Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey

This paper provides a tutorial and survey for a specific kind of illustrative visualization technique: feature lines. We examine different feature line methods. For this, we provide the differential geometry behind these concepts and adapt this mathematical field to the discrete differential geometry. All discrete differential geometry terms are explained for triangulated surface meshes. These utilities serve as basis for the feature line methods. We provide the reader with all knowledge to re-implement every feature line method. Furthermore, we summarize the methods and suggest a guideline for which kind of surface which feature line algorithm is best suited. Our work is motivated by, but not restricted to, medical and biological surface models.Comment: 33 page

Geometric deep learning

The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas

Diffeomorphic Deformation via Sliced Wasserstein Distance Optimization for Cortical Surface Reconstruction

Mesh deformation is a core task for 3D mesh reconstruction, but defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing two randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance. Nevertheless, the set-based approach still has limitations such as lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose a novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via \textit{varifold} representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is an effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. Furthermore, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics

Conformal electromagnetic wave propagation using primal mimetic finite elements

Elektromagnetische Wellenausbreitung bildet die physikalische Grundlage für unzählige Anwendungen in verschiedenen Bereichen der heutigen Welt. Um räumliche Szenarien zu modellieren, muss der kontinuierliche Raum in geeigneter Weise in ein Rechengebiet umgewandelt werden. Üblich diskretisierte Modelle – welche auf verschiedenen Größen beruhen – berücksichtigen die Beziehungen zwischen Feldvariablen mittels Relationen, welche durch partielle Differentialgleichungen repräsentiert werden. Um mathematische Beziehungen zwischen abhängigen Variablen in zweckdienlicher Art nachzubilden, schaffen hyperkomplexe Zahlensysteme ein passendes alternatives Rahmenwerk. Dieser Ansatz bezweckt das Einbinden bestimmter Systemeigenschaften und umfasst zusätzlich zur Modellierung von Feldproblemen, bei denen alle Variablen vorkommen, auch vereinfachte Modelle. Um eine wettbewerbsfähige Alternative zur üblichen numerischen Behandlung elektromagnetischer Felder in beobachtungsorientierter Weise darzubieten, wird das elektrische und magnetische Feld elektromagnetischer Wellenfelder als eine zusammengefasste Feldgröße, eingebettet im Funktionenraum, verstanden. Dieses Vorgehen ist intuitiv, da beide Felder in der Elektrodynamik gemeinsam auftreten und direkt messbar sind. Der Schwerpunkt dieser Arbeit ist in zwei Ziele untergliedert. Auf der einen Seite wird ein umformuliertes Maxwell-System in einer metrikfreien Umgebung mittels dem sogenannten „bikomplexen Ansatz“ umfassend untersucht. Auf der anderen Seite wird eine mögliche numerische Implementierung hinsichtlich der Finite-Elemente-Methode auf modernem Wege durch Nutzung der diskreten äußeren Analysis mit Fokus auf Genauigkeitsbelange bewertet. Hinsichtlich der numerischen Genauigkeitsbewertung wird demonstriert, dass der vorgelegte Ansatz grundsätzlich eine höhere Exaktheit zeigt, wenn man ihn mit Formulierungen vergleicht, welche auf der Helmholtz-Gleichung beruhen. Diese Dissertation trägt eine generalisierte hyperkomplexe alternative Darstellung von gewöhnlichen elektrodynamischen Ausdrucksweisen zum Themengebiet der Wellenausbreitung bei. Durch die Nutzung einer direkten Formulierung des elektrischen Feldes in Verbindung mit dem magnetischen Feld wird die Rechengenauigkeit von Randwertproblemen erhöht. Um diese Genauigkeitserhöhung zu erreichen, wird eine geeignete Erweiterung der de Rham-Kohomologie unterbreitet.Electromagnetic wave propagation provides the physical basis for countless applications in various subjects of today’s world. In order to model spatial scenarios, the continuous space must be converted to an appropriate computational domain. Ordinarily discretized models – which are based on distinct quantities – consider the connection between field variables by relations which are represented by partial differential equations. To reproduce mathematical relationships between dependent variables in a convenient manner, hypercomplex number systems build a suitable alternative framework. This approach aims to incorporate certain system properties and covers, in addition to the modeling of field problems where all variables are present, also simplified models. To provide a competitive alternative to the ordinary numerical handling of electromagnetic fields in an observation-based way, the electric and magnetic field of electromagnetic wave fields is understood as only one combined field variable embedded in the function space. This procedure is intuitive since both fields occur together in electrodynamics and are directly measureable. The focus of this thesis is twofold. On the one side, a reformulated Maxwell system is broadly investigated in a metric-free environment by the use of the so-called ”bicomplex approach”. On the other side, a possible numerical implementation concerning the Finite Element Method is evaluated in a modern way by the use of discrete exterior calculus with focus on accuracy matters. Regarding the numerical accuracy evaluation, it is demonstrated that the presented approach yields a higher exactness in general when comparing it to formulations which are based on the Helmholtz equation. This thesis contributes generalized hypercomplex alternative representations of ordinary electrodynamic expressions to the topic of wave propagation. By the use of a direct formulation of the electric field in conjunction with the magnetic field, the computational accuracy of boundary value problems is improved. In order to achieve this increase of accuracy, an appropriate enhancement of the de Rham cohomology is proposed