Laura Papaleo

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.

Morse-Smale decompositions for modeling terrain knowledge

In this paper, we describe, analyze and compare techniques for extracting spatial knowledge from a terrain model. Specifically, we investigate techniques for extracting a morphological representation from a terrain model based on an approximation of a Morse-Smale complex. A Morse-Smale complex defines a decomposition of a topographic surface into regions with vertices at the critical points and bounded by integral lines which connect passes to pits and peaks. This provides a terrain representation which encompasses the knowledge on the salient characteristics of the terrain. We classify the various techniques for computing a Morse-Smale complexe based on the underlying terrain model, a Regular Square Grid (RSG) or a Triangulated Irregular Network (TIN), and based on the algorithmic approach they apply. Finally, we discuss hierarchical terrain representations based on a Morse-Smale decomposition.

A multi-resolution representation for terrain morphology

Mesh-based terrain representations provide accurate descriptions of a terrain, but fail in capturing its morphological structure. The morphology of a terrain is defined by its critical points and by the critical lines joining them, which form a so-called surface network. Because of the large size of current terrain data sets, a multi-resolution representation of the terrain morphology is crucial. Here, we address the problem of representing the morphology of a terrain at different resolutions. The basis of the multi-resolution terrain model, that we call a Multi-resolution Surface Network (MSN), is a generalization operator on a surface network, which produces a simplified representation incrementally. An MSN is combined with a multi-resolution mesh-based terrain model, which encompasses the terrain morphology at different resolutions. We show how variable-resolution representations can be extracted from an MSN, and we present also an implementation of an MSN in a compact encoding data structure.

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.

Semantic-Based Segmentation and Annotation of 3D Models

3D objects have become widely available and used in different application domains. Thus, it is becoming fundamental to use, integrate and develop techniques for extracting and maintaining their embedded knowledge. These techniques should be encapsulated in portable and intelligent systems able to semantically annotate the 3D object models in order to improve their usability and indexing, especially in innovative web cooperative environments. Lately, we are moving in this direction, with the definition and development of data structures, methods and interfaces for structuring and semantically annotating 3D complex models (and scenes) - even changing in time - according to ontology-driven metadata and following ontology-driven processes. Here, we concentrate on the tools for segmenting manifold 3D models and on the underline structural representation that we build and manipulate. We also describe the first prototype of an annotation tool which allows a hierarchical semantic-driven tagging of the segmented model and provides an interface from which the user can inspect and browse the entire segmentation graph.

Efficient on-line mosaicing from 3D acoustical images

This paper presents a system for the 3D reconstruction of an underwater environment on the basis of multiple range views from an acoustical camera. The challenge is to provide the reconstruction on-line, as the range views are obtained from the sensor. The final target of the work is to improve the understanding of a human operator driving an underwater remotely operated vehicle (ROV). The acoustic camera provides a sequence of 3D images in real time (about 5 fps in the current version). Data must be registered and fused to generate a unique 3D mosaic in the form of a triangle mesh, which is rendered through a graphical interface. Available technologies for registration and meshing have been modified to match time constraints. We report experiments on real data.

Knowledge-Based Representation Of 3D Media

In recent years, 3D media have become more and more widespread and have been made available in numerous online repositories. A systematic and formal approach for representing and organizing shape-related information is needed to share 3D media, to communicate the knowledge associated to shape modeling processes and to facilitate its reuse in useful cross-domain usage scenarios. In this paper we present an initial attempt to formalize an ontology for digital shapes, called the Common Shape Ontology (CSO). We discuss about the rationale, the requirements and the scope of this ontology, we present in detail its structure and describe the most relevant choices related to its development. Finally, we show how the CSO conceptualization is used in domain-specific application scenarios.

A Java3D framework for inspecting and segmenting 3D models

Models of 3D objects have become widely accessible in several disciplines within academia and industry, spanning from scientific visualization to entertainment. In the last few years, 3D models are often organized into digital libraries accessible over the network, and thus semantic annotation of such models becomes an important issue. A fundamental step in annotating a 3D model is to segment it into meaningful parts. In this work, we present a Java3D framework for inspecting and segmenting 3D objects represented in X3D format. In particular, we present a combination of segmentation and merging techniques for producing a feasible decomposition of the boundary of a 3D object. We represent such decomposition as a graph, that we call the segmentation graph which is the basis for semantic annotation. We describe also the interface we have developed to allow visualization and browsing of both the decomposition and the segmentation graph in order to understand the topological structure of the resulting decomposition.

A Discrete Approach to Compute Terrain Morphology

We consider the problem of extracting morphology of a terrain represented as a Triangulated Irregular Network (TIN). We propose a new algorithm and compare it with representative algorithms of the main approaches existing in the literature to this problem. The new algorithm has the advantage of being simple, using only comparisons (and no floating-point computations), and of being suitable for an extension to higher dimensions. Our experiments consider both real data and artificial test data. We evaluate the difference in the results produced on the same terrain data, as well as the impact of resolution level on such a difference, by considering representations of the same terrain at different resolutions.

A complete system for on-line 3D modelling from acoustic images

This paper presents a system for the three-dimensional (3D) reconstruction of an underwater environment on the basis of multiple range views from an acoustical camera. The challenge is to provide the reconstruction on-line, as the range views are obtained from the sensor. The final target of the work is to improve the understanding of a human operator driving an underwater Remotely Operated Vehicle. The acoustic camera provides a sequence of 3D images in real time. Data must be registered and fused to generate a unique 3D mosaic in the form of a triangle mesh, which is rendered through a graphical interface. Available technologies for registration and meshing have been modified and extended to match time constraints. Some experiments on real data are reported.