Parallel Computation of Piecewise Linear Morse-Smale Segmentations

This paper presents a well-scaling parallel algorithm for the computation of Morse-Smale (MS) segmentations, including the region separators and region boundaries. The segmentation of the domain into ascending and descending manifolds, solely defined on the vertices, improves the computational time using path compression and fully segments the border region. Region boundaries and region separators are generated using a multi-label marching tetrahedra algorithm. This enables a fast and simple solution to find optimal parameter settings in preliminary exploration steps by generating an MS complex preview. It also poses a rapid option to generate a fast visual representation of the region geometries for immediate utilization. Two experiments demonstrate the performance of our approach with speedups of over an order of magnitude in comparison to two publicly available implementations. The example section shows the similarity to the MS complex, the useability of the approach, and the benefits of this method with respect to the presented datasets. We provide our implementation with the paper.Comment: Journal: IEEE Transactions on Visualization and Computer Graphics / Submitted: 22-Jun-2022 / Accepted: 13-Mar-202

Conforming Morse-Smale complexes

pre-printMorse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing application-dependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topology-based representation

Operatori za multi-rezolucione komplekse Morza i ćelijske komplekse

The topic of the thesis is analysis of the topological structure of scalar fields and shapes represented through Morse and cell complexes, respectively. This is achieved by defining simplification and refinement operators on these complexes. It is shown that the defined operators form a basis for the set of operators that modify Morse and cell complexes. Based on the defined operators, a multi-resolution model for Morse and cell complexes is constructed, which contains a large number of representations at uniform and variable resolution.Тема дисертације је анализа тополошке структуре скаларних поља и облика представљених у облику комплекса Морза и ћелијских комплекса, редом. То се постиже дефинисањем оператора за симплификацију и рафинацију тих комплекса. Показано је да дефинисани оператори чине базу за скуп оператора на комплексима Морза и ћелијским комплексима. На основу дефинисаних оператора конструисан је мулти-резолуциони модел за комплексе Морза и ћелијске комплексе, који садржи велики број репрезентација униформне и варијабилне резолуције.Tema disertacije je analiza topološke strukture skalarnih polja i oblika predstavljenih u obliku kompleksa Morza i ćelijskih kompleksa, redom. To se postiže definisanjem operatora za simplifikaciju i rafinaciju tih kompleksa. Pokazano je da definisani operatori čine bazu za skup operatora na kompleksima Morza i ćelijskim kompleksima. Na osnovu definisanih operatora konstruisan je multi-rezolucioni model za komplekse Morza i ćelijske komplekse, koji sadrži veliki broj reprezentacija uniformne i varijabilne rezolucije

The Discrete Morse Complex of Images: Algorithms, Modeling and Applications

The Morse complex can be used for studying the topology of a function, e.g., an image or terrain height field when understood as bivariate functions. We present an algorithm for the computation of the discrete Morse complex of two-dimensional images using an edge-based data structure. By using this data structure, it is possible to perform local operations efficiently, which is important to construct the complex and make the structure useful for areas like visualization, persistent homology computation, or construction of a topological hierarchy. We present theoretical and applied results to demonstrate benefits and use of our method

Operatori za multi-rezolucione komplekse Morza i ćelijske komplekse

The topic of the thesis is analysis of the topological structure of scalar fields and shapes represented through Morse and cell complexes, respectively. This is achieved by defining simplification and refinement operators on these complexes. It is shown that the defined operators form a basis for the set of operators that modify Morse and cell complexes. Based on the defined operators, a multi-resolution model for Morse and cell complexes is constructed, which contains a large number of representations at uniform and variable resolution.Тема дисертације је анализа тополошке структуре скаларних поља и облика представљених у облику комплекса Морза и ћелијских комплекса, редом. То се постиже дефинисањем оператора за симплификацију и рафинацију тих комплекса. Показано је да дефинисани оператори чине базу за скуп оператора на комплексима Морза и ћелијским комплексима. На основу дефинисаних оператора конструисан је мулти-резолуциони модел за комплексе Морза и ћелијске комплексе, који садржи велики број репрезентација униформне и варијабилне резолуције.Tema disertacije je analiza topološke strukture skalarnih polja i oblika predstavljenih u obliku kompleksa Morza i ćelijskih kompleksa, redom. To se postiže definisanjem operatora za simplifikaciju i rafinaciju tih kompleksa. Pokazano je da definisani operatori čine bazu za skup operatora na kompleksima Morza i ćelijskim kompleksima. Na osnovu definisanih operatora konstruisan je multi-rezolucioni model za komplekse Morza i ćelijske komplekse, koji sadrži veliki broj reprezentacija uniformne i varijabilne rezolucije

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

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

Topological analysis of discrete scalar data

This thesis presents a novel computational framework that allows for a robust extraction and quantification of the Morse-Smale complex of a scalar field given on a 2- or 3- dimensional manifold. The proposed framework is based on Forman\u27s discrete Morse theory, which guarantees the topological consistency of the computed complex. Using a graph theoretical formulation of this theory, we present an algorithmic library that computes the Morse-Smale complex combinatorially with an optimal complexity of $O(n^2)$ and efficiently creates a multi-level representation of it. We explore the discrete nature of this complex, and relate it to the smooth counterpart. It is often necessary to estimate the feature strength of the individual components of the Morse-Smale complex -- the critical points and separatrices. To do so, we propose a novel output-sensitive strategy to compute the persistence of the critical points. We also extend this wellfounded concept to separatrices by introducing a novel measure of feature strength called separatrix persistence. We evaluate the applicability of our methods in a wide variety of application areas ranging from computer graphics to planetary science to computer and electron tomography.In dieser Dissertation präsentieren wir ein neues System zur robusten Berechnung des Morse-Smale Komplexes auf 2- oder 3-dimensionalen Mannigfaltigkeiten. Das vorgestellte System basiert auf Forman’s diskreter Morsetheorie und garantiert damit die topologische Konsistenz des berechneten Komplexes. Basierend auf einer graphentheoretischer Formulierung präesentieren wir eine Bibliothek von Algorithmen, die es erlaubt, den Morse-Smale Komplex mit einer optimalen Kompliztät von $O(n^2)$ kombinatorisch zu berechnen und effizient eine mehrskalige Repräsentation davon erstellt. Wir untersuchen die diskrete Natur dieses Komplexes und vergleichen ihn zu seinem kontinuierlichen Gegenstück. Es ist häufig notwendig, die Merkmalsstärke einzelner Bestandteile des Komplexes -- der kritischen Punkte und Separatrizen -- abzuschätzen. Hierfür stellen wir eine neue outputsensitive Strategie vor, um die Persistenz von kritischen Punkten zu berechen. Wir erweitern dieses fundierte Konzept auf Separatrizen durch die Einführung des Wichtigkeitsmaßes Separatrixpersistenz. Wir evaluieren die Anwendbarkeit unserer Methoden anhand vielfältiger Anwendungen aus den Gebieten der Computergrafik, Planetologie, Computer- und Elektronentomographie

Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings

International audienceThis application paper presents a novel framework based on topological data analysis for the automatic evaluation and ranking of viscous finger simulation runs in an ensemble with respect to a reference acquisition. Individual fingers in a given time-step are associated with critical point pairs in the distance field to the injection point, forming persistence diagrams. Different metrics, based on optimal transport, for comparing time-varying persistence diagrams in this specific applicative case are introduced. We evaluate the relevance of the rankings obtained with these metrics, both qualitatively thanks to a lightweight web visual interface, and quantitatively by studying the deviation from a reference ranking suggested by experts. Extensive experiments show the quantitative superiority of our approach compared to traditional alternatives. Our web interface allows experts to conveniently explore the produced rankings. We show a complete viscous fingering case study demonstrating the utility of our approach in the context of porous media fluid flow, where our framework can be used to automatically discard physically-irrelevant simulation runs from the ensemble and rank the most plausible ones. We document an in-situ implementation to lighten I/O and performance constraints arising in the context of parametric studies