3D Shape Matching through Topological Structures

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Analysis and Parameterization of Triangulated Surfaces

This dissertation deals with the analysis and parameterization of surfaces represented by triangle meshes, that is, piecewise linear surfaces which enable a simple representation of 3D models commonly used in mathematics and computer science. Providing equivalent and high-level representations of a 3D triangle mesh M is of basic importance for approaching different computational problems and applications in the research fields of Computational Geometry, Computer Graphics, Geometry Processing, and Shape Modeling. The aim of the thesis is to show how high-level representations of a given surface M can be used to find other high-level or equivalent descriptions of M and vice versa. Furthermore, this analysis is related to the study of local and global properties of triangle meshes depending on the information that we want to capture and needed by the application context. The local analysis of an arbitrary triangle mesh M is based on a multi-scale segmentation of M together with the induced local parameterization, where we replace the common hypothesis of decomposing M into a family of disc-like patches (i.e., 0-genus and one boundary component) with a feature-based segmentation of M into regions of 0-genus without constraining the number of boundary components of each patch. This choice and extension is motivated by the necessity of identifying surface patches with features, of reducing the parameterization distortion, and of better supporting standard applications of the parameterization such as remeshing or more generally surface approximation, texture mapping, and compression. The global analysis, characterization, and abstraction of M take into account its topological and geometric aspects represented by the combinatorial structure of M (i.e., the mesh connectivity) with the associated embedding in R^3. Duality and dual Laplacian smoothing are the first characterizations of M presented with the final aim of a better understanding of the relations between mesh connectivity and geometry, as discussed by several works in this research area, and extended in the thesis to the case of 3D parameterization. The global analysis of M has been also approached by defining a real function on M which induces a Reeb graph invariant with respect to affine transformations and best suited for applications such as shape matching and comparison. Morse theory and the Reeb graph were also used for supporting a new and simple method for solving the global parameterization problem, that is, the search of a cut graph of an arbitrary triangle mesh M. The main characteristics of the proposed approach with respect to previous work are its capability of defining a family of cut graphs, instead of just one cut, of bordered and closed surfaces which are treated with a unique approach. Furthermore, each cut graph is smooth and the way it is built is based on the cutting procedure of 0-genus surfaces that was used for the local parameterization of M. As discussed in the thesis, defining a family of cut graphs provides a great flexibility and effective simplifications of the analysis, modeling, and visualization of (time-depending) scalar and vector fields; in fact, the global parameterization of M enables to reduce th

Computer-aided Visualization of Colonoscopy

Colonoscopy is the most widely used medical technique to examine the human large intestine (colon) and eliminate precancerous or malignant lesions, i.e., polyps. It uses a high-definition camera to examine the inner surface of the colon. Very often, a portion of the colon surface is not visualized during the procedure. Unsurveyed portions of the colon can harbor polyps that then progress to colorectal cancer. Unfortunately, it is hard for the endoscopist to realize there is unsurveyed surface from the video as it is formed. A system to alert endoscopists to missed surface area could thus more fully protect patients from colorectal cancer following colonoscopy. In this dissertation computer-aided visualization techniques were developed in order to solve this problem:1. A novel Simultaneous Localization and Mapping (SLAM) algorithm called RNNSLAM was proposed to address the difficulties of applying a traditional SLAM system on colonic images. I improved a standard SLAM system with a previously proposed Recurrent Neural Network for Depth and Pose Estimation (RNN-DP). The combination of SLAM’s optimization mechanism and RNN-DP’s prior knowledge achieved state-of-the-art performance on colonoscopy, especially addressing the drift problem in both SLAM and RNN-DP. A fusion module was added to this system to generate a dense 3D surface.2. I conducted exploration research on recognizing colonic places that have been visited based on video frames. This technique called image relocalization or retrieval is needed for helping the endoscopist to fully survey the previously unsurveyed regions. A benchmark testing dataset was created for colon image retrieval. Deep neural networks were successfully trained using Structure from Motion results on colonoscopy and achieved promising results.3. To visualize highly-curved portions of a colon or the whole colon, a generalized cylinder deformation algorithm was proposed to semi-flatten the geometry of the colon model for more succinct and global visualization.Doctor of Philosoph

3D Shape Similarity Through Structural Descriptors

Due to the recent improvements to 3D object acquisition, visualization and modeling techniques, the number of 3D models available is more and more growing, and there is an increasing demand for tools supporting the automatic search for 3D objects and their sub-parts in digital archives. Whilst there are already techniques for rapidly extracting knowledge from massive volumes of texts (like Google [htt]) it is harder to structure, filter, organize, retrieve and maintain archives of digital shapes like images, 3D objects, 3D animations and virtual or augmented reality. This situations suggests that in the future a primary challenge in computer graphics will be how to find models having a similar global and/or local appearance. Shape descriptors and the methodologies used to compare them, occupy an important role for achieving this task. For this reason a first contribution of this thesis is to provide a critical analysis of the most representative geometric and structural shape descriptors with respect to a set of properties that shape descriptors should have. This analysis is targeted at highlighting the differences between descriptors in order to better understand where a descriptor fails and another succeed. As a second contribution, the thesis investigates the problem of using a structural descriptor for shape comparison purposes. A large class of structural shape descriptors can be easily encoded as directed, a-cyclic and attributed graphs, thus the problem of comparing structural descriptors is approached as a graph matching problem. The techniques used for graph comparison have an exponential computational complexity and it is therefore necessary to define an algorithmic approximation of the optimal solution. The methods for structural descriptors comparison, commonly used in the computer graphics community, consist of heuristic graph matching algorithms for specific application tasks, while it is lacking a general approach suitable for incorporating different heuristics applicable in different application tasks. The second contribution presented in this thesis is aimed at defining a framework for expressing the optimal algorithm for the computation of the maximal common subgraph in a formalization which makes it straightforward usable for plugging heuristics in it, in order to achieving different approximations of the optimal solution according to the specific case. Implemented heuristics for robust graph matching with respect to graph structural noise are discussed and experimented on sub-part correspondence between similar 3D objects, and shape retrieval application with respect to different structural graph descriptors

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