Das weltweite Ausmaß der Urbanisierung zählt zu den großen ökologischen Herausforderungen des 21. Jahrhunderts. Die Fernerkundung bietet die Möglichkeit das Verständnis dieses Prozesses und seiner Auswirkungen zu erweitern. Der Fokus dieser Arbeit lag in der Quantifizierung der städtischen Landbedeckung mittels Maschinellen Lernens und räumlich unterschiedlich aufgelöster Hyperspektraldaten. Untersuchungen berücksichtigten innovative methodische Entwicklungen und neue Möglichkeiten, die durch die bevorstehende Satellitenmission EnMAP geschaffen werden. Auf Basis von Bilder des flugzeugestützten HyMap Sensors mit Auflösungen von 3,6 m und 9 m sowie simulierten EnMAP-Daten mit einer Auflösung von 30 m wurde eine Kartierung entlang des Stadt-Umland-Gradienten Berlins durchgeführt. Im ersten Teil der Arbeit wurde die Kombination von Support Vektor Regression mit synthetischen Trainingsdaten für die Subpixelkartierung eingeführt. Ergebnisse zeigen, dass sich der Ansatz gut zur Quantifizierung thematisch relevanter und spektral komplexer Oberflächenarten eignet, dass er verbesserte Ergebnisse gegenüber weiteren Subpixelverfahren erzielt, und sich als universell einsetzbar hinsichtlich der räumlichen Auflösung erweist. Im zweiten Teil der Arbeit wurde der Wert zukünftiger EnMAP-Daten für die städtische Fernerkundung abgeschätzt. Detaillierte Untersuchungen unterstreichen deren Eignung für eine verbesserte und erweiterte Beschreibung der Stadt nach dem bewährten Vegetation-Impervious-Soil-Schema. Analysen der Möglichkeiten und Grenzen zeigen sowohl Nachteile durch die höhere Anzahl von Mischpixel im Vergleich zu hyperspektralen Flugzeugdaten als auch Vorteile aufgrund der verbesserten Differenzierung städtischer Materialien im Vergleich zu multispektralen Daten. Insgesamt veranschaulicht diese Arbeit, dass die Kombination von hyperspektraler Satellitenbildfernerkundung mit Methoden des Maschinellen Lernens eine neue Qualität in die städtische Fernerkundung bringen kann. / The global dimension of urbanization constitutes a great environmental challenge for the 21st century. Remote sensing is a valuable Earth observation tool, which helps to better understand this process and its ecological implications. The focus of this work was to quantify urban land cover by means of machine learning and imaging spectrometer data at multiple spatial scales. Experiments considered innovative methodological developments and novel opportunities in urban research that will be created by the upcoming hyperspectral satellite mission EnMAP. Airborne HyMap data at 3.6 m and 9 m resolution and simulated EnMAP data at 30 m resolution were used to map land cover along an urban-rural gradient of Berlin. In the first part of this work, the combination of support vector regression with synthetically mixed training data was introduced as sub-pixel mapping technique. Results demonstrate that the approach performs well in quantifying thematically meaningful yet spectrally challenging surface types. The method proves to be both superior to other sub-pixel mapping approaches and universally applicable with respect to changes in spatial scales. In the second part of this work, the value of future EnMAP data for urban remote sensing was evaluated. Detailed explorations on simulated data demonstrate their suitability for improving and extending the approved vegetation-impervious-soil mapping scheme. Comprehensive analyses of benefits and limitations of EnMAP data reveal both challenges caused by the high numbers of mixed pixels, when compared to hyperspectral airborne imagery, and improvements due to the greater material discrimination capability when compared to multispectral spaceborne imagery. In summary, findings demonstrate how combining spaceborne imaging spectrometry and machine learning techniques could introduce a new quality to the field of urban remote sensing.

The aim of this thesis is to present new results concerning duality in scalar optimization. We show how the theory can be applied to optimization problems arising in the theory of risk measures, portfolio optimization and machine learning. First we give some notations and preliminaries we need within the thesis. After that we recall how the well-known Lagrange dual problem can be derived by using the general perturbation theory and give some generalized interior point regularity conditions used in the literature. Using these facts we consider some special scalar optimization problems having a composed objective function and geometric (and cone) constraints. We derive their duals, give strong duality results and optimality condition using some regularity conditions. Thus we complete and/or extend some results in the literature especially by using the mentioned regularity conditions, which are weaker than the classical ones. We further consider a scalar optimization problem having single chance constraints and a convex objective function. We also derive its dual, give a strong duality result and further consider a special case of this problem. Thus we show how the conjugate duality theory can be used for stochastic programming problems and extend some results given in the literature. In the third chapter of this thesis we consider convex risk and deviation measures. We present some more general measures than the ones given in the literature and derive formulas for their conjugate functions. Using these we calculate some dual representation formulas for the risk and deviation measures and correct some formulas in the literature. Finally we proof some subdifferential formulas for measures and risk functions by using the facts above. The generalized deviation measures we introduced in the previous chapter can be used to formulate some portfolio optimization problems we consider in the fourth chapter. Their duals, strong duality results and optimality conditions are derived by using the general theory and the conjugate functions, respectively, given in the second and third chapter. Analogous calculations are done for a portfolio optimization problem having single chance constraints using the general theory given in the second chapter. Thus we give an application of the duality theory in the well-developed field of portfolio optimization. We close this thesis by considering a general Support Vector Machines problem and derive its dual using the conjugate duality theory. We give a strong duality result and necessary as well as sufficient optimality conditions. By considering different cost functions we get problems for Support Vector Regression and Support Vector Classification. We extend the results given in the literature by dropping the assumption of invertibility of the kernel matrix. We use a cost function that generalizes the well-known Vapnik's ε-insensitive loss and consider the optimization problems that arise by using this. We show how the general theory can be applied for a real data set, especially we predict the concrete compressive strength by using a special Support Vector Regression problem.

The aim of this thesis is to present new results concerning duality in scalar optimization. We show how the theory can be applied to optimization problems arising in the theory of risk measures, portfolio optimization and machine learning. First we give some notations and preliminaries we need within the thesis. After that we recall how the well-known Lagrange dual problem can be derived by using the general perturbation theory and give some generalized interior point regularity conditions used in the literature. Using these facts we consider some special scalar optimization problems having a composed objective function and geometric (and cone) constraints. We derive their duals, give strong duality results and optimality condition using some regularity conditions. Thus we complete and/or extend some results in the literature especially by using the mentioned regularity conditions, which are weaker than the classical ones. We further consider a scalar optimization problem having single chance constraints and a convex objective function. We also derive its dual, give a strong duality result and further consider a special case of this problem. Thus we show how the conjugate duality theory can be used for stochastic programming problems and extend some results given in the literature. In the third chapter of this thesis we consider convex risk and deviation measures. We present some more general measures than the ones given in the literature and derive formulas for their conjugate functions. Using these we calculate some dual representation formulas for the risk and deviation measures and correct some formulas in the literature. Finally we proof some subdifferential formulas for measures and risk functions by using the facts above. The generalized deviation measures we introduced in the previous chapter can be used to formulate some portfolio optimization problems we consider in the fourth chapter. Their duals, strong duality results and optimality conditions are derived by using the general theory and the conjugate functions, respectively, given in the second and third chapter. Analogous calculations are done for a portfolio optimization problem having single chance constraints using the general theory given in the second chapter. Thus we give an application of the duality theory in the well-developed field of portfolio optimization. We close this thesis by considering a general Support Vector Machines problem and derive its dual using the conjugate duality theory. We give a strong duality result and necessary as well as sufficient optimality conditions. By considering different cost functions we get problems for Support Vector Regression and Support Vector Classification. We extend the results given in the literature by dropping the assumption of invertibility of the kernel matrix. We use a cost function that generalizes the well-known Vapnik's ε-insensitive loss and consider the optimization problems that arise by using this. We show how the general theory can be applied for a real data set, especially we predict the concrete compressive strength by using a special Support Vector Regression problem.