DIST / UNIGE IMATI-GE / CNR Surface Remeshing and Applications by Marco Attene DIST, Università di Genova CNR – IMATI, Sez. di Genova Via All'Opera Pia 13, 16145, Genova, Italy Via De Marini 6, 16149, Genova, Italy http://www.dist.unige.it http://www.ima.ge.cnr.it Università degli Studi di Genova Consiglio Nazionale delle Ricerche Dipartimento di Informatica, Istituto di Matematica Applicata e Sistemistica e Telematica Tecnologie Informatiche Sezione di Genova Dottorato di Ricerca in Ingegneria Elettronica e Informatica (XVI ciclo) Dissertation Surface Remeshing and Applications by Marco Attene Advisors Dott. Bianca Falcidieno – IMATI-GE / CNR Dott. Michela Spagnuolo – IMATI-GE / CNR March, 2004 Abstract Due to the focus of popular graphic accelerators, triangle meshes remain the primary representation for 3D surfaces. They are the simplest form of interpolation between surface samples, which may have been acquired with a laser scanner, computed from a 3D scalar field resolved on a regular grid, or identified on slices of medical data. Typical methods for the generation of triangle meshes from raw data attempt to lose as less information as possible, so that the resulting surface models can be used in the widest range of scenarios. When such a general-purpose model has to be used in a particular application context, however, a pre-processing is often worth to be considered. In some cases, it is convenient to slightly modify the geometry and/or the connectivity of the mesh, so that further processing can take place more easily. Other applications may require the mesh to have a pre-defined structure, which is often different from the one of the original general-purpose mesh. The central focus of this thesis is the automatic remeshing of highly detailed surface triangulations. Besides a thorough discussion of state-of-the-art applications such as real-time rendering and simulation, new approaches are proposed which use remeshing for topological analysis, flexible mesh generation and 3D compression. Furthermore, innovative methods are introduced to post- process polygonal models in order to recover information which was lost, or hidden, by a prior remeshing process. Besides the technical contributions, this thesis aims at showing that surface remeshing is much more useful than it may seem at a first sight, as it represents a nearly fundamental step for making several applications feasible in practice. Sommario Data la specializzazione dei più diffusi acceleratori grafici, le mesh triangolari restano la rappresentazione principale per le superfici 3D. Esse sono la forma più semplice di interpolazione fra campioni di superficie, i quali possono venire acquisiti tramite uno scanner laser, calcolati a partire da un campo scalare elaborato su una griglia regolare, oppure estratti da immagini 3D biomedicali. Tipicamente, i metodi usati per la generazione di mesh triangolari a partire da dati grezzi tentano di perdere la minor quantità di informazioni possibile, in modo che i modelli di superficie risultanti possano essere utilizzati in un insieme di scenari il più ampio possibile. Quando tali modelli general-purpose devono essere utilizzati in un particolare contesto di applicazione, comunque, vale spesso la pena di considerare un pre-processing. In alcuni casi, ad esempio, è conveniente modificare leggermente la geometria e/o la connettività della mesh, in modo da facilitarne ulteriori elaborazioni. Altre applicazioni potrebbero richiedere che la mesh abbia una struttura pre-definita, che spesso non coincide con quella della mesh general-purpose originale. Il tema trattato da questa tesi è il remeshing automatico di superfici 3D triangolate ad alto livello di dettaglio. Oltre a discutere i campi di applicazione ed i metodi proposti in letteratura, come il rendering in tempo reale e la simulazione, questa tesi propone nuovi approcci che utilizzano il remeshing per l'analisi della topologia, per rendere più flessibili i processi di generazione di mesh e per la compressione di superfici 3D. Inoltre, viene mostrato come elaborare modelli poligonali in modo da recuperare informazioni perdute, o nascoste, da un precedente processo di remeshing. Oltre all'aspetto tecnico-scientifico, questa tesi si propone di mostrare che il remeshing di superfici è molto più utile di quanto possa sembrare ad una prima analisi, poiché rappresenta un passo, spesso fondamentale, per rendere molte applicazioni fattibili in pratica e migliorarne l'impatto. Acknowledgements This work would not have been possible without the support of some key persons in my life. I first thank my parents for their encouragement in proceeding straight ahead on my studies. I also thank them for their precious suggestions, which never turned into constraints, about what could have been interesting to study (they have been prophetic !). Many thanks are also due to my advisors Bianca Falcidieno and Michela Spagnuolo. Their assistance during my years at IMATI has been invaluable. They spent a lot of time for reading, correcting and suggesting modifications to my manuscripts, and their comments undoubtedly improved my publications as well as this dissertation. I would also like to thank Prof. Geoff Wyvill for his help in preparing my work about parametric surfaces, including my Siggraph talk. I also owe a lot to Prof. Jarek Rossignac, who introduced me in the world of 3D compression and made it possible to publish our joint work on a prestigious journal such as ACM TOG. Finally, I must express my appreciation to all of my colleagues at IMATI who really helped me and made it possible to turn abstract ideas into scientific papers and to efficiently organize my work within the institute. In particular, I thank my colleague Silvia Biasotti who co-authored the work about topological analysis, and took care of most of the mathematical stuff. Index M. Attene Index 1 Introduction ...............................................................................................13 1.1 Motivation .............................................................................................14 1.2 Contributions .........................................................................................16 1.3 Overview of Material.............................................................................17 2 Background................................................................................................19 2.1 Notions from Point-Set Topology .........................................................19 Equivalence Relations.................................................................................19 Topological Spaces .....................................................................................20 Metric Spaces..............................................................................................20 Euclidean Space ..........................................................................................21 Subspace Topology .....................................................................................22 Homeomorphisms of topological spaces ....................................................23 Regular sets (r-sets).....................................................................................24 Manifolds ....................................................................................................25 2.2 Triangle Meshes ....................................................................................26 Simplicial Complexes .................................................................................26 Euler Characteristic.....................................................................................28 3 Related Work.............................................................................................30 3.1 Structured and unstructured meshes......................................................30 3.2 Mesh generation and Remeshing...........................................................31 3.3 Polygonal Mesh Simplification .............................................................33 3.4 Generation of Finite Element Meshes ...................................................35 3.5 Mesh Parameterization ..........................................................................36 3.6 Compression..........................................................................................37 3.7 Surface Fairing ......................................................................................39 3.8 Mesh Repairing .....................................................................................40 4 Surface Reconstruction..............................................................................43 4.1 Context of Application and Prior Research...........................................43 4.2 Problem statement and Related Work ...................................................45 4.3 Constrained Sculpturing ........................................................................48 The Extended Gabriel Hypergraph .............................................................49 Using constraints.........................................................................................51 Complexity Analysis...................................................................................55 4.4 Implementation......................................................................................55 4.5 Results and discussion...........................................................................56 5 Triangulation of Parametric Surfaces........................................................59