Nonrigid deformation modeling and estimation from images is a technically challenging task due to its nonlinear, nonconvex and high-dimensional nature. Traditional optimization procedures often rely on good initializations and give locally optimal solutions. On the other hand, learning-based methods that directly model the relationship between deformed images and their parameters either cannot handle complicated forms of mapping, or suffer from the Nyquist Limit and the curse of dimensionality due to high degrees of freedom in the deformation space. In particular, to achieve a worst-case guarantee of ∈ error for a deformation with d degrees of freedom, the sample complexity required is O(1/∈d). In this thesis, a generative model for deformation is established and analyzed using a unified theoretical framework. Based on the framework, three algorithms, Data-Driven Descent, Top-down and Bottom-up Hierarchical Models, are designed and constructed to solve the generative model. Under Lipschitz conditions that rule out unsolvable cases (e.g., deformation of a blank image), all algorithms achieve globally optimal solutions to the specific generative model. The sample complexity of these methods is substantially lower than that of learning-based approaches, which are agnostic to deformation modeling. To achieve global optimality guarantees with lower sample complexity, the structureembedded in the deformation model is exploited. In particular, Data-driven Descentrelates two deformed images that are far away in the parameter space by compositionalstructures of deformation and reduce the sample complexity to O(Cd log 1/∈).Top-down Hierarchical Model factorizes the local deformation into patches once theglobal deformation has been estimated approximately and further reduce the samplecomplexity to O(Cd/1+C2 log 1/∈). Finally, the Bottom-up Hierarchical Model buildsrepresentations that are invariant to local deformation. With the representations, theglobal deformation can be estimated independently of local deformation, reducingthe sample complexity to O((C/∈)d0) (d0 ≪ d). From the analysis, this thesis showsthe connections between approaches that are traditionally considered to be of verydifferent nature. New theoretical conjectures on approaches like Deep Learning, arealso provided. practice, broad applications of the proposed approaches have also been demonstrated to estimate water distortion, air turbulence, cloth deformation and human pose with state-of-the-art results. Some approaches even achieve near real-time performance. Finally, application-dependent physics-based models are built with good performance in document rectification and scene depth recovery in turbulent media.

Σε αυτήν την εργασία, παρουσιάζουμε τις βασικότερες κλασικές προσεγγίσεις επίλυσης Πολυ-αντικειμενικών Προβλημάτων Βελτιστοποίησης(ΠΠΒ)καθώς και ένα από τα πιο δημοφιλή λογισμικά για επίλυση ΠΠΒ, το NIMBUS. Συγκεκριμένα, δίνουμε τον ορισμό ενός ΠΠΒ, το θεωρητικό υπόβαθρο -- για την καλύτερη κατανόηση των μεθόδων που θα ακολουθήσουν - και τις διαφορές των ΠΠΒ με τα κλασσικά Μονο-αντικειμενικά προβλήματα Βελτιστοποίησης. Επιπλέον, παρουσιάζουμε τις τρεις κύριες κατηγορίες προσέγγισης των ΠΠΒ (μη-αλληλεπιδραστικές, αλληλεπιδραστικές, εξελικτικές) ο διαχωρισμός των οποίων γίνεται ανάλογα με την άμεση ή έμμεση εμπλοκή του Λήπτη Απόφασης. Η μελέτη μας εστιάζεται κυρίως στην κατηγορία των μη-αλληλεπιδραστικών προσεγγίσεων, στην οποία ο ΛΑ εμπλέκεται έμμεσα. Τέλος, ολοκληρώνουμε την μελέτη μας με την αναλυτική παρουσίαση της επίλυσης ενός ΠΠB με την χρήση του λογισμικού NIMBUS. / In this contribution, we study the classical approaches for solving Multi-objective Optimization Problems (MOOP) as well as one of the most popular software that solves MOOP, namely NIMBUS. More specifically, we present the definition and the theoretical background around MOOP and we discuss the differences between MOOP and the classical single-objective optimization problems. We also present the three main categories of approaches of solving MOOP (non-interactive, interactive, evolutionary) that are characterized by the way the Decision Maker participates in the solution. We focus on the first category by analyzing each of the non-interactive approaches. Finally, we conclude by presenting an analytic illustration of an example that solves a MOOP using the NIMBUS software.

Diploma thesis solves what to do with the old flat in the long-term horizon considering all possible risks. Thesis analyses current status of the flat and calculates its approximate price in the reality market. Then it compares rentability of hiring with the option of selling and investing to the three most common bank products. It analyses risks associated with all options. The output is the recommendation of the most profitable option considering the owners interests.