Constructing Desirable Scalar Fields for Morse Analysis on Meshes

Morse theory is a powerful mathematical tool that uses the local differential properties of a manifold to make conclusions about global topological aspects of the manifold. Morse theory has been proven to be a very useful tool in computer graphics, geometric data processing and understanding. This work is divided into two parts. The first part is concerned with constructing geometry and symmetry aware scalar functions on a triangulated $2$-manifold. To effectively apply Morse theory to discrete manifolds, one needs to design scalar functions on them with certain properties such as respecting the symmetry and the geometry of the surface and having the critical points of the scalar function coincide with feature or symmetry points on the surface. In this work, we study multiple methods that were suggested in the literature to construct such functions such as isometry invariant scalar functions, Poisson fields and discrete conformal factors. We suggest multiple refinements to each family of these functions and we propose new methods to construct geometry and symmetry-aware scalar functions using mainly the theory of the Laplace-Beltrami operator. Our proposed methods are general and can be applied in many areas such as parametrization, shape analysis, symmetry detection and segmentation. In the second part of the thesis we utilize Morse theory to give topologically consistent segmentation algorithms

Spectral Geometric Methods for Deformable 3D Shape Retrieval

As 3D applications ranging from medical imaging to industrial design continue to grow, so does the importance of developing robust 3D shape retrieval systems. A key issue in developing an accurate shape retrieval algorithm is to design an efficient shape descriptor for which an index can be built, and similarity queries can be answered efficiently. While the overwhelming majority of prior work on 3D shape analysis has concentrated primarily on rigid shape retrieval, many real objects such as articulated motions of humans are nonrigid and hence can exhibit a variety of poses and deformations. In this thesis, we present novel spectral geometric methods for analyzing and distinguishing between deformable 3D shapes. First, we comprehensively review recent shape descriptors based on the spectral decomposition of the Laplace-Beltrami operator, which provides a rich set of eigenbases that are invariant to intrinsic isometries. Then we provide a general and flexible framework for the analysis and design of shape signatures from the spectral graph wavelet perspective. In a bid to capture the global and local geometry, we propose a multiresolution shape signature based on a cubic spline wavelet generating kernel. This signature delivers best-in-class shape retrieval performance. Second, we investigate the ambiguity modeling of codebook for the densely distributed low-level shape descriptors. Inspired by the ability of spatial cues to improve discrimination between shapes, we also propose to adopt the isocontours of the second eigenfunction of the Laplace-Beltrami operator to perform surface partition, which can significantly ameliorate the retrieval performance of the time-scaled local descriptors. To further enhance the shape retrieval accuracy, we introduce an intrinsic spatial pyramid matching approach. Extensive experiments are carried out on two 3D shape benchmarks to assess the performance of the proposed spectral geometric approaches in comparison with state-of-the-art methods

3D object retrieval and segmentation: various approaches including 2D poisson histograms and 3D electrical charge distributions.

Nowadays 3D models play an important role in many applications: viz. games, cultural heritage, medical imaging etc. Due to the fast growth in the number of available 3D models, understanding, searching and retrieving such models have become interesting fields within computer vision. In order to search and retrieve 3D models, we present two different approaches: one is based on solving the Poisson Equation over 2D silhouettes of the models. This method uses 60 different silhouettes, which are automatically extracted from different viewangles. Solving the Poisson equation for each silhouette assigns a number to each pixel as its signature. Accumulating these signatures generates a final histogram-based descriptor for each silhouette, which we call a SilPH (Silhouette Poisson Histogram). For the second approach, we propose two new robust shape descriptors based on the distribution of charge density on the surface of a 3D model. The Finite Element Method is used to calculate the charge density on each triangular face of each model as a local feature. Then we utilize the Bag-of-Features and concentric sphere frameworks to perform global matching using these local features. In addition to examining the retrieval accuracy of the descriptors in comparison to the state-of-the-art approaches, the retrieval speeds as well as robustness to noise and deformation on different datasets are investigated. On the other hand, to understand new complex models, we have also utilized distribution of electrical charge for proposing a system to decompose models into meaningful parts. Our robust, efficient and fully-automatic segmentation approach is able to specify the segments attached to the main part of a model as well as locating the boundary parts of the segments. The segmentation ability of the proposed system is examined on the standard datasets and its timing and accuracy are compared with the existing state-of-the-art approaches

Retrieval and classification methods for textured 3D models: a comparative study

International audienceThis paper presents a comparative study of six methods for the retrieval and classification of tex-tured 3D models, which have been selected as representative of the state of the art. To better analyse and control how methods deal with specific classes of geometric and texture deformations, we built a collection of 572 synthetic textured mesh models, in which each class includes multiple texture and geometric modifications of a small set of null models. Results show a challenging, yet lively, scenario and also reveal interesting insights in how to deal with texture information according to different approaches, possibly working in the CIELab as well as in modifications of the RGB colour space

Inflammation evaluation through low-cost 3D scanning of human body parts

[CATALÀ] S'ha creat un programa que permet carregar models 3D adquirits a partir d'articulacions reals. Utilitzant diferents tècniques com l'alineació de models, la comparació visual i el càlcul de diferents mesures l'eina proporciona informació valuosa per a facilitar l'avaluació del nivell d'inflamació.[ANGLÈS] A program capable of loading 3D models acquired from real joints has been created. Using different techniques such as model alignment, visual comparison and computation of different measures the tool gives valuable information to facilitate the assessment of the inflammation level

Machine Learning Approaches to Human Body Shape Analysis

Soft biometrics, biomedical sciences, and many other fields of study pay particular attention to the study of the geometric description of the human body, and its variations. Although multiple contributions, the interest is particularly high given the non-rigid nature of the human body, capable of assuming different poses, and numerous shapes due to variable body composition. Unfortunately, a well-known costly requirement in data-driven machine learning, and particularly in the human-based analysis, is the availability of data, in the form of geometric information (body measurements) with related vision information (natural images, 3D mesh, etc.). We introduce a computer graphics framework able to generate thousands of synthetic human body meshes, representing a population of individuals with stratified information: gender, Body Fat Percentage (BFP), anthropometric measurements, and pose. This contribution permits an extensive analysis of different bodies in different poses, avoiding the demanding, and expensive acquisition process. We design a virtual environment able to take advantage of the generated bodies, to infer the body surface area (BSA) from a single view. The framework permits to simulate the acquisition process of newly introduced RGB-D devices disentangling different noise components (sensor noise, optical distortion, body part occlusions). Common geometric descriptors in soft biometric, as well as in biomedical sciences, are based on body measurements. Unfortunately, as we prove, these descriptors are not pose invariant, constraining the usability in controlled scenarios. We introduce a differential geometry approach assuming body pose variations as isometric transformations of the body surface, and body composition changes covariant to the body surface area. This setting permits the use of the Laplace-Beltrami operator on the 2D body manifold, describing the body with a compact, efficient, and pose invariant representation. We design a neural network architecture able to infer important body semantics from spectral descriptors, closing the gap between abstract spectral features, and traditional measurement-based indices. Studying the manifold of body shapes, we propose an innovative generative adversarial model able to learn the body shapes. The method permits to generate new bodies with unseen geometries as a walk on the latent space, constituting a significant advantage over traditional generative methods

Composite Finite Elements for Trabecular Bone Microstructures

In many medical and technical applications, numerical simulations need to be performed for objects with interfaces of geometrically complex shape. We focus on the biomechanical problem of elasticity simulations for trabecular bone microstructures. The goal of this dissertation is to develop and implement an efficient simulation tool for finite element simulations on such structures, so-called composite finite elements. We will deal with both the case of material/void interfaces (complicated domains) and the case of interfaces between different materials (discontinuous coefficients). In classical finite element simulations, geometric complexity is encoded in tetrahedral and typically unstructured meshes. Composite finite elements, in contrast, encode geometric complexity in specialized basis functions on a uniform mesh of hexahedral structure. Other than alternative approaches (such as e.g. fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes, and extended finite element methods), the composite finite elements are tailored to geometry descriptions by 3D voxel image data and use the corresponding voxel grid as computational mesh, without introducing additional degrees of freedom, and thus making use of efficient data structures for uniformly structured meshes. The composite finite element method for complicated domains goes back to Wolfgang Hackbusch and Stefan Sauter and restricts standard affine finite element basis functions on the uniformly structured tetrahedral grid (obtained by subdivision of each cube in six tetrahedra) to an approximation of the interior. This can be implemented as a composition of standard finite element basis functions on a local auxiliary and purely virtual grid by which we approximate the interface. In case of discontinuous coefficients, the same local auxiliary composition approach is used. Composition weights are obtained by solving local interpolation problems for which coupling conditions across the interface need to be determined. These depend both on the local interface geometry and on the (scalar or tensor-valued) material coefficients on both sides of the interface. We consider heat diffusion as a scalar model problem and linear elasticity as a vector-valued model problem to develop and implement the composite finite elements. Uniform cubic meshes contain a natural hierarchy of coarsened grids, which allows us to implement a multigrid solver for the case of complicated domains. Besides simulations of single loading cases, we also apply the composite finite element method to the problem of determining effective material properties, e.g. for multiscale simulations. For periodic microstructures, this is achieved by solving corrector problems on the fundamental cells using affine-periodic boundary conditions corresponding to uniaxial compression and shearing. For statistically periodic trabecular structures, representative fundamental cells can be identified but do not permit the periodic approach. Instead, macroscopic displacements are imposed using the same set as before of affine-periodic Dirichlet boundary conditions on all faces. The stress response of the material is subsequently computed only on an interior subdomain to prevent artificial stiffening near the boundary. We finally check for orthotropy of the macroscopic elasticity tensor and identify its axes.Zusammengesetzte finite Elemente für trabekuläre Mikrostrukturen in Knochen In vielen medizinischen und technischen Anwendungen werden numerische Simulationen für Objekte mit geometrisch komplizierter Form durchgeführt. Gegenstand dieser Dissertation ist die Simulation der Elastizität trabekulärer Mikrostrukturen von Knochen, einem biomechanischen Problem. Ziel ist es, ein effizientes Simulationswerkzeug für solche Strukturen zu entwickeln, die sogenannten zusammengesetzten finiten Elemente. Wir betrachten dabei sowohl den Fall von Interfaces zwischen Material und Hohlraum (komplizierte Gebiete) als auch zwischen verschiedenen Materialien (unstetige Koeffizienten). In klassischen Finite-Element-Simulationen wird geometrische Komplexität typischerweise in unstrukturierten Tetraeder-Gittern kodiert. Zusammengesetzte finite Elemente dagegen kodieren geometrische Komplexität in speziellen Basisfunktionen auf einem gleichförmigen Würfelgitter. Anders als alternative Ansätze (wie zum Beispiel fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes und extended finite element methods) sind die zusammengesetzten finiten Elemente zugeschnitten auf die Geometriebeschreibung durch dreidimensionale Bilddaten und benutzen das zugehörige Voxelgitter als Rechengitter, ohne zusätzliche Freiheitsgrade einzuführen. Somit können sie effiziente Datenstrukturen für gleichförmig strukturierte Gitter ausnutzen. Die Methode der zusammengesetzten finiten Elemente geht zurück auf Wolfgang Hackbusch und Stefan Sauter. Man schränkt dabei übliche affine Finite-Element-Basisfunktionen auf gleichförmig strukturierten Tetraedergittern (die man durch Unterteilung jedes Würfels in sechs Tetraeder erhält) auf das approximierte Innere ein. Dies kann implementiert werden durch das Zusammensetzen von Standard-Basisfunktionen auf einem lokalen und rein virtuellen Hilfsgitter, durch das das Interface approximiert wird. Im Falle unstetiger Koeffizienten wird die gleiche lokale Hilfskonstruktion verwendet. Gewichte für das Zusammensetzen erhält man hier, indem lokale Interpolationsprobleme gelöst werden, wozu zunächst Kopplungsbedingungen über das Interface hinweg bestimmt werden. Diese hängen ab sowohl von der lokalen Geometrie des Interface als auch von den (skalaren oder tensorwertigen) Material-Koeffizienten auf beiden Seiten des Interface. Wir betrachten Wärmeleitung als skalares und lineare Elastizität als vektorwertiges Modellproblem, um die zusammengesetzten finiten Elemente zu entwickeln und zu implementieren. Gleichförmige Würfelgitter enthalten eine natürliche Hierarchie vergröberter Gitter, was es uns erlaubt, im Falle komplizierter Gebiete einen Mehrgitterlöser zu implementieren. Neben Simulationen einzelner Lastfälle wenden wir die zusammengesetzten finiten Elemente auch auf das Problem an, effektive Materialeigenschaften zu bestimmen, etwa für mehrskalige Simulationen. Für periodische Mikrostrukturen wird dies erreicht, indem man Korrekturprobleme auf der Fundamentalzelle löst. Dafür nutzt man affin-periodische Randwerte, die zu uniaxialem Druck oder zu Scherung korrespondieren. In statistisch periodischen trabekulären Mikrostrukturen lassen sich ebenfalls Fundamentalzellen identifizieren, sie erlauben jedoch keinen periodischen Ansatz. Stattdessen werden makroskopische Verschiebungen zu denselben affin-periodischen Randbedingungen vorgegeben, allerdings durch Dirichlet-Randwerte auf allen Seitenflächen. Die Spannungsantwort des Materials wird anschließend nur auf einem inneren Teilbereich berechnet, um künstliche Versteifung am Rand zu verhindern. Schließlich prüfen wir den makroskopischen Elastizitätstensor auf Orthotropie und identifizieren deren Achsen

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Toward Controllable and Robust Surface Reconstruction from Spatial Curves

Reconstructing surface from a set of spatial curves is a fundamental problem in computer graphics and computational geometry. It often arises in many applications across various disciplines, such as industrial prototyping, artistic design and biomedical imaging. While the problem has been widely studied for years, challenges remain for handling different type of curve inputs while satisfying various constraints. We study studied three related computational tasks in this thesis. First, we propose an algorithm for reconstructing multi-labeled material interfaces from cross-sectional curves that allows for explicit topology control. Second, we addressed the consistency restoration, a critical but overlooked problem in applying algorithms of surface reconstruction to real-world cross-sections data. Lastly, we propose the Variational Implicit Point Set Surface which allows us to robustly handle noisy, sparse and non-uniform inputs, such as samples from spatial curves