This dissertation deals with customised image analysis tools in cancer research. In the field of biomedical sciences, mathematical imaging has become crucial in order to account for advancements in technical equipment and data storage by sound mathematical methods that can process and analyse imaging data in an automated way. This thesis contributes to the development of such mathematically sound imaging models in four ways: (i) automated cell segmentation and tracking. In cancer drug development, time-lapse light microscopy experiments are conducted for performance validation. The aim is to monitor behaviour of cells in cultures that have previously been treated with chemotherapy drugs, since atypical duration and outcome of mitosis, the process of cell division, can be an indicator of successfully working drugs. As an imaging modality we focus on phase contrast microscopy, hence avoiding phototoxicity and influence on cell behaviour. As a drawback, the common halo- and shade-off effect impede image analysis. We present a novel workflow uniting both automated mitotic cell detection with the Hough transform and subsequent cell tracking by a tailor-made level-set method in order to obtain statistics on length of mitosis and cell fates. The proposed image analysis pipeline is deployed in a MATLAB software package called MitosisAnalyser. For the detection of mitotic cells we use the circular Hough transform. This concept is investigated further in the framework of image regularisation in the general context of imaging inverse problems, in which circular objects should be enhanced, (ii) exploiting sparsity of first-order derivatives in combination with the linear circular Hough transform operation. Furthermore, (iii) we present a new unified higher-order derivative-type regularisation functional enforcing sparsity of a vector field related to an image to be reconstructed using curl, divergence and shear operators. The model is able to interpolate between well-known regularisers such as total generalised variation and infimal convolution total variation. Finally, (iv) we demonstrate how we can learn sparsity promoting parametrised regularisers via quotient minimisation, which can be motivated by generalised Eigenproblems. Learning approaches have recently become very popular in the field of inverse problems. However, the majority aims at fitting models to favourable training data, whereas we incorporate knowledge about both fit and misfit data. We present results resembling behaviour of well-established derivative-based sparse regularisers, introduce novel families of non-derivative-based regularisers and extend this framework to classification problems.

Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char. / This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.

After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Em suma, esta tese propõe uma metodologia de acompanhamento de interfaces móveis que baseia-se no método dos conjuntos de nível aqui chamado de método das curvas de nível, uma denominação baseada nas aplicações em que as interfaces são representadas por curvas acoplado a uma implementação semi-Lagrangeana, para problemas em meios porosos. Embora esta técnica possa, em princípio, ser aplicada a qualquer problema físico que apresente uma interface móvel, nesta tese são focados escoamentos em meios porosos consolidados e saturados por um ou dois fluidos imiscíveis e incompressíveis. Adicionalmente, um método iterativo paralelizável para a resolução de sistemas de equações lineares definidos em redes, que podem ser reduzidos à forma das equações fundamentais de equilíbrio, é empregado na determinação dos campos de velocidade associados aos escoamentos em meios porosos. O cenário semi-Lagrangeano acoplado ao método das curvas de nível é comparado com a implementação utilizando o bem conhecido esquema up-wind. Um exaustivo estudo realizado revela a superioridade da metodologia proposta frente à concorrente utilizando o up-wind. Finalmente, o método das curvas de nível com implementação semi-Lagrangeana (método semi-Lagrangeano das curvas de nível), e o método iterativo para a determinação do campo de velocidades são aplicados no estudo de problemas transientes em meios porosos que apresentam instabilidades dos tipos Saffman-Taylor e Rayleigh-Taylor. Este estudo envolve uma análise de estabilidade linear, a introdução de diversas perturbações trigonométricas na interface e a sua evolução não-linear. / Briefy, this thesis proposes a method for capturing moving interfaces based on the level set method coupled to a Semi-Lagrangian implementation for problems in porous media. Although this method could, in principle, be applied to any physical problem with moving interfaces, we foccus, in this thesis, on flows inside a consolidated porous media saturated by one or two imiscible and incompressible fluids. Besides, a parallelizable iterative method for solving linear systems defined on a network that can be reduced to the fundamental equilibrium equations, is employed to determine the velocity field associated with the flow in a porous medium. The semi-Lagrangian scheme coupled with the level set method is compared with the well-known implementation with the up-wind scheme. An exhaustive study is performed and reveals the superiority of the proposed scheme in relation to the competing one using the up-wind method. Finally, the level set method with semi-Lagrangian implementation and the iterative method for determining the velocity field are applied to the study of transient problems in porous media which present Saffman-Taylor and Rayleigh-Taylor instabilities. This study involves the application of a linear stability analysis, the introduction of several trigonometric perturbations to the interface and its non-linear evolution.

The present thesis deals with two-phase electrohydrodynamic simulations of bubble and droplet dynamics under externally applied electric fields. We used the Coupled Level-Set and Volume-of-fluid method (CLSVOF) and two different electrohydrody-namic formulations to study the process of bubble and drop formation from orifices and needles, the interactions of two conducting drops immersed in a dielectric medium, and the oscillations of sessile drops under two different ways of applying external elec-tric field. For the process of bubble formation in dielectric liquids due to the injection of air from submerged orifices and needles, we show that a non-uniform electric field pro-duces smaller bubbles while a uniform electric field changes only the bubble shape. We further explain the reason behind the bubble volume reduction under a non-uniform electric field. We show that the distribution of the electric stresses on the bubble inter-face is such that very high electric stresses act on the bubble base due to a non-uniform electric field. This causes a premature neck formation and bubble detachment lead-ing to the formation of smaller bubbles. We also observe that the non-uniform elec-tric stresses pull the bubble interface contact line inside the needle. With oscillatory electric fields, we show that a further reduction in bubble sizes is possible, but only at certain electric field oscillation frequencies. At other frequencies, bubbles bigger than those under a constant electric field of strength equal to the amplitude of the AC electric field, are produced. We further study the bubble oscillation modes under an oscillatory electric field. We implemented a Volume-of-fluid method based charge advection scheme which is charge conservative and non-diffusive. With the help of this scheme, we were able to simulate the electrohydrodynamic interactions of conducting-dielectric fluid pairs. For two conducting drops inside a dielectric fluid, we observe that they fail to coalesce when the strength of the applied electric field is beyond a critical value. We observe that the non-coalescence between the two drops occur due to the charge transfer upon drop-drop contact. The electric forces which initially bring the two drops closer, switch direction upon charge transfer and pull the drops away from each other. The factors governing the non-coalescence are the electric conductivity of the drop’s liquid which governs the time scale of charge transfer relative to the capillary time scale and the magnitude of the electric forces relative to the capillary and the viscous forces. Similar observations are recorded for the interactions of a charged conducting drop with an interface between a dielectric fluid and a conducting fluid which is the same as the drop’s liquid. For the case of a pendant conducting drop attached to a capillary and without any influx of liquid from the capillary, we observed that the drop undergoes oscillations at lower values of electric potential when subjected to a step change in the applied electric potential. At higher values of electric potential, we observed the phenomenon of cone-jet formation which results due to the accumulation of the electric charges and thus the electric forces at the drop tip. For the formation of a pendant conducting drops from a charged capillary due to liquid injection, we observed that the drops are elongated in presence of an electric field. This happens because the free charge which appears at the drop tip is attracted towards the grounded electrode. This also leads to the formation of elongated liquid threads which connect the drop to the capillary during drop detachment. We plotted the variation of total electric charge inside the drops with respect to time and found the charge increases steeply as the drop becomes elongated and moves towards the grounded electrode. For sessile drop oscillations under an alternating electric field, two different modes of operations are studied. In the so called ‘Contact mode’ case, the droplet is placed on a dielectric coated grounded electrode and the charged needle electrode remains in direct contact with the drop as it oscillates. In the ‘Non-contact mode’ case, the drop is placed directly on the grounded electrode and electric potential is applied to a needle electrode which now remains far from the drop. We show that the drop oscillations in the contact mode are caused by concentration of electric forces near the three phase contact line where the electric charge accumulates because of the repulsion from the needle. For the non-contact mode, we observe that the electric charge is attracted by the needle towards the drop apex resulting in a concentration of the electric forces in that region. So the drop oscillates due to the electric forces acting on a region near the drop tip. We also present the variation of the total electric charge inside the drop with respect to time for the two cases studied.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Em suma, esta tese propõe uma metodologia de acompanhamento de interfaces móveis que baseia-se no método dos conjuntos de nível aqui chamado de método das curvas de nível, uma denominação baseada nas aplicações em que as interfaces são representadas por curvas acoplado a uma implementação semi-Lagrangeana, para problemas em meios porosos. Embora esta técnica possa, em princípio, ser aplicada a qualquer problema físico que apresente uma interface móvel, nesta tese são focados escoamentos em meios porosos consolidados e saturados por um ou dois fluidos imiscíveis e incompressíveis. Adicionalmente, um método iterativo paralelizável para a resolução de sistemas de equações lineares definidos em redes, que podem ser reduzidos à forma das equações fundamentais de equilíbrio, é empregado na determinação dos campos de velocidade associados aos escoamentos em meios porosos. O cenário semi-Lagrangeano acoplado ao método das curvas de nível é comparado com a implementação utilizando o bem conhecido esquema up-wind. Um exaustivo estudo realizado revela a superioridade da metodologia proposta frente à concorrente utilizando o up-wind. Finalmente, o método das curvas de nível com implementação semi-Lagrangeana (método semi-Lagrangeano das curvas de nível), e o método iterativo para a determinação do campo de velocidades são aplicados no estudo de problemas transientes em meios porosos que apresentam instabilidades dos tipos Saffman-Taylor e Rayleigh-Taylor. Este estudo envolve uma análise de estabilidade linear, a introdução de diversas perturbações trigonométricas na interface e a sua evolução não-linear. / Briefy, this thesis proposes a method for capturing moving interfaces based on the level set method coupled to a Semi-Lagrangian implementation for problems in porous media. Although this method could, in principle, be applied to any physical problem with moving interfaces, we foccus, in this thesis, on flows inside a consolidated porous media saturated by one or two imiscible and incompressible fluids. Besides, a parallelizable iterative method for solving linear systems defined on a network that can be reduced to the fundamental equilibrium equations, is employed to determine the velocity field associated with the flow in a porous medium. The semi-Lagrangian scheme coupled with the level set method is compared with the well-known implementation with the up-wind scheme. An exhaustive study is performed and reveals the superiority of the proposed scheme in relation to the competing one using the up-wind method. Finally, the level set method with semi-Lagrangian implementation and the iterative method for determining the velocity field are applied to the study of transient problems in porous media which present Saffman-Taylor and Rayleigh-Taylor instabilities. This study involves the application of a linear stability analysis, the introduction of several trigonometric perturbations to the interface and its non-linear evolution.

La reconstruction tomographique par temps de première arrivée est rendue difficile par son caractère mal posé et par la non-linéarité du problème direct associé. Dans cette thèse, on se propose d'employer un modèle déformable, permettant d'introduire un a priori global sur la forme des objets à reconstruire, pour obtenir des solutions plus stables et de meilleure qualité. Dans un premier temps, nous introduisons des contraintes de forme de haut niveau en reconstruction tomographique d'émission, modalité où le problème direct est linéaire. Dans un second temps, différentes stratégies de résolution du problème non linéaire de reconstruction en temps d'arrivée sont envisagées. La solution retenue approche le problème direct par une suite de problèmes linéaires, conduisant à un algorithme par minimisations successives simples, au coursdesquelles l'a priori de forme est introduit. L'efficacité de la méthode est montrée en simulation et à partir de données réelles, acquises sur un banc développé par l'IFSTTAR pour le contrôle non destructif de structures de génie civil. / Image reconstruction from first time arrival is a difficult task due to its ill-posedness nature and to the non linearity of the direct problem associated. In this thesis, the purpose is to use a deformable model because it enables to introduce a global shape prior on the objects to reconstruct, which leads to more stable solutions with better quality. First, high level shape constraints are introduced in Computerized Tomography for which the direct problem is linear. Secondly, different strategies to solve the image reconstruction problem with a non linearity hypothesis are considered. The chosen strategy approximates the direct problem by a series of linear problems, which leads to a simple successive minimization algorithm with the introduction of the shape prior along the minimization. The efficiency of the method is demonstrated for simulated data as for real data obtained from a specific measurement device developped by IFSTTAR for non destructive evaluation of civil engineering structures.

The macroscopic behavior of complex heterogeneous materials is strongly governed by the interactions between their elementary constituents within their microstructure. Beside experimental efforts characterizing the behaviors of such materials, there is growing interest, in view of the increasing computational power available, in building models representing their microstructural systems integrating the elementary behaviors of their constituents and their geometrical organization. While a large number of contributions on this aspect focus on the investigation of advanced physics in material parameter studies using rather simple geometries to represent the spatial organization of heterogeneities, few are dedicated to the exploration of the role of microstructural geometries by means of morphological parameter studies.<p>The critical ingredients of this second type of investigation are (I) the generation of sets of representative volume elements ( RVE ) describing the geometry of microstructures with a satisfying control on the morphology relevant to the material of interest and (II) the discretization of governing equations of a model representing the investigated physics on those RVEs domains. One possible reason for the under-representation of morphologically detailed RVEs in the related literature may be related to several issues associated with the geometrical complexity of the microstructures of considered materials in both of these steps. Based on this hypothesis, this work is aimed at bringing contributions to advanced techniques for the generation and discretization of microstructures of complex heterogeneous materials, focusing on geometrical issues. In particular, a special emphasis is put on the consistent geometrical representation of RVEs across generation and discretization methodologies and the accommodation of a quantitative control on specific morphological features characterizing the microstructures of the covered materials.<p>While several promising recent techniques are dedicated to the discretization of arbitrary complex geometries in numerical models, the literature on RVEs generation methodologies does not provide fully satisfying solutions for most of the cases. The general strategy in this work consisted in selecting a promising state-of-the-art discretization method and in designing improved RVE generation techniques with the concern of guaranteeing their seamless collaboration. The chosen discretization technique is a specific variation of the generalized / extended finite element method that accommodates the representation of arbitrary input geometries represented by level set functions. The RVE generation techniques were designed accordingly, using level set functions to define and manipulate the RVEs geometries. <p>The RVE methodologies developed are mostly morphologically motivated, incorporating governing parameters allowing the reproduction and the quantitative control of specific morphological features of the considered materials. These developments make an intensive use of distance fields and level set functions to handle the geometrical complexity of microstructures. Valuable improvements were brought to the RVE generation methodologies for several materials, namely granular and particle-based materials, coated and cemented geomaterials, polycrystalline materials, cellular materials and textile-based materials. RVEs produced using those developments have allowed extensive testing of the investigated discretization method, using complex microstructures in proof-of-concept studies involving the main ingredients of RVE-based morphological parameter studies of complex heterogeneous materials. In particular, the illustrated approach offers the possibility to address three crucial aspects of those kinds of studies: (I) to easily conduct simulations on a large number of RVEs covering a significant range of morphological variations for a material, (II) to use advanced constituent material behaviors and (III) to discretize large 3D RVEs. Based on those illustrations and the experience gained from their realization, the main strengths and limitations of the considered discretization methods were clearly identified. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Vacuum handling is a widespread technology in automated production systems for gripping of workpieces. Unfortunately, this solution accounts for a considerable share of industrial energy consumption. This consumption is mainly due to the losses in the involved steps like air compression, distribution, vacuum generation and gripper suction. However, the energy efficiency of vacuum handling systems is still relatively unexplored. The consortium partners are working together in the BiVaS project, which is funded by the Federal Ministry for Economic Affairs and Energy (BMWi). In this project, the drawback of high energy consumption is approached by the development of an energyefficient ejector, a biomimetic suction gripper as well as on system integration and the development of energy-efficient operating strategies and their energetic balancing in order to reduce the consumption of compressed air by 20 %. This reduction will be quantified experimentally in a pilot plant benchmark, where a realistic handling process is developed and examined. This paper shows the state of the art focusing on energy consumption of vacuum handling technology and the behaviour of the involved components during different handling operations. Furthermore, first biomimetic concepts and an estimation of the increase of energy efficiency will be presented for the running project.

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