Jacobi Fiber Surfaces for Bivariate Reeb Space Computation

This paper presents an efficient algorithm for the computation of the Reeb space of an input bivariate piecewise linear scalar function f defined on a tetrahedral mesh. By extending and generalizing algorithmic concepts from the univariate case to the bivariate one, we report the first practical, output-sensitive algorithm for the exact computation of such a Reeb space. The algorithm starts by identifying the Jacobi set of f , the bivariate analogs of critical points in the univariate case. Next, the Reeb space is computed by segmenting the input mesh along the new notion of Jacobi Fiber Surfaces, the bivariate analog of critical contours in the univariate case. We additionally present a simplification heuristic that enables the progressive coarsening of the Reeb space. Our algorithm is simple to implement and most of its computations can be trivially parallelized. We report performance numbers demonstrating orders of magnitude speedups over previous approaches, enabling for the first time the tractable computation of bivariate Reeb spaces in practice. Moreover, unlike range-based quantization approaches (such as the Joint Contour Net), our algorithm is parameter-free. We demonstrate the utility of our approach by using the Reeb space as a semi-automatic segmentation tool for bivariate data. In particular, we introduce continuous scatterplot peeling, a technique which enables the reduction of the cluttering in the continuous scatterplot, by interactively selecting the features of the Reeb space to project. We provide a VTK-based C++ implementation of our algorithm that can be used for reproduction purposes or for the development of new Reeb space based visualization techniques

Task-based Augmented Reeb Graphs with Dynamic ST-Trees

International audienceThis paper presents, to the best of our knowledge, the first parallel algorithm for the computation of the augmented Reeb graph of piecewise linear scalar data. Such augmented Reeb graphs have a wide range of applications , including contour seeding and feature based segmentation. Our approach targets shared-memory multi-core workstations. For this, it completely revisits the optimal, but sequential, Reeb graph algorithm, which is capable of handing data in arbitrary dimension and with optimal time complexity. We take advantage of Fibonacci heaps to exploit the ST-Tree data structure through independent local propagations, while maintaining the optimal, linearithmic time complexity of the sequential reference algorithm. These independent propagations can be expressed using OpenMP tasks, hence benefiting in parallel from the dynamic load balancing of the task runtime while enabling us to increase the parallelism degree thanks to a dual sweep. We present performance results on triangulated surfaces and tetrahedral meshes. We provide comparisons to related work and show that our new algorithm results in superior time performance in practice, both in sequential and in parallel. An open-source C++ implementation is provided for reproducibility

Computational Topology Methods for Shape Modelling Applications

This thesis deals with computational topology, a recent branch of research that involves both mathematics and computer science, and tackles the problem of discretizing the Morse theory to functions defined on a triangle mesh. The application context of Morse theory in general, and Reeb graphs in particular, deals with the analysis of geometric shapes and the extraction of skeletal structures that synthetically represents shape, preserving the topological properties and the main morphological characteristics. Regarding Computer Graphics, shapes, that is a one-, two- or higher- dimensional connected, compact space having a visual appearance, are typically approximated by digital models. Since topology focuses on the qualitative properties of spaces, such as the connectedness and how many and what type of holes it has, topology is the best tool to describe the shape of a mathematical model at a high level of abstraction. Geometry, conversely, is mainly related to the quantitative characteristics of a shape. Thus, the combination of topology and geometry creates a new generation of tools that provide a computational description of the most representative features of the shape along with their relationship. Extracting qualitative information, that is the information related to semantic of the shape and its morphological structure, from discrete models is a central goal in shape modeling. In this thesis a conceptual model is proposed which represents a given surface based on topological coding that defines a sketch of the surface, discarding irrelevant details and classifying its topological type. The approach is based on Morse theory and Reeb graphs, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the geometry of the discrete shape model. To fully develop the method, both theoretical and computational aspects have been considered, related to the definition and the extension of the Reeb graph to the discrete domain. For the definition and automatic construction of the conceptual model, a new method has been developed that analyzes and characterizes a triangle mesh with respect to the behavior of a real and at least continuous function defined on the mesh. The proposed solution handles also degenerate critical points, such as non-isolated critical points. To do that, the surface model is characterized using a contour-based strategy, recognizing critical areas instead of critical points and coding the evolution of the contour levels in a graph-like structure, named Extended Reeb Graph, (ERG), which is a high-level abstract model suitable for representing and manipulating piece-wise linear surfaces. The descriptive power of the (ERG) has been also augmented with the introduction of geometric information together with the topological ones, and it has been also studied the relation between the extracted topological and morphological features with respect to the real characteristics of the surface, giving and evaluation of the dimension of the discarded details. Finally, the effectiveness of our description framework has been evaluated in several application contexts

Exploring 3D Shapes through Real Functions

This thesis lays in the context of research on representation, modelling and coding knowledge related to digital shapes, where by shape it is meant any individual object having a visual appareance which exists in some two-, three- or higher dimensional space. Digital shapes are digital representations of either physically existing or virtual objects that can be processed by computer applications. While the technological advances in terms of hardware and software have made available plenty of tools for using and interacting with the geometry of shapes, to manipulate and retrieve huge amount of data it is necessary to define methods able to effectively code them. In this thesis a conceptual model is proposed which represents a given 3D object through the coding of its salient features and defines an abstraction of the object, discarding irrelevant details. The approach is based on the shape descriptors defined with respect to real functions, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the discrete shape model. A distinctive feature of these shape descriptors is their capability of combining topological and geometrical information properties of the shape, giving an abstraction of the main shape features. To fully develop this conceptual model, both theoretical and computational aspects have been considered, related to the definition and the extension of the different shape descriptors to the computational domain. Main emphasis is devoted to the application of these shape descriptors in computational settings; to this aim we display a number of application domains that span from shape retrieval, to shape classification and to best view selection.Questa tesi si colloca nell\u27ambito di ricerca riguardante la rappresentazione, la modellazione e la codifica della conoscenza connessa a forme digitali, dove per forma si intende l\u27aspetto visuale di ogni oggetto che esiste in due, tre o pi? dimensioni. Le forme digitali sono rappresentazioni di oggetti sia reali che virtuali, che possono essere manipolate da un calcolatore. Lo sviluppo tecnologico degli ultimi anni in materia di hardware e software ha messo a disposizione una grande quantit? di strumenti per acquisire, rappresentare e processare la geometria degli oggetti; tuttavia per gestire questa grande mole di dati ? necessario sviluppare metodi in grado di fornirne una codifica efficiente. In questa tesi si propone un modello concettuale che descrive un oggetto 3D attraverso la codifica delle caratteristiche salienti e ne definisce una bozza ad alto livello, tralasciando dettagli irrilevanti. Alla base di questo approccio ? l\u27utilizzo di descrittori basati su funzioni reali in quanto forniscono un\u27astrazione della forma molto utile per analizzare e strutturare l\u27informazione contenuta nel modello discreto della forma. Una peculiarit? di tali descrittori di forma ? la capacit? di combinare propriet? topologiche e geometriche consentendo di astrarne le principali caratteristiche. Per sviluppare questo modello concettuale, ? stato necessario considerare gli aspetti sia teorici che computazionali relativi alla definizione e all\u27estensione in ambito discreto di vari descrittori di forma. Particolare attenzione ? stata rivolta all\u27applicazione dei descrittori studiati in ambito computazionale; a questo scopo sono stati considerati numerosi contesti applicativi, che variano dal riconoscimento alla classificazione di forme, all\u27individuazione della posizione pi? significativa di un oggetto