Doctor of Philosophy

dissertationInverse Electrocardiography (ECG) aims to noninvasively estimate the electrophysiological activity of the heart from the voltages measured at the body surface, with promising clinical applications in diagnosis and therapy. The main challenge of this emerging technique lies in its mathematical foundation: an inverse source problem governed by partial differential equations (PDEs) which is severely ill-conditioned. Essential to the success of inverse ECG are computational methods that reliably achieve accurate inverse solutions while harnessing the ever-growing complexity and realism of the bioelectric simulation. This dissertation focuses on the formulation, optimization, and solution of the inverse ECG problem based on finite element methods, consisting of two research thrusts. The first thrust explores the optimal finite element discretization specifically oriented towards the inverse ECG problem. In contrast, most existing discretization strategies are designed for forward problems and may become inappropriate for the corresponding inverse problems. Based on a Fourier analysis of how discretization relates to ill-conditioning, this work proposes refinement strategies that optimize approximation accuracy o f the inverse ECG problem while mitigating its ill-conditioning. To fulfill these strategies, two refinement techniques are developed: one uses hybrid-shaped finite elements whereas the other adapts high-order finite elements. The second research thrust involves a new methodology for inverse ECG solutions called PDE-constrained optimization, an optimization framework that flexibly allows convex objectives and various physically-based constraints. This work features three contributions: (1) fulfilling optimization in the continuous space, (2) formulating rigorous finite element solutions, and (3) fulfilling subsequent numerical optimization by a primal-dual interiorpoint method tailored to the given optimization problem's specific algebraic structure. The efficacy o f this new method is shown by its application to localization o f cardiac ischemic disease, in which the method, under realistic settings, achieves promising solutions to a previously intractable inverse ECG problem involving the bidomain heart model. In summary, this dissertation advances the computational research of inverse ECG, making it evolve toward an image-based, patient-specific modality for biomedical research

A Modular Approach to Large-scale Design Optimization of Aerospace Systems.

Gradient-based optimization and the adjoint method form a synergistic combination that enables the efficient solution of large-scale optimization problems. Though the gradient-based approach struggles with non-smooth or multi-modal problems, the capability to efficiently optimize up to tens of thousands of design variables provides a valuable design tool for exploring complex tradeoffs and finding unintuitive designs. However, the widespread adoption of gradient-based optimization is limited by the implementation challenges for computing derivatives efficiently and accurately, particularly in multidisciplinary and shape design problems. This thesis addresses these difficulties in two ways. First, to deal with the heterogeneity and integration challenges of multidisciplinary problems, this thesis presents a computational modeling framework that solves multidisciplinary systems and computes their derivatives in a semi-automated fashion. This framework is built upon a new mathematical formulation developed in this thesis that expresses any computational model as a system of algebraic equations and unifies all methods for computing derivatives using a single equation. The framework is applied to two engineering problems: the optimization of a nanosatellite with 7 disciplines and over 25,000 design variables; and simultaneous allocation and mission optimization for commercial aircraft involving 330 design variables, 12 of which are integer variables handled using the branch-and-bound method. In both cases, the framework makes large-scale optimization possible by reducing the implementation effort and code complexity. The second half of this thesis presents a differentiable parametrization of aircraft geometries and structures for high-fidelity shape optimization. Existing geometry parametrizations are not differentiable, or they are limited in the types of shape changes they allow. This is addressed by a novel parametrization that smoothly interpolates aircraft components, providing differentiability. An unstructured quadrilateral mesh generation algorithm is also developed to automate the creation of detailed meshes for aircraft structures, and a mesh convergence study is performed to verify that the quality of the mesh is maintained as it is refined. As a demonstration, high-fidelity aerostructural analysis is performed for two unconventional configurations with detailed structures included, and aerodynamic shape optimization is applied to the truss-braced wing, which finds and eliminates a shock in the region bounded by the struts and the wing.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111567/1/hwangjt_1.pd

Visual Odometry and Sparse Scene Reconstruction for UAVs with a Multi-Fisheye Camera System

Autonomously operating UAVs demand a fast localization for navigation, to actively explore unknown areas and to create maps. For pose estimation, many UAV systems make use of a combination of GPS receivers and inertial sensor units (IMU). However, GPS signal coverage may go down occasionally, especially in the close vicinity of objects, and precise IMUs are too heavy to be carried by lightweight UAVs. This and the high cost of high quality IMU motivate the use of inexpensive vision based sensors for localization using visual odometry or visual SLAM (simultaneous localization and mapping) techniques. The first contribution of this thesis is a more general approach to bundle adjustment with an extended version of the projective coplanarity equation which enables us to make use of omnidirectional multi-camera systems which may consist of fisheye cameras that can capture a large field of view with one shot. We use ray directions as observations instead of image points which is why our approach does not rely on a specific projection model assuming a central projection. In addition, our approach allows the integration and estimation of points at infinity, which classical bundle adjustments are not capable of. We show that the integration of far or infinitely far points stabilizes the estimation of the rotation angles of the camera poses. In its second contribution, we employ this approach to bundle adjustment in a highly integrated system for incremental pose estimation and mapping on light-weight UAVs. Based on the image sequences of a multi-camera system our system makes use of tracked feature points to incrementally build a sparse map and incrementally refines this map using the iSAM2 algorithm. Our system is able to optionally integrate GPS information on the level of carrier phase observations even in underconstrained situations, e.g. if only two satellites are visible, for georeferenced pose estimation. This way, we are able to use all available information in underconstrained GPS situations to keep the mapped 3D model accurate and georeferenced. In its third contribution, we present an approach for re-using existing methods for dense stereo matching with fisheye cameras, which has the advantage that highly optimized existing methods can be applied as a black-box without modifications even with cameras that have field of view of more than 180 deg. We provide a detailed accuracy analysis of the obtained dense stereo results. The accuracy analysis shows the growing uncertainty of observed image points of fisheye cameras due to increasing blur towards the image border. Core of the contribution is a rigorous variance component estimation which allows to estimate the variance of the observed disparities at an image point as a function of the distance of that point to the principal point. We show that this improved stochastic model provides a more realistic prediction of the uncertainty of the triangulated 3D points.Autonom operierende UAVs benötigen eine schnelle Lokalisierung zur Navigation, zur Exploration unbekannter Umgebungen und zur Kartierung. Zur Posenbestimmung verwenden viele UAV-Systeme eine Kombination aus GPS-Empfängern und Inertial-Messeinheiten (IMU). Die Verfügbarkeit von GPS-Signalen ist jedoch nicht überall gewährleistet, insbesondere in der Nähe abschattender Objekte, und präzise IMUs sind für leichtgewichtige UAVs zu schwer. Auch die hohen Kosten qualitativ hochwertiger IMUs motivieren den Einsatz von kostengünstigen bildgebenden Sensoren zur Lokalisierung mittels visueller Odometrie oder SLAM-Techniken zur simultanen Lokalisierung und Kartierung. Im ersten wissenschaftlichen Beitrag dieser Arbeit entwickeln wir einen allgemeineren Ansatz für die Bündelausgleichung mit einem erweiterten Modell für die projektive Kollinearitätsgleichung, sodass auch omnidirektionale Multikamerasysteme verwendet werden können, welche beispielsweise bestehend aus Fisheyekameras mit einer Aufnahme einen großen Sichtbereich abdecken. Durch die Integration von Strahlrichtungen als Beobachtungen ist unser Ansatz nicht von einem kameraspezifischen Abbildungsmodell abhängig solange dieses der Zentralprojektion folgt. Zudem erlaubt unser Ansatz die Integration und Schätzung von unendlich fernen Punkten, was bei klassischen Bündelausgleichungen nicht möglich ist. Wir zeigen, dass durch die Integration weit entfernter und unendlich ferner Punkte die Schätzung der Rotationswinkel der Kameraposen stabilisiert werden kann. Im zweiten Beitrag verwenden wir diesen entwickelten Ansatz zur Bündelausgleichung für ein System zur inkrementellen Posenschätzung und dünnbesetzten Kartierung auf einem leichtgewichtigen UAV. Basierend auf den Bildsequenzen eines Mulitkamerasystems baut unser System mittels verfolgter markanter Bildpunkte inkrementell eine dünnbesetzte Karte auf und verfeinert diese inkrementell mittels des iSAM2-Algorithmus. Unser System ist in der Lage optional auch GPS Informationen auf dem Level von GPS-Trägerphasen zu integrieren, wodurch sogar in unterbestimmten Situation - beispielsweise bei nur zwei verfügbaren Satelliten - diese Informationen zur georeferenzierten Posenschätzung verwendet werden können. Im dritten Beitrag stellen wir einen Ansatz zur Verwendung existierender Methoden für dichtes Stereomatching mit Fisheyekameras vor, sodass hoch optimierte existierende Methoden als Black Box ohne Modifzierungen sogar mit Kameras mit einem Gesichtsfeld von mehr als 180 Grad verwendet werden können. Wir stellen eine detaillierte Genauigkeitsanalyse basierend auf dem Ergebnis des dichten Stereomatchings dar. Die Genauigkeitsanalyse zeigt, wie stark die Genauigkeit beobachteter Bildpunkte bei Fisheyekameras zum Bildrand aufgrund von zunehmender Unschärfe abnimmt. Das Kernstück dieses Beitrags ist eine Varianzkomponentenschätzung, welche die Schätzung der Varianz der beobachteten Disparitäten an einem Bildpunkt als Funktion von der Distanz dieses Punktes zum Hauptpunkt des Bildes ermöglicht. Wir zeigen, dass dieses verbesserte stochastische Modell eine realistischere Prädiktion der Genauigkeiten der 3D Punkte ermöglicht

Self-adaptive isogeometric spatial discretisations of the first and second-order forms of the neutron transport equation with dual-weighted residual error measures and diffusion acceleration

As implemented in a new modern-Fortran code, NURBS-based isogeometric analysis (IGA) spatial discretisations and self-adaptive mesh refinement (AMR) algorithms are developed in the application to the first-order and second-order forms of the neutron transport equation (NTE). These AMR algorithms are shown to be computationally efficient and numerically accurate when compared to standard approaches. IGA methods are very competitive and offer certain unique advantages over standard finite element methods (FEM), not least of all because the numerical analysis is performed over an exact representation of the underlying geometry, which is generally available in some computer-aided design (CAD) software description. Furthermore, mesh refinement can be performed within the analysis program at run-time, without the need to revisit any ancillary mesh generator. Two error measures are described for the IGA-based AMR algorithms, both of which can be employed in conjunction with energy-dependent meshes. The first heuristically minimises any local contributions to the global discretisation error, as per some appropriate user-prescribed norm. The second employs duality arguments to minimise important local contributions to the error as measured in some quantity of interest; this is commonly known as a dual-weighted residual (DWR) error measure and it demands the solution to both the forward (primal) and the adjoint (dual) NTE. Finally, convergent and stable diffusion acceleration and generalised minimal residual (GMRes) algorithms, compatible with the aforementioned AMR algorithms, are introduced to accelerate the convergence of the within-group self-scattering sources for scattering-dominated problems for the first and second-order forms of the NTE. A variety of verification benchmark problems are analysed to demonstrate the computational performance and efficiency of these acceleration techniques.Open Acces

Model-Based Environmental Visual Perception for Humanoid Robots

The visual perception of a robot should answer two fundamental questions: What? and Where? In order to properly and efficiently reply to these questions, it is essential to establish a bidirectional coupling between the external stimuli and the internal representations. This coupling links the physical world with the inner abstraction models by sensor transformation, recognition, matching and optimization algorithms. The objective of this PhD is to establish this sensor-model coupling

Fast and robust image feature matching methods for computer vision applications

Service robotic systems are designed to solve tasks such as recognizing and manipulating objects, understanding natural scenes, navigating in dynamic and populated environments. It's immediately evident that such tasks cannot be modeled in all necessary details as easy as it is with industrial robot tasks; therefore, service robotic system has to have the ability to sense and interact with the surrounding physical environment through a multitude of sensors and actuators. Environment sensing is one of the core problems that limit the deployment of mobile service robots since existing sensing systems are either too slow or too expensive. Visual sensing is the most promising way to provide a cost effective solution to the mobile robot sensing problem. It's usually achieved using one or several digital cameras placed on the robot or distributed in its environment. Digital cameras are information rich sensors and are relatively inexpensive and can be used to solve a number of key problems for robotics and other autonomous intelligent systems, such as visual servoing, robot navigation, object recognition, pose estimation, and much more. The key challenges to taking advantage of this powerful and inexpensive sensor is to come up with algorithms that can reliably and quickly extract and match the useful visual information necessary to automatically interpret the environment in real-time. Although considerable research has been conducted in recent years on the development of algorithms for computer and robot vision problems, there are still open research challenges in the context of the reliability, accuracy and processing time. Scale Invariant Feature Transform (SIFT) is one of the most widely used methods that has recently attracted much attention in the computer vision community due to the fact that SIFT features are highly distinctive, and invariant to scale, rotation and illumination changes. In addition, SIFT features are relatively easy to extract and to match against a large database of local features. Generally, there are two main drawbacks of SIFT algorithm, the first drawback is that the computational complexity of the algorithm increases rapidly with the number of key-points, especially at the matching step due to the high dimensionality of the SIFT feature descriptor. The other one is that the SIFT features are not robust to large viewpoint changes. These drawbacks limit the reasonable use of SIFT algorithm for robot vision applications since they require often real-time performance and dealing with large viewpoint changes. This dissertation proposes three new approaches to address the constraints faced when using SIFT features for robot vision applications, Speeded up SIFT feature matching, robust SIFT feature matching and the inclusion of the closed loop control structure into object recognition and pose estimation systems. The proposed methods are implemented and tested on the FRIEND II/III service robotic system. The achieved results are valuable to adapt SIFT algorithm to the robot vision applications

Computational Physics II

E-kursuse materjal

Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

A Fully Parallelized and Budgeted Multi-level Monte Carlo Framework for Partial Differential Equations: From Mathematical Theory to Automated Large-Scale Computations

All collected data on any physical, technical or economical process is subject to uncertainty. By incorporating this uncertainty in the model and propagating it through the system, this data error can be controlled. This makes the predictions of the system more trustworthy and reliable. The multi-level Monte Carlo (MLMC) method has proven to be an effective uncertainty quantification tool, requiring little knowledge about the problem while being highly performant. In this doctoral thesis we analyse, implement, develop and apply the MLMC method to partial differential equations (PDEs) subject to high-dimensional random input data. We set up a unified framework based on the software M++ to approximate solutions to elliptic and hyperbolic PDEs with a large selection of finite element methods. We combine this setup with a new variant of the MLMC method. In particular, we propose a budgeted MLMC (BMLMC) method which is capable to optimally invest reserved computing resources in order to minimize the model error while exhausting a given computational budget. This is achieved by developing a new parallelism based on a single distributed data structure, employing ideas of the continuation MLMC method and utilizing dynamic programming techniques. The final method is theoretically motivated, analyzed, and numerically well-tested in an automated benchmarking workflow for highly challenging problems like the approximation of wave equations in randomized media