volume visualization of unstructured tetrahedral meshes

We address the problem of the efficient visualization of large irregular volume data sets by exploiting a multiresolution model based on tetrahedral meshes. Multiresolution models, also called Level-Of-Detail (LOD) models, allow encoding the whole data set at a virtually continuous range of different resolutions. We have identified a set of queries for extracting meshes at variable resolution from a multiresolution model, based on field values, domain location, or opacity of the transfer function. Such queries allow trading off between resolution and speed in visualization. We define a new compact data structure for encoding a multiresolution tetrahedral mesh built through edge collapses to support selective refinement efficiently and show that such a structure has a storage cost from 3 to 5.5 times lower than standard data structures used for tetrahedral meshes. The data structures and variable resolution queries have been implemented together with state-of-the art visualization techniques in a system for the interactive visualization of three-dimensional scalar fields defined on tetrahedral meshes. Experimental results show that selective refinement queries can support interactive visualization of large data sets.

Constrained Delaunay triangulation for multiresolution surface description

The problem of building a constrained Delaunay triangulation (CDT) at different levels of resolution is considered for the hierarchical description of topographic surfaces. The surface is approximated at each level by a network of planar triangular faces having vertices at a subset of surface-specific points, such as peaks, pits, or passes, and including edges that describe surface-specific lines, such as ridges or valleys. Each approximation is built based on a Delaunay triangulation of the data points that includes the given constraint segments. A dynamic algorithm for constrained Delaunay triangulation is proposed. The algorithm is based on the stepwise refinement of a CDT by the incremental insertion of points and constraint segments.<<ETX>>

Building a feature-based object description from a boundary model

Abstract Form features, like protrusions or depressions on a face, through-holes or handles, can be extracted from a relational boundary model of a solid object, called the ‘symmetric boundary graph’ by loop identification and connected component labelling. The result is a decomposition of the object boundary into volumetric components describing features, which is represented as a directed labelled multigraph, called the ‘object decomposition graph’. Based on such a model, issues such as representation uniqueness and matching of object descriptions are discussed.

Dynamic view-dependent multiresolution on a client–server architecture

Abstract We consider the problem of transmitting huge triangle meshes in the context of a Web-like client–server architecture. Approximations of the original mesh are transmitted by applying selective refinement. A multiresolution geometric model is maintained by the server. A client may query the server for a mesh at an arbitrary, continuously variable, level of detail. The client makes repeated queries over time with different query parameters. The server answers to queries by traversing the multiresolution model and transmitting updates to the client, which uses them to progressively modify a current mesh. We study this problem in the context of a vertex-based multiresolution model, which is a special instance of the Multi-Triangulation (a model that was developed in an earlier work), based on vertex insertion and removal. We define a compact data structure for such a model that exploits the specific update rule. We propose a dynamic algorithm for selective refinement and we discuss in detail its implementation as a client–server application. In order to reduce memory requirements and channel traffic, we develop a compressed representation which allows us to express mesh updates with a code of small size. We also address client caching to further limit bandwidth occupancy. Experimental results show that the Multi-Triangulation can be a key Web technology for triangle mesh manipulation.

Constant-time neighbor finding in hierarchical tetrahedral meshes

Techniques are presented for moving between adjacent tetrahedra in a tetrahedral mesh. The tetrahedra result from a recursive decomposition of a cube into six initial congruent tetrahedra. A new technique is presented for labeling the triangular faces. The labeling enables the implementation of a binary-like decomposition of each tetrahedron which is represented using a pointerless representation. Outlines of algorithms are given for traversing adjacent triangular faces of equal size in constant time.

Simplex and Diamond Hierarchies: Models and Applications

Hierarchical spatial decompositions are a basic modelling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through regular simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multi‐resolution models of scalar fields, such as terrains, and static or time‐varying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplices, called diamonds, as the modelling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension‐independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.

Multiresolution tetrahedral meshes: an analysis and a comparison

We deal with the problem of analyzing and visualizing large-size volume data sets. To this aim, we consider multiresolution representations based on a decomposition of the field domain into tetrahedral cells. We compare two types of multiresolution representations that differ on the rule applied to refine an initial coarse mesh: one is based on tetrahedron bisection, and one based on vertex split. The two representations can be viewed as instances of a common multiresolution model, that we call a multiresolution mesh. Encoding data structures for the two representations are briefly described. An experimental comparison on structured volume data sets is presented.

A Survey of Topology-based Methods in Visualization

This paper presents the state of the art in the area of topology‐based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.