Nichtlineare Verzerrungen von Rasterkraftmikroskopie (engl.: scanning probe microscopy, Abk.: SPM) Bildern beeinträchtigen die Qualität von Nanotomographiebildern und SPM Bildsequenzen. In dieser Arbeit wird ein neues, nichtlineares Registrierungsverfahren vorgestellt, das auf einem für medizinische Anwendungen entwickelten Algorithmus aufbaut und diesen für die Behandlung von SPM Daten erweitert. Die nichtlineare Registrierung ermöglicht es, verschiedene nanostrukturierte Materialen über große Bereiche (1 µm x 1 µm) mit einer Auflösung von 10 nm abzubilden. Dies erlaubt eine wesentlich detailliertere quantitative Analyse der Daten. Hierfür wurde eine neue Datenreduktions- und Visualisierungsmethode für Mikrodomänennetzwerke von Blockcopolymeren eingeführt. Zwei- und dreidimensionale Mikrodomänenstrukturen werden zu ihrem Skelett reduziert, Verzweigungspunkte farblich codiert und der entstandene Graph visualisiert. Die Anzahl verschiedener Skelettverzweigungen lässt sich über die Zeit verfolgen. Die Methode wurde mit lokalen Minkowskimaßen der ursprünglichen Graustufenbilder verglichen. Sie liefert morphologische und geometrische Informationen auf unterschiedlichen Längenskalen.

Les fonctionnelles de Minkowski définissent des mesures topologiques et géométriques d'ensembles, insuffisantes pour la caractérisation, des ensembles différents pouvant avoir les mêmes fonctionnelles. D'autres fonctionnelles de forme, géométriques et morphométriques, sont donc utilisées. Un diagramme de forme, défini grâce à deux fonctionnelles morphométriques, donne une représentation permettant d'étudier les formes d'ensembles. En analyse d'image, ces fonctionnelles et diagrammes sont souvent limités aux images binaires et déterminés de manière globale et mono-échelle. Les Voisinages Adaptatifs Généraux (VAG) simultanément adaptatifs avec les échelles d'analyse, structures spatiales et intensités des images, permettent de pallier ces limites. Une analyse locale, adaptative et multi-échelle des images à tons de gris est proposée sous forme de cartographies des fonctionnelles de forme à VAG.Les VAG, définis en tout point du support spatial d'une image à tons de gris, sont homogènes par rapport à un critère d'analyse représenté dans un modèle vectoriel, suivant une tolérance d'homogénéité. Les fonctionnelles de forme calculées pour chaque VAG de l'image définissent les cartographies des fonctionnelles de forme à VAG. Les histogrammes et diagrammes de ces cartographies donnent des distributions statistiques des formes des structures locales de l'image contrairement aux histogrammes classiques qui donnent une distribution globale des intensités de l'image. L'impact de la variation des critères axiomatiques des VAG est analysé à travers ces cartographies, histogrammes et diagrammes. Des cartographies multi-échelles sont construites, définissant des fonctions de forme à VAG. / Minkowski functionals define set topological and geometrical measurements, insufficient for the characterization, because different sets may have the same functionals. Thus, other shape functionals, geometrical and morphometrical are used. A shape diagram, defined thanks to two morphometrical functionals, provides a representation allowing the study of set shapes. In quantitative image analysis, these functionals and diagrams are often limited to binary images and achieved in a global and monoscale way. The General Adaptive Neighborhoods (GANs) simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities, enable to overcome these limitations. The GAN-based Minkowski functionals are introduced, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way.The GANs, defined around each point of the spatial support of a gray-tone image, are homogeneous with respect to an analyzing criterion function represented in an algebraic model, according to an homogeneity tolerance. The shape functionals computed on the GAN of each point of the spatial support of the image, define the so-called GAN-based shape maps. The map histograms and diagrams provide statistical distributions of the shape of the gray-tone image local structures, contrary to the classical histogram that provides a global distribution of image intensities. The impact of axiomatic criteria variations is analyzed through these maps, histograms and diagrams. Thus, multiscale maps are built, defining GAN-based shape functions.

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

Les tessellations, aussi appelées mosaïques, permettent de modéliser de nombreuses structures, comme des assemblages de cellules en biologie ou de grains en science des matériaux. La tessellation aléatoire la plus connue est le diagramme de Voronoï qui à partir d'un ensemble de points, appelés germes, partitionne le plan. L'approche innovante de cette thèse est d'utiliser des champs aléatoires gaussiens pour générer des germes et des distances aléatoires, qui vont permettre de simuler une grande variété de tessellations en termes de formes et de tailles des cellules.Pour connaître les propriétés des tessellations simulées à partir de champs aléatoires gaussiens, celles-ci vont être caractérisées et comparées à d'autres tessellations. Tout d'abord par une approche ponctuelle en étudiant les germes, dont leur distribution spatiale. Puis par une approche par région, en étudiant la géométrie et la morphométrie des cellules.L'endothélium cornéen humain est une monocouche de cellules formant un pavage hexagonal régulier à la naissance, et perdant de sa régularité ensuite. La qualité du greffon cornéen est donnée par certaines observations, comme la densité, l'homogénéité de la forme et des tailles des cellules endothéliales.L'évolution avec l'âge de cette mosaïque cornéenne va être caractérisée à partir d’une base d’images de l’endothélium. L'originalité est ensuite d'effectuer une estimation de l'âge d’un endothélium à partir des différentes mesures permettant de caractériser les tessellations, et enfin de mettre en place une méthode prometteuse afin de savoir si une cornée a une évolution normale. / Tessellations, also called mosaics, are used to model many structures, for example cellular arrangements in biology or grains in material science. The most known tessellation is the Voronoï diagram which partitions the space from a set of points, called germs. The innovative approach of this thesis is to use Gaussian random fields to generate germs and random distances. The use of random fields allows to simulate a great variety of tessellations in terms of cells forms and sizes.To study the properties of each type of tessellation, they are characterized: first, by studying the germs, including their spatial distribution, and then by analyzing the cells geometry and morphometry. These tessellations are also compared to other known tessellations.The human corneal endothelium is a mono-layer of cells forming a regular hexagonal mosaic at birth, and losing his regularity later. The corneal graft quality is given by some observations made on the endothelial mosaic (cells density, the homogeneity of cells sizes and shapes).A database of endothelium images allows to characterize the evolution with age of the corneal mosaic. The originality is to estimate the age of an endothelium based on the measures computed to characterize the tessellations, and finally to set up a promising method to evaluate if a corneal evolution is normal.

Cette thèse est divisée en deux parties : la partie I traite de combinatoire additive, la partie II s’est portée sur des questions de théorie élémentaire des nombres. Dans le chapitre 1, on généralise la transformée de Davenport pour prouver que si S\mathbb A=(A, +)S est un demi-groupe cancellatif (éventuellement non commutatif) et SX, YS sont des sous-ensembles non vides de SAS tels que le sous semi groupe engendré par SYS est commutatif, on a SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, où S\gamma(\ctlot)S dénote la constante de Cauchy-Davenport d’un ensemble. On en obtient une extension des théorèmes de Chowla et Pillai pour les groupes cycliques et une version plus forte d’un théorème additif de Karolyi et Hamidoune. Dans le chapitre 2, on montre que si S(A,+)S est un semi-groupe cancellatif et si SX, Y\subsetcq AS alors SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. Cela donne une généralisation de l’inégalité de Kemperman pour les groupes sans torsion et une version plus forte du théorème d’Hamidoune-Karolyi. Dans le chapitre 3, on généralise des résultats par Freiman et al., en prouvant que si S(A,\ctlot)S est un semi-groupe linéairement ordonnable et SSS est un sous-ensemble fini de SAS engendrant un sous-semi-groupe non-abélien, alors S|S^2-\gc3|S|-2S. Dans le chapitre 4, on prouve des résultats liés à une conjecture par Gyorgy et Smyth sur la finitude des entiers Sn\gc1S tels que Sn^kS divise Sa^a \pmb^nS pour des entiers fixés SaS, SbS et SkS avec Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. Enfin, dans le chapitre 5, on considère une question de divisibilité dans les entiers, en quelque sorte liée au problème de Znam et à la conjecture d’Agoh-Giuga / This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla’s and Pillai’s theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman’s inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam’s problem and the Agoh-Giuga conjecture

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

La Tesis Doctoral está dedicada al estudio de ciertas propiedades de los radios sucesivos de los cuerpos convexos (funcionales definidos a partir de circunradios e inradios de proyecciones o secciones del cuerpo). Comenzamos estableciendo las nociones básicas necesarias para el desarrollo de los contenidos. A continuación calculamos los radios sucesivos de familias particulares de conjuntos (p-bolas, anchura constante, cuerpos tangenciales), y estudiamos la conexión existente entre estos funcionales y los números de Gelfand y Kolmogorov. En el tercer capítulo consideramos el problema de Pukhov-Perel'man sobre la mejor cota superior para un cierto cociente de radios, determinando desigualdades para problemas de este tipo que van a permitir mejorar los resultados existentes en ciertos casos. Finalmente, estudiamos cómo se relacionan los radios sucesivos de la suma de Minkowski (Firey) de dos cuerpos convexos con los correspondientes funcionales de los conjuntos, obteniendo los resultados óptimos en todos los casos. / The Doctoral Thesis is focused in the study of some properties of the successive radii of convex bodies (functionals defined by means of circumradii and inradii of projections or sections of the set). We start establishing the basic notions that will be needed further on. Next, we compute the successive radii of particular families of sets (p-balls, constant width sets and tangential bodies), and study the connection between these functionals and the Gelfand and Kolmogorov numbers. In the third chapter we consider the Pukhov-Perel'man problem on the best upper bound for a particular ratio of radii, determining inequalities for some problems of this type which will allow to improve the known results in particular cases. Finally we study how the successive radii of the (Firey)-Minkowski addition of two convex bodies are related with the corresponding functionals of the sets, obtaining the optimal results in all cases.

This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla's and Pillai's theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman's inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam's problem and the Agoh-Giuga conjecture