3D time series analysis of cell shape using Laplacian approaches

Background: Fundamental cellular processes such as cell movement, division or food uptake critically depend on cells being able to change shape. Fast acquisition of three-dimensional image time series has now become possible, but we lack efficient tools for analysing shape deformations in order to understand the real three-dimensional nature of shape changes. Results: We present a framework for 3D+time cell shape analysis. The main contribution is three-fold: First, we develop a fast, automatic random walker method for cell segmentation. Second, a novel topology fixing method is proposed to fix segmented binary volumes without spherical topology. Third, we show that algorithms used for each individual step of the analysis pipeline (cell segmentation, topology fixing, spherical parameterization, and shape representation) are closely related to the Laplacian operator. The framework is applied to the shape analysis of neutrophil cells. Conclusions: The method we propose for cell segmentation is faster than the traditional random walker method or the level set method, and performs better on 3D time-series of neutrophil cells, which are comparatively noisy as stacks have to be acquired fast enough to account for cell motion. Our method for topology fixing outperforms the tools provided by SPHARM-MAT and SPHARM-PDM in terms of their successful fixing rates. The different tasks in the presented pipeline for 3D+time shape analysis of cells can be solved using Laplacian approaches, opening the possibility of eventually combining individual steps in order to speed up computations

Discrete conformal mappings via circle patterns

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples

Numerical and variational aspects of mesh parameterization and editing

A surface parameterization is a smooth one-to-one mapping between the surface and a parametric domain. Typically, surfaces with disk topology are mapped onto the plane and genus-zero surfaces onto the sphere. As any attempt to flatten a non-trivial surface onto the plane will inevitably induce a certain amount of distortion, the main concern of research on this topic is to minimize parametric distortion. This thesis aims at presenting a balanced blend of mathematical rigor and engineering intuition to address the challenges raised by the mesh parameterization problem. We study the numerical aspects of mesh parameterization in the light of parallel developments in both mathematics and engineering. Furthermore, we introduce the concept of quasi-harmonic maps for reducing distortion in the fixed boundary case and extend it to both the free boundary and the spherical case. Thinking of parameterization in a more general sense as the construction of one or several scalar fields on a surface, we explore the potential of this construction for mesh deformation and surface matching. We propose an \u27;on-surface parameterization\u27; for guiding the deformation process and performing surface matching. A direct harmonic interpolation in the quaternion domain is also shown to give promising results for deformation transfer.Eine Flächenparameterisierung ist eine globale bijektive Abbildung zwischen der Fläche und einem zugehörigen parametrischen Gebiet. Gewöhnlich werden Flächen mit scheibenförmiger Topologie auf eine Kreisscheibe und Flächen mit Genus Null auf eine Sphäre abgebildet. Das Hauptinteresse der Forschung an diesem Thema ist die Minimierung der parametrischen Verzerrung, die unweigerlich bei jedem Versuch, eine nicht triviale Fläche über einer Ebene zu parameterisieren, erzeugt wird. Diese Arbeit strebt zur Behandlung des Parametrisierungsproblems eine ausgeglichene Mischung zwischen mathematischer Präzision und ingenieurwissenschaftlicher Intuition an. Wir behandeln dabei die numerischen Aspekte des Parameterisierungsproblems im Hinblick auf die aktuellen parallelen Entwicklungen in der Mathematik und den Ingenieurwissenschaften. Weiterhin führen wir das Konzept der quasi-harmonischen Abbildungen ein, um die Verzerrung bei gegebenen Randbedingungen zu verringern. Anschließend verallgemeinern wir dieses Konzept auf den sphärischen Fall und auf den Fall mit freien Randbedingungen. Durch allgemeinere Betrachtung der Parameterisierung als Konstruktion eines oder mehrerer skalarer Felder auf einer Fläche ergibt sich ein neuer Ansatz zur Netzdeformation und der Erzeugung von Flächenkorrespondenzen. Wir stellen eine \u27;on-surface parameterization\u27; vor, welche den Deformationsprozess leitet und Flächenkorrespondenzen erstellt. Darüber hinaus zeigt eine direkte harmonische Interpolation in der Domäne der Quaternionen auch vielversprechende Resultate für die Übertragung von Deformationen

Understanding the Structure of 3D Shapes

Compact representations of three dimensional objects are very often used in computer graphics to create effective ways to analyse, manipulate and transmit 3D models. Their ability to abstract from the concrete shapes and expose their structure is important in a number of applications, spanning from computer animation, to medicine, to physical simulations. This thesis will investigate new methods for the generation of compact shape representations. In the first part, the problem of computing optimal PolyCube base complexes will be considered. PolyCubes are orthogonal polyhedra used in computer graphics to map both surfaces and volumes. Their ability to resemble the original models and at the same time expose a very simple and regular structure is important in a number of applications, such as texture mapping, spline fitting and hex-meshing. The second part will focus on medial descriptors. In particular, two new algorithms for the generation of curve-skeletons will be presented. These methods are completely based on the visual appearance of the input, therefore they are independent from the type, number and quality of the primitives used to describe a shape, determining, thus, an advancement to the state of the art in the field

Understanding the Structure of 3D Shapes

Compact representations of three dimensional objects are very often used in computer graphics to create effective ways to analyse, manipulate and transmit 3D models. Their ability to abstract from the concrete shapes and expose their structure is important in a number of applications, spanning from computer animation, to medicine, to physical simulations. This thesis will investigate new methods for the generation of compact shape representations. In the first part, the problem of computing optimal PolyCube base complexes will be considered. PolyCubes are orthogonal polyhedra used in computer graphics to map both surfaces and volumes. Their ability to resemble the original models and at the same time expose a very simple and regular structure is important in a number of applications, such as texture mapping, spline fitting and hex-meshing. The second part will focus on medial descriptors. In particular, two new algorithms for the generation of curve-skeletons will be presented. These methods are completely based on the visual appearance of the input, therefore they are independent from the type, number and quality of the primitives used to describe a shape, determining, thus, an advancement to the state of the art in the field

Métamorphose de maillage 3D

Cette thèse de doctorat aborde spécifiquement le problème de la métamorphose entre différents maillages 3D, qui peut assurer un niveau élevé de qualité pour la séquence de transition, qui devrait être aussi lisse et progressive que possible, cohérente par rapport à la géométrie et la topologie, et visuellement agréable. Les différentes étapes impliquées dans le processus de transformation sont développées dans cette thèse. Nos premières contributions concernent deux approches différentes des paramétrisations: un algorithme de mappage barycentrique basé sur la préservation des rapports de longueur et une technique de paramétrisation sphérique, exploitant la courbure Gaussien. L'évaluation expérimentale, effectuées sur des modèles 3D de formes variées, démontré une amélioration considérable en termes de distorsion maillage pour les deux méthodes. Afin d aligner les caractéristiques des deux modèles d'entrée, nous avons considéré une technique de déformation basée sur la fonction radial CTPS C2a approprié pour déformer le mappage dans le domaine paramétrique et maintenir un mappage valide a travers le processus de mouvement. La dernière contribution consiste d une une nouvelle méthode qui construit un pseudo metamaillage qui évite l'exécution et le suivi des intersections d arêtes comme rencontrées dans l'état-of-the-art. En outre, notre méthode permet de réduire de manière drastique le nombre de sommets normalement nécessaires dans une structure supermesh. Le cadre générale de métamorphose a été intégré dans une application prototype de morphing qui permet à l'utilisateur d'opérer de façon interactive avec des modèles 3D et de contrôler chaque étape du processusThis Ph.D. thesis specifically deals with the issue of metamorphosis of 3D objects represented as 3D triangular meshes. The objective is to elaborate a complete 3D mesh morphing methodology which ensures high quality transition sequences, smooth and gradual, consistent with respect to both geometry and topology, and visually pleasant. Our first contributions concern the two different approaches of parameterization: a new barycentric mapping algorithm based on the preservation of the mesh length ratios, and a spherical parameterization technique, exploiting a Gaussian curvature criterion. The experimental evaluation, carried out on 3D models of various shapes, demonstrated a considerably improvement in terms of mesh distortion for both methods. In order to align the features of the two input models, we have considered a warping technique based on the CTPS C2a radial basis function suitable to deform the models embeddings in the parametric domain maintaining a valid mapping through the entire movement process. We show how this technique has to be adapted in order to warp meshes specified in the parametric domains. A final contribution consists of a novel algorithm for constructing a pseudo-metamesh that avoids the complex process of edge intersections encountered in the state-of-the-art. The obtained mesh structure is characterized by a small number of vertices and it is able to approximate both the source and target shapes. The entire mesh morphing framework has been integrated in an interactive application that allows the user to control and visualize all the stages of the morphing processEVRY-INT (912282302) / SudocSudocFranceF

Unconstrained Spherical Parameterization

We introduce a novel approach for the construction of spherical parameterizations based on energy minimization. The energies are derived in a general manner from classic formulations well known in the planar parameterization setting (e.g., conformal, Tutte, area, stretch energies, etc.), based on the following principles: the energy should (1) be a measure of spherical triangles; (2) treat energies independently of the triangle location on the sphere; and (3) converge to the continuous energy from above under refinement. Based on these considerations we give a very simple non-linear modification of standard formulas that fulfills all these requirements. The method avoids the usual collapse of flat energies when they are transferred to the spherical setting without additional constraints (e.g., fixing three or more vertices). Our unconstrained energy minimization problem is amenable to the use of standard solvers. Consequently the implementation effort is minimal while still achieving excellent robustness and performance through the use of widely available numerical minimization software

Approximation of Surfaces by Normal Meshes

This thesis introduces a novel geometry processing pipeline based on unconstrained spherical parameterization and normal remeshing. We claim three main contributions: First we show how to increase the stability of Normal Mesh construction, while speeding it up by decomposing the process into two stages: parameterization and remeshing. We show that the remeshing step can be seen as resampling under a small perturbation of the given parameterization. Based on this observation we describe a novel algorithm for efficient and stable (interpolatory) normal mesh construction via parameterization perturbation. Our second contribution is the introduction of Variational Normal Meshes. We describe a novel algorithm for encoding these meshes, and use our implementation to argue that variational normal meshes have a higher approximation quality than interpolating normal meshes, as expected. In particular we demonstrate that interpolating normal meshes have about 60 percent higher Hausdorff approximation error for the same number of vertices than our novel variational normal meshes. We also show that variational normal meshes have less aliasing artifacts than interpolatory normal meshes. The third contribution is on creating parameterizations for unstructured genus zero meshes. Previous approaches could only avoid collapses by introducing artificial constraints or continuous reprojections, which are avoided by our method. The key idea is to define upper bound energies that are still good approximations. We achieve this by dividing classical planar triangle energies by the minimum distance to the sphere center. We prove that these simple modifaction provides the desired upper bounds and are good approximations in the finite element sense. We have implemented all algorithms and provide example results and statistical data supporting our theoretical observations