Thesis (M.S.)--University of North Carolina Wilmington, 2008. / Title from PDF title page (viewed May 27, 2009) Includes bibliographical references (p. 31-34)

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

This thesis investigates two well defined problems in image segmentation, viz. interactive and semantic image segmentation. Interactive segmentation involves power assisting a user in cutting out objects from an image, whereas semantic segmentation involves partitioning pixels in an image into object categories. We investigate various models and energy formulations for both these problems in this thesis. In order to improve the performance of interactive systems, low level texture features are introduced as a replacement for the more commonly used RGB features. To quantify the improvement obtained by using these texture features, two annotated datasets of images are introduced (one consisting of natural images, and the other consisting of camouflaged objects). A significant improvement in performance is observed when using texture features for the case of monochrome images and images containing camouflaged objects. We also explore adding mid-level cues such as shape constraints into interactive segmentation by introducing the idea of geodesic star convexity, which extends the existing notion of a star convexity prior in two important ways: (i) It allows for multiple star centres as opposed to single stars in the original prior and (ii) It generalises the shape constraint by allowing for Geodesic paths as opposed to Euclidean rays. Global minima of our energy function can be obtained subject to these new constraints. We also introduce Geodesic Forests, which exploit the structure of shortest paths in implementing the extended constraints. These extensions to star convexity allow us to use such constraints in a practical segmentation system. This system is evaluated by means of a “robot user” to measure the amount of interaction required in a precise way, and it is shown that having shape constraints reduces user effort significantly compared to existing interactive systems. We also introduce a new and harder dataset which augments the existing GrabCut dataset with more realistic images and ground truth taken from the PASCAL VOC segmentation challenge. In the latter part of the thesis, we bring in object category level information in order to make the interactive segmentation tasks easier, and move towards fully automated semantic segmentation. An algorithm to automatically segment humans from cluttered images given their bounding boxes is presented. A top down segmentation of the human is obtained using classifiers trained to predict segmentation masks from local HOG descriptors. These masks are then combined with bottom up image information in a local GrabCut like procedure. This algorithm is later completely automated to segment humans without requiring a bounding box, and is quantitatively compared with other semantic segmentation methods. We also introduce a novel way to acquire large quantities of segmented training data relatively effortlessly using the Kinect. In the final part of this work, we explore various semantic segmentation methods based on learning using bottom up super-pixelisations. Different methods of combining multiple super-pixelisations are discussed and quantitatively evaluated on two segmentation datasets. We observe that simple combinations of independently trained classifiers on single super-pixelisations perform almost as good as complex methods based on jointly learning across multiple super-pixelisations. We also explore CRF based formulations for semantic segmentation, and introduce novel visual words based object boundary description in the energy formulation. The object appearance and boundary parameters are trained jointly using structured output learning methods, and the benefit of adding pairwise terms is quantified on two different datasets.

This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol.

A measurement-independent formulation of quantum mechanics called ‘regular histories’ (RH) is presented, able to reproduce the predictions of the standard formalism without the need to for a quantum-classical divide or the presence of an observer. It applies to closed systems and features no wave-function collapse. Weights are assigned only to histories satisfying a criterion called ‘regularity’. As the set of regular histories is not closed under the Boolean operations this requires a new con- cept of weight, called ‘likelihood’. Remarkably, this single change is enough to overcome many of the well-known obstacles to a sensible interpretation of quantum mechanics. For example, Bell’s theorem, which makes essential use of probabilities, places no constraints on the locality properties of a theory based on likelihoods. Indeed, RH is both counter- factually definite and free from action-at-a-distance. Moreover, in RH the meaningful histories are exactly those that can be witnessed at least in principle. Since it is especially difficult to make sense of the concept of probability for histories whose occurrence is intrinsically indeterminable, this makes likelihoods easier to justify than probabilities. Interaction with the environment causes the kinds of histories relevant at the macroscopic scale of human experience to be witnessable and indeed to generate Boolean algebras of witnessable histories, on which likelihoods reduce to ordinary probabilities. Further- more, a formal notion of inference defined on regular histories satisfies, when restricted to such Boolean algebras, the classical axioms of implication, explaining our perception of a largely classical world. Even in the context of general quantum histories the rules of reasoning in RH are remark- ably intuitive. Classical logic must only be amended to reflect the fundamental premise that one cannot meaningfully talk about the occurrence of unwitnessable histories. Crucially, different histories with the same ‘physical content’ can be interpreted in the same way and independently of the family in which they are expressed. RH thereby rectifies a critical flaw of its inspiration, the consistent histories (CH) approach, which requires either an as yet unknown set selection rule or a paradigm shift towards an un- conventional picture of reality whose elements are histories-with-respect-to-a-framework. It can be argued that RH compares favourably with other proposed interpretations of quantum mechanics in that it resolves the measurement problem while retaining an essentially classical worldview without parallel universes, a framework-dependent reality or action-at-a-distance.

Cerebrospinal fluid (CSF) fills a system of cavities at the centre of the brain, known as ventricles, and the subarachnoid space surrounding the brain and the spinal cord. In addition, CSF is in free communication with the interstitial fluid of the brain tissue. Disturbances in CSF dynamics can lead to diseases that cause severe brain damage or even death. So-called infusion tests are frequently performed in the diagnosis of such diseases. In this type of test, changes in average CSF pressure are related to changes in CSF volume through infusion of known volumes of additional fluid. Traditionally, infusion tests are analysed with single compartment models, which treat all CSF as part of one compartment and balance fluid inflow, outflow and storage through a single ordinary differential equation. Poroelastic models of the brain, on the other hand, have been used to simulate spatial changes with disease, particularly of the ventricle size, on larger time scales of days, weeks or months. Wirth and Sobey (2008) developed a two-fluid poroelastic model of the brain in which CSF pressure pulsations are linked to arterial blood pressure pulsations. In this thesis, this model is developed further and simulation results are compared to clinical data. At first, the functional form of the compliance, which governs the storage of CSF in single compartment models, is examined by comparison of two different compliance models with clinical data. The derivations of a single-fluid and a two-fluid poroelastic model of the brain in spherical symmetry are laid out in detail and some of the parameters are related to the compliance functions considered earlier. The finite element implementation of the two-fluid model is described and finally simulation results of the average CSF pressure response and the pressure pulsations are compared to clinical data.

Introduction: Antimalarial resistance, particularly artemisinin resistance, is a major threat to P. falciparum malaria elimination efforts worldwide. Urgent intervention is required to tackle artemisinin resistance but field data on which to base planning of strategies are limited. The aims were to collect available field data and develop population level mathematical models of P. falciparum malaria treatment and artemisinin resistance in order to determine the optimal strategies for elimination of artemisinin resistant malaria in Cambodia and treatment of pre-hospital and severe malaria in Cambodia and Bangladesh. Methods: Malaria incidence and parasite clearance data from Cambodia and Bangladesh were collected and analysed and modelling parameters derived. Population dynamic mathematical models of P. falciparum malaria were produced. Results: The modelling demonstrated that elimination of artemisinin resistant P. falciparum malaria would be achievable in Cambodia in the context of artemisinin resistance using high coverages with ACT treatment, ideally combined with LLITNs and adjunctive single dose primaquine. Sustained efforts would be necessary to achieve elimination and effective surveillance is essential, both to identify the baseline malaria burden and to monitor parasite prevalence as interventions are implemented. A modelled policy change to rectal and intravenous artesunate in the context of pre-existing artemisinin resistance would not compromise the efficacy of ACT for malaria elimination. Conclusions: By being developed rapidly in response to specific questions the models presented here are helping to inform planning efforts to combat artemisinin resistance. As further field data become available, their planned on-going development will produce increasingly realistic and informative models which can be expected to play a central role in planning efforts for years to come.

Krivelevich and Patkós conjectured in 2009 that χ(G(n, p)) ∼ χ=(G(n, p)) ∼ χ∗=(G(n, p)) for C/n < p < 1 − ε, where ε > 0. We prove this conjecture for n−1+ε1 < p < 1 − ε2 where ε1, ε2 > 0. We investigate several measures that have been proposed to indicate centrality of nodes in networks, and find examples of networks where they fail to distinguish any of the vertices nodes from one another. We develop a new method to investigate core-periphery structure, which entails identifying densely-connected core nodes and sparsely-connected periphery nodes. Finally, we present an experiment and an analysis of empirical networks, functional human brain networks. We found that reconfiguration patterns of dynamic communities can be used to classify nodes into a stiff core, a flexible periphery, and a bulk. The separation between this stiff core and flexible periphery changes as a person learns a simple motor skill and, importantly, it is a good predictor of how successful the person is at learning the skill. This temporally defined core-periphery organisation corresponds well with the core- periphery detected by the method that we proposed earlier the static networks created by averaging over the subjects dynamic functional brain networks.

Colorectal cancer (CRC) arises in part from the dysregulation of cellular proliferation, associated with the canonical Wnt pathway, and differentiation, effected by the Notch signalling network. In this thesis, we develop a mathematical model of ordinary differential equations (ODEs) for the coupled interaction of the Notch and Wnt pathways in cells of the human intestinal epithelium. Our central aim is to understand the role of such crosstalk in the genesis and treatment of CRC. An embedding of this model in cells of a simulated colonic tissue enables computational exploration of the cell fate response to spatially inhomogeneous growth cues in the healthy intestinal epithelium. We also examine an alternative, rule-based model from the literature, which employs a simple binary approach to pathway activity, in which the Notch and Wnt pathways are constitutively on or off. Comparison of the two models demonstrates the substantial advantages of the equation-based paradigm, through its delivery of stable and robust cell fate patterning, and its versatility for exploring the multiscale consequences of a variety of subcellular phenomena. Extension of the ODE-based model to include mutant cells facilitates the study of Notch-mediated therapeutic approaches to CRC. We find a marked synergy between the application of &gamma;-secretase inhibitors and Hath1 stabilisers in the treatment of early-stage intestinal polyps. This combined treatment is an efficient means of inducing mitotic arrest in the cell population of the intestinal epithelium through enforced conversion to a secretory phenotype and is highlighted as a viable route for further theoretical, experimental and clinical study.

Die Dissertation mit dem Titel 'Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing' beschäftigt sich mit modernen Verfahren der Signalverarbeitung in der Bildgebung sowie in der Bildverarbeitung. Hierzu werden Riesz-Transformationen und translationsinvariante Wavelet Frames zu monogenen Frames vereint und angewandt. Bekannte Techniken wie der Strukturtensor und der Energieoperator werden mit Hilfe der neuen Verfahren verbessert und für die Orientierungsbestimmung in Bildern genutzt. Eine weitere Anwendung stellt der Algorithmus 'Equalization of Brightness' dar. Er wird mit einigen Anpassungen verwendet, um eine Implementierung der monogenen Wavelet Frames mit Hilfe des NVIDIA CUDA Frameworks vorzustellen. Bei einem empirischen Vergleich der vorgestellten Techniken mit den ursprünglichen Verfahren konnten präzisere Ergebnisse mit niedrigerer Rauschanfälligkeit nachgewiesen werden. Ein weiterer Punkt der Arbeit beschäftigt sich mit den Möglichkeiten, monogene Wavelet Frames als Filter in optischen Systemen einzusetzen.:Preface Introduction 1. Time Frequency Analysis for Signal Processing 2. The Riesz Transform and Monogenic Wavelet Frames 3. Applications of Monogenic Wavelet Frames Conclusion A. Mathematical Appendix B. Source Code Listings Bibliography List of Figures