## Doctoral Thesis

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In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.

This dissertation focuses on the evaluation of technical and environmental sustainability of water distribution systems based on scenario analysis. The decision support system is created to assist in the decision making-process and to visualize the results of the sustainability assessment for current and future populations and scenarios. First, a methodology is developed to assess the technical and environmental sustainability for the current and future water distribution system scenarios. Then, scenarios are produced to evaluate alternative solutions for the current water distribution system as well as future populations and water demand variations. Finally, a decision support system is proposed using a combination of several visualization approaches to increase the data readability and robustness for the sustainability evaluations of the water distribution system.
The technical sustainability of a water distribution system is measured using the sustainability index methodology which is based on the reliability, resiliency and vulnerability performance criteria. Hydraulic efficiency and water quality requirements are represented using the nodal pressure and water age parameters, respectively. The U.S. Environmental Protection Agency EPANET software is used to simulate hydraulic (i.e. nodal pressure) and water quality (i.e. water age) analysis in a case study. In addition, the environmental sustainability of a water network is evaluated using the “total fresh water use” and “total energy intensity” indicators. For each scenario, multi-criteria decision analysis is used to combine technical and environmental sustainability criteria for the study area.
The technical and environmental sustainability assessment methodology is first applied to the baseline scenario (i.e. the current water distribution system). Critical locations where hydraulic efficiency and water quality problems occur in the current system are identified. There are two major scenario options that are considered to increase the sustainability at these critical locations. These scenarios focus on creating alternative systems in order to test and verify the technical and environmental sustainability methodology rather than obtaining the best solution for the current and future water distribution systems. The first scenario is a traditional approach in order to increase the hydraulic efficiency and water quality. This scenario includes using additional network components such as booster pumps, valves etc. The second scenario is based on using reclaimed water supply to meet the non-potable water demand and fire flow. The fire flow simulation is specifically included in the sustainability assessment since regulations have significant impact on the urban water infrastructure design. Eliminating the fire flow need from potable water distribution systems would assist in saving fresh water resources as well as to reduce detention times.
The decision support system is created to visualize the results of each scenario and to effectively compare these results with each other. The EPANET software is a powerful tool used to conduct hydraulic and water quality analysis but for the decision support system purposes the visualization capabilities are limited. Therefore, in this dissertation, the hydraulic and water quality simulations are completed using EPANET software and the results for each scenario are visualized by combining several visualization techniques in order to provide a better data readability. The first technique introduced here is using small multiple maps instead of the animation technique to visualize the nodal pressure and water age parameters. This technique eliminates the change blindness and provides easy comparison of time steps. In addition, a procedure is proposed to aggregate the nodes along the edges in order to simplify the water network. A circle view technique is used to visualize two values of a single parameter (i.e. the nodal pressure or water age). The third approach is based on fitting the water network into a grid representation which assists in eliminating the irregular geographic distribution of the nodes and improves the visibility of each circle view. Finally, a prototype for an interactive decision support tool is proposed for the current population and water demand scenarios. Interactive tools enable analyzing of the aggregated nodes and provide information about the results of each of the current water distribution scenarios.

The recognition of day-to-day activities is still a very challenging and important research topic. During recent years, a lot of research has gone into designing and realizing smart environ- ments in different application areas such as health care, maintenance, sports or smart homes. As a result, a large amount of sensor modalities were developed, different types of activity and context recognition services were implemented and the resulting systems were benchmarked using state-of-the-art evaluation techniques. However, so far hardly any of these approaches have found their way into the market and consequently into the homes of real end-users on a large scale. The reason for this is, that almost all systems have one or more of the following characteristics in common: expensive high-end or prototype sensors are used which are not af- fordable or reliable enough for mainstream applications; many systems are deployed in highly instrumented environments or so-called "living labs", which are far from real-life scenarios and are often evaluated only in research labs; almost all systems are based on complex system con- figurations and/or extensive training data sets, which means that a large amount of data must be collected in order to install the system. Furthermore, many systems rely on a user and/or environment dependent training, which makes it even more difficult to install them on a large scale. Besides, a standardized integration procedure for the deployment of services in existing environments and smart homes has still not been defined. As a matter of fact, service providers use their own closed systems, which are not compatible with other systems, services or sensors. It is clear, that these points make it nearly impossible to deploy activity recognition systems in a real daily-life environment, to make them affordable for real users and to deploy them in hundreds or thousands of different homes.
This thesis works towards the solution of the above mentioned problems. Activity and context recognition systems designed for large-scale deployment and real-life scenarios are intro- duced. Systems are based on low-cost, reliable sensors and can be set up, configured and trained with little effort, even by technical laymen. It is because of these characteristics that we call our approach "minimally invasive". As a consequence, large amounts of training data, that are usu- ally required by many state-of-the-art approaches, are not necessary. Furthermore, all systems were integrated unobtrusively in real-world/similar to real-world environments and were evalu- ated under real-life, as well as similar to real-life conditions. The thesis addresses the following topics: First, a sub-room level indoor positioning system is introduced. The system is based on low-cost ceiling cameras and a simple computer vision tracking approach. The problem of user identification is solved by correlating modes of locomotion patterns derived from the trajectory of unidentified objects and on-body motion sensors. Afterwards, the issue of recognizing how and what mainstream household devices have been used for is considered. Based on a low-cost microphone, the water consumption of water-taps can be approximated by analyzing plumbing noise. Besides that, operating modes of mainstream electronic devices were recognized by using rule-based classifiers, electric current features and power measurement sensors. As a next step, the difficulty of spotting subtle, barely distinguishable hand activities and the resulting object interactions, within a data set containing a large amount of background data, is addressed. The problem is solved by introducing an on-body core system which is configured by simple, one-time physical measurements and minimal data collections. The lack of large training sets is compensated by fusing the system with activity and context recognition systems, that are able to reduce the search space observed. Amongst other systems, previously introduced approaches and ideas are revisited in this section. An in-depth evaluation shows the impact of each fusion procedure on the performance and run-time of the system. The approaches introduced are able to provide significantly better results than a state-of-the-art inertial system using large amounts of training data. The idea of using unobtrusive sensors has also been applied to the field of behavior analysis. Integrated smartphone sensors are used to detect behavioral changes of in- dividuals due to medium-term stress periods. Behavioral parameters related to location traces, social interactions and phone usage were analyzed to detect significant behavioral changes of individuals during stressless and stressful time periods. Finally, as a closing part of the the- sis, a standardization approach related to the integration of ambient intelligence systems (as introduced in this thesis) in real-life and large-scale scenarios is shown.

Test rig optimization
(2014)

Designing good test rigs for fatigue life tests is a common task in the auto-
motive industry. The problem to find an optimal test rig configuration and
actuator load signals can be formulated as a mathematical program. We in-
troduce a new optimization model that includes multi-criteria, discrete and
continuous aspects. At the same time we manage to avoid the necessity to
deal with the rainflow-counting (RFC) method. RFC is an algorithm, which
extracts load cycles from an irregular time signal. As a mathematical func-
tion it is non-convex and non-differentiable and, hence, makes optimization
of the test rig intractable.
The block structure of the load signals is assumed from the beginning.
It highly reduces complexity of the problem without decreasing the feasible
set. Also, we optimize with respect to the actuators’ positions, which makes
it possible to take torques into account and thus extend the feasible set. As
a result, the new model gives significantly better results, compared with the
other approaches in the test rig optimization.
Under certain conditions, the non-convex test rig problem is a union of
convex problems on cones. Numerical methods for optimization usually need
constraints and a starting point. We describe an algorithm that detects each
cone and its interior point in a polynomial time.
The test rig problem belongs to the class of bilevel programs. For every
instance of the state vector, the sum of functions has to be maximized. We
propose a new branch and bound technique that uses local maxima of every
summand.

In the theory of option pricing one is usually concerned with evaluating expectations under the risk-neutral measure in a continuous-time model.
However, very often these values cannot be calculated explicitly and numerical methods need to be applied to approximate the desired quantity. Monte Carlo simulations, numerical methods for PDEs and the lattice approach are the methods typically employed. In this thesis we consider the latter approach, with the main focus on binomial trees.
The binomial method is based on the concept of weak convergence. The discrete-time model is constructed so as to ensure convergence in distribution to the continuous process. This means that the expectations calculated in the binomial tree can be used as approximations of the option prices in the continuous model. The binomial method is easy to implement and can be adapted to options with different types of payout structures, including American options. This makes the approach very appealing. However, the problem is that in many cases, the convergence of the method is slow and highly irregular, and even a fine discretization does not guarantee accurate price approximations. Therefore, ways of improving the convergence properties are required.
We apply Edgeworth expansions to study the convergence behavior of the lattice approach. We propose a general framework, that allows to obtain asymptotic expansion for both multinomial and multidimensional trees. This information is then used to construct advanced models with superior convergence properties.
In binomial models we usually deal with triangular arrays of lattice random vectors. In this case the available results on Edgeworth expansions for lattices are not directly applicable. Therefore, we first present Edgeworth expansions, which are also valid for the binomial tree setting. We then apply these result to the one-dimensional and multidimensional Black-Scholes models. We obtain third order expansions
for general binomial and trinomial trees in the 1D setting, and construct advanced models for digital, vanilla and barrier options. Second order expansion are provided for the standard 2D binomial trees and advanced models are constructed for the two-asset digital and the two-asset correlation options. We also present advanced binomial models for a multidimensional setting.

The demand of sustainability is continuously increasing. Therefore, thermoplastic
composites became a focus of research due to their good weight to performance
ratio. Nevertheless, the limiting factor of their usage for some processes is the loss of
consolidation during re-melting (deconsolidation), which reduces the part quality.
Several studies dealing with deconsolidation are available. These studies investigate
a single material and process, which limit their usefulness in terms of general
interpretations as well as their comparability to other studies. There are two main
approaches. The first approach identifies the internal void pressure as the main
cause of deconsolidation and the second approach identifies the fiber reinforcement
network as the main cause. Due to of their controversial results and limited variety of
materials and processes, there is a big need of a more comprehensive investigation
on several materials and processes.
This study investigates the deconsolidation behavior of 17 different materials and
material configurations considering commodity, engineering, and performance
polymers as well as a carbon and two glass fiber fabrics. Based on the first law of
thermodynamics, a deconsolidation model is proposed and verified by experiments.
Universal applicable input parameters are proposed for the prediction of
deconsolidation to minimize the required input measurements. The study revealed
that the fiber reinforcement network is the main cause of deconsolidation, especially
for fiber volume fractions higher than 48 %. The internal void pressure can promote
deconsolidation, when the specimen was recently manufactured. In other cases the
internal void pressure as well as the surface tension prevents deconsolidation.
During deconsolidation the polymer is displaced by the volume increase of the void.
The polymer flow damps the progress of deconsolidation because of the internal
friction of the polymer. The crystallinity and the thermal expansion lead to a
reversible thickness increase during deconsolidation. Moisture can highly accelerate
deconsolidation and can increase the thickness by several times because of the
vaporization of water. The model is also capable to predict reconsolidation under the
defined boundary condition of pressure, time, and specimen size. For high pressure
matrix squeeze out occur, which falsifies the accuracy of the model.The proposed model was applied to thermoforming, induction welding, and
thermoplastic tape placement. It is demonstrated that the load rate during
thermoforming is the critical factor of achieving complete reconsolidation. The
required load rate can be determined by the model and is dependent on the cooling
rate, the forming length, the extent of deconsolidation, the processing temperature,
and the final pressure. During induction welding deconsolidation can tremendously
occur because of the left moisture in the polymer at the molten state. The moisture
cannot fully diffuse out of the specimen during the faster heating. Therefore,
additional pressure is needed for complete reconsolidation than it would be for a dry
specimen. Deconsolidation is an issue for thermoplastic tape placement, too. It limits
the placement velocity because of insufficient cooling after compaction. If the
specimen after compaction is locally in a molten state, it deconsolidates and causes
residual stresses in the bond line, which decreases the interlaminar shear strength. It
can be concluded that the study gains new knowledge and helps to optimize these
processes by means of the developed model without a high number of required
measurements.
Aufgrund seiner guten spezifischen Festigkeit und Steifigkeit ist der
endlosfaserverstärkte Thermoplast ein hervorragender Leichtbauwerkstoff. Allerdings
kann es während des Wiederaufschmelzens durch Dekonsolidierung zu einem
Verlust der guten mechanischen Eigenschaften kommen, daher ist Dekonsolidierung
unerwünscht. In vielen Studien wurde die Dekonsolidierung mit unterschiedlichen
Ergebnissen untersucht. Dabei wurde meist ein Material und ein Prozess betrachtet.
Eine allgemeine Interpretation und die Vergleichbarkeit unter den Studien sind
dadurch nur begrenzt möglich. Aus der Literatur sind zwei Ansätze bekannt. Dem
ersten Ansatz liegt der Druckunterschied zwischen Poreninnendruck und
Umgebungsdruck als Hauptursache der Dekonsolidierung zu Grunde. Beim zweiten
Ansatz wird die Faserverstärkung als Hauptursache identifiziert. Aufgrund der
kontroversen Ergebnisse und der begrenzten Anzahl der Materialien und
Verarbeitungsverfahren, besteht die Notwendigkeit einer umfassenden Untersuchung
über mehrere Materialien und Prozesse. Diese Studie umfasst drei Polymere
(Polypropylen, Polycarbonat und Polyphenylensulfid), drei Gewebe (Köper, Atlas und
Unidirektional) und zwei Prozesse (Autoklav und Heißpressen) bei verschiedenen
Faservolumengehalten.
Es wurde der Einfluss des Porengehaltes auf die interlaminare Scherfestigkeit
untersucht. Aus der Literatur ist bekannt, dass die interlaminare Scherfestigkeit mit
der Zunahme des Porengehaltes linear sinkt. Dies konnte für die Dekonsolidierung
bestätigt werden. Die Reduktion der interlaminaren Scherfestigkeit für
thermoplastische Matrizes ist kleiner als für duroplastische Matrizes und liegt im
Bereich zwischen 0,5 % bis 1,5 % pro Prozent Porengehalt. Außerdem ist die
Abnahme signifikant vom Matrixpolymer abhängig.
Im Falle der thermisch induzierten Dekonsolidierung nimmt der Porengehalt
proportional zu der Dicke der Probe zu und ist ein Maß für die Dekonsolidierung. Die
Pore expandiert aufgrund der thermischen Gasexpansion und kann durch äußere
Kräfte zur Expansion gezwungen werden, was zu einem Unterdruck in der Pore
führt. Die Faserverstärkung ist die Hauptursache der Dickenzunahme
beziehungsweise der Dekonsolidierung. Die gespeicherte Energie, aufgebaut während der Kompaktierung, wird während der Dekonsolidierung abgegeben. Der
Dekompaktierungsdruck reicht von 0,02 MPa bis 0,15 MPa für die untersuchten
Gewebe und Faservolumengehalte. Die Oberflächenspannung behindert die
Porenexpansion, weil die Oberfläche vergrößert werden muss, die zusätzliche
Energie benötigt. Beim Kontakt von benachbarten Poren verursacht die
Oberflächenspannung ein Verschmelzen der Poren. Durch das bessere Volumen-
Oberfläche-Verhältnis wird Energie abgebaut. Der Polymerfluss bremst die
Entwicklung der Dickenzunahme aufgrund der erforderlichen Energie (innere
Reibung) der viskosen Strömung. Je höher die Temperatur ist, desto niedriger ist die
Viskosität des Polymers, wodurch weniger Energie für ein weiteres Porenwachstum
benötigt wird. Durch den reversiblen Einfluss der Kristallinität und der
Wärmeausdehnung des Verbundes wird während der Erwärmung die Dicke erhöht
und während der Abkühlung wieder verringert. Feuchtigkeit kann einen enormen
Einfluss auf die Dekonsolidierung haben. Ist noch Feuchtigkeit über der
Schmelztemperatur im Verbund vorhanden, verdampft diese und kann die Dicke um
ein Vielfaches der ursprünglichen Dicke vergrößern.
Das Dekonsolidierungsmodell ist in der Lage die Rekonsolidierung vorherzusagen.
Allerdings muss der Rekonsolidierungsdruck unter einem Grenzwert liegen
(0,15 MPa für 50x50 mm² und 1,5 MPa für 500x500 mm² große Proben), da es sonst
bei der Probe zu einem Polymerfluss aus der Probe von mehr als 2 % kommt. Die
Rekonsolidierung ist eine inverse Dekonsolidierung und weist die gleichen
Mechanismen in der entgegengesetzten Richtung auf.
Das entwickelte Modell basiert auf dem ersten Hauptsatz der Thermodynamik und
kann die Dicke während der Dekonsolidierung und der Rekonsolidierung
vorhersagen. Dabei wurden eine homogene Porenverteilung und eine einheitliche,
kugelförmige Porengröße angenommen. Außerdem wurde die Massenerhaltung
angenommen. Um den Aufwand für die Bestimmung der Eingangsgrößen zu
reduzieren, wurden allgemein gültige Eingabeparameter bestimmt, die für eine
Vielzahl von Konfigurationen gelten. Das simulierte Materialverhalten mit den
allgemein gültigen Eingangsparametern erzielte unter den definierten
Einschränkungen eine gute Übereinstimmung mit dem tatsächlichen
Materialverhalten. Nur bei Konfigurationen mit einer Viskositätsdifferenz von mehr als 30 % zwischen der Schmelztemperatur und der Prozesstemperatur sind die
allgemein gültigen Eingangsparameter nicht anwendbar. Um die Relevanz für die
Industrie aufzuzeigen, wurden die Effekte der Dekonsolidierung für drei weitere
Verfahren simuliert. Es wurde gezeigt, dass die Kraftzunahmegeschwindigkeit
während des Thermoformens ein Schlüsselfaktor für eine vollständige
Rekonsolidierung ist. Wenn die Kraft zu langsam appliziert wird oder die finale Kraft
zu gering ist, ist die Probe bereits erstarrt, bevor eine vollständige Konsolidierung
erreicht werden kann. Auch beim Induktionsschweißen kann Dekonsolidierung
auftreten. Besonders die Feuchtigkeit kann zu einer starken Zunahme der
Dekonsolidierung führen, verursacht durch die sehr schnellen Heizraten von mehr als
100 K/min. Die Feuchtigkeit kann während der kurzen Aufheizphase nicht vollständig
aus dem Polymer ausdiffundieren, sodass die Feuchtigkeit beim Erreichen der
Schmelztemperatur in der Probe verdampft. Beim Tapelegen wird die
Ablegegeschwindigkeit durch die Dekonsolidierung begrenzt. Nach einer scheinbar
vollständigen Konsolidierung unter der Walze kann die Probe lokal dekonsolidieren,
wenn das Polymer unter der Oberfläche noch geschmolzen ist. Die daraus
resultierenden Poren reduzieren die interlaminare Scherfestigkeit drastisch um 5,8 %
pro Prozent Porengehalt für den untersuchten Fall. Ursache ist die Kristallisation in
der Verbindungszone. Dadurch werden Eigenspannungen erzeugt, die in der
gleichen Größenordnung wie die tatsächliche Scherfestigkeit sind.

In automotive testrigs we apply load time series to components such that the outcome is as close as possible to some reference data. The testing procedure should in general be less expensive and at the same time take less time for testing. In my thesis, I propose a testrig damage optimization problem (WSDP). This approach improves upon the testrig stress optimization problem (TSOP) used as a state of the art by industry experts.
In both (TSOP) and (WSDP), we optimize the load time series for a given testrig configuration. As the name suggests, in (TSOP) the reference data is the stress time series. The detailed behaviour of the stresses as functions of time are sometimes not the most important topic. Instead the damage potential of the stress signals are considered. Since damage is not part of the objectives in the (TSOP) the total damage computed from the optimized load time series is not optimal with respect to the reference damage. Additionally, the load time series obtained is as long as the reference stress time series and the total damage computation needs cycle counting algorithms and Goodmann corrections. The use of cycle counting algorithms makes the computation of damage from load time series non-differentiable.
To overcome the issues discussed in the previous paragraph this thesis uses block loads for the load time series. Using of block loads makes the damage differentiable with respect to the load time series. Additionally, in some special cases it is shown that damage is convex when block loads are used and no cycle counting algorithms are required. Using load time series with block loads enables us to use damage in the objective function of the (WSDP).
During every iteration of the (WSDP), we have to find the maximum total damage over all plane angles. The first attempt at solving the (WSDP) uses discretization of the interval for plane angle to find the maximum total damage at each iteration. This is shown to give unreliable results and makes maximum total damage function non-differentiable with respect to the plane angle. To overcome this, damage function for a given surface stress tensor due to a block load is remodelled by Gaussian functions. The parameters for the new model are derived.
When we model the damage by Gaussian function, the total damage is computed as a sum of Gaussian functions. The plane with the maximum damage is similar to the modes of the Gaussian Mixture Models (GMM), the difference being that the Gaussian functions used in GMM are probability density functions which is not the case in the damage approximation presented in this work. We derive conditions for a single maximum for Gaussian functions, similar to the ones given for the unimodality of GMM by Aprausheva et al. in [1].
By using the conditions for a single maximum we give a clustering algorithm that merges the Gaussian functions in the sum as clusters. Each cluster obtained through clustering is such that it has a single maximum in the absence of other Gaussian functions of the sum. The approximate point of the maximum of each cluster is used as the starting point for a fixed point equation on the original damage function to get the actual maximum total damage at each iteration.
We implement the method for the (TSOP) and the two methods (with discretization and with clustering) for (WSDP) on two example problems. The results obtained from the (WSDP) using discretization is shown to be better than the results obtained from the (TSOP). Furthermore we show that, (WSDP) using clustering approach to finding the maximum total damage, takes less number of iterations and is more reliable than using discretization.

According to the domain specific models of speech perception, speech is supposed to be processed distinctively compared to non-speech. This assumption is supported by many studies dealing with the processing of speech and non-speech stimuli. However, the complexity of both stimulus classes is not matched in most studies, which might be a confounding factor, according to the cue specific models of speech perception. One solution is spectrally rotated speech, which has already been used in a range of fMRI and PET studies. In order to be able to investigate the role of stimulus complexity, vowels, spectrally rotated vowels and a second non-speech condition with two bands of sinusoidal waves, representing the first two formants of the vowels, were used in the present thesis. A detailed description of the creation and the properties of the whole stimulus set are given in Chapter 2 (Experiment 1) of this work. These stimuli were used to investigate the auditory processing of speech and non-speech sounds in a group of dyslexic adults and age matched controls (Experiment 2). The results support the assumption of a general auditory deficit in dyslexia. In order to compare the sensory processing of speech and non-speech in healthy adults on the electrophysiological level, stimuli were also presented within a multifeature oddball paradigm (Experiment 3). Vowels evoked a larger mismatch negativity (MMN) compared to both non-speech stimulus types. The MMN evoked by tones and spectrally rotated tones were compared in Experiment 4, to investigate the role of harmony. No difference in the area of MMN was found, indicating that the results found in Experiment 3 were not moderated by the harmonic structure of the vowels. All results are discussed in the context of the domain and cue specific models of speech perception.

This thesis is devoted to the computational aspects of intersection theory and enumerative geometry. The first results are a Sage package Schubert3 and a Singular library schubert.lib which both provide the key functionality necessary for computations in intersection theory and enumerative geometry. In particular, we describe an alternative method for computations in Schubert calculus via equivariant intersection theory. More concretely, we propose an explicit formula for computing the degree of Fano schemes of linear subspaces on hypersurfaces. As a special case, we also obtain an explicit formula for computing the number of linear subspaces on a general hypersurface when this number is finite. This leads to a much better performance than classical Schubert calculus.
Another result of this thesis is related to the computation of Gromov-Witten invariants. The most powerful method for computing Gromov-Witten invariants is the localization of moduli spaces of stable maps. This method was introduced by Kontsevich in 1995. It allows us to compute Gromov-Witten invariants via Bott's formula. As an insightful application, we computed the numbers of rational curves on general complete intersection Calabi-Yau threefolds in projective spaces up to degree six. The results are all in agreement with predictions made from mirror symmetry.

Three dimensional (3d) point data is used in industry for measurement and reverse engineering. Precise point data is usually acquired with triangulating laser scanners or high precision structured light scanners. Lower precision point data is acquired by real-time structured light devices or by stereo matching with multiple cameras. The basic principle of all these methods is the so-called triangulation of 3d coordinates from two dimensional (2d) camera images.
This dissertation contributes a method for multi-camera stereo matching that uses a system of four synchronized cameras. A GPU based stereo matching method is presented to achieve a high quality reconstruction at interactive frame rates. Good depth resolution is achieved by allowing large disparities between the images. A multi level approach on the GPU allows a fast processing of these large disparities. In reverse engineering, hand-held laser scanners are used for the scanning of complex shaped objects. The operator of the scanner can scan complex regions slower, multiple times, or from multiple angles to achieve a higher point density. Traditionally, computer aided design (CAD) geometry is reconstructed in a separate step after the scanning. Errors or missing parts in the scan prevent a successful reconstruction. The contribution of this dissertation is an on-line algorithm that allows the reconstruction during the scanning of an object. Scanned points are added to the reconstruction and improve it on-line. The operator can detect the areas in the scan where the reconstruction needs additional data.
First, the point data is thinned out using an octree based data structure. Local normals and principal curvatures are estimated for the reduced set of points. These local geometric values are used for segmentation using a region growing approach. Implicit quadrics are fitted to these segments. The canonical form of the quadrics provides the parameters of basic geometric primitives.
An improved approach uses so called accumulated means of local geometric properties to perform segmentation and primitive reconstruction in a single step. Local geometric values can be added and removed on-line to these means to get a stable estimate over a complete segment. By estimating the shape of the segment it is decided which local areas are added to a segment. An accumulated score estimates the probability for a segment to belong to a certain type of geometric primitive. A boundary around the segment is reconstructed using a growing algorithm that ensures that the boundary is closed and avoids self intersections.