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Ulm, Univ. Diss., 2007.

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Imaging and image processing have been used in variety of applications, such as medical, astronomy, forensic, and industry. Numerous techniques have been proposed to solve speci c problems faced in particular applications which are comprised of a series of processes such as, image enhancement, ltering, segmentation, representation, and recognition. However, there is no a universal algorithm which can be applied to variant image modalities with corresponding applications. With the aim of learning image processing tasks, as supervised learning techniques, we can develop e ective algorithms which are image and task oriented. Learning image processing comprises two main phases, namely: training and testing phases. During training phase, the algorithm has the capability of discovering and adjusting an optimum transformation function or optimal mathematical morphology chain by utilizing a user-prepared ground-truth (gold) sample. Later on, in testing phase, the obtained transform function of morphological chain is applied to untrained test images. The current thesis has three main parts as follows. In the rst part, genetic programming (GP) is employed to obtain an optimum transformation function. The GP utilizes one user-prepared gold sample to learn from. The magni cent feature of this method is that it does not require a prior knowledge or large training set to learn from. The performance of the proposed approach has been examined on 147 X-ray lung images. The results for image thresholding (i.e., Otsu's method) after applying optimal transformation are promising. In the second part, an optimum mathematical morphology (MM) chain is obtained by applying GP to localize the object of interest in a binary image. Morphology operations use 27 regular structuring elements along with commonly used morphological operations (i.e., erosion, dilation, opening, and closing) to build an optimal MM chain. The obtained chains are tested against challenging test cases, such as, object translation, scaling, and rotation. In the third part, a hybrid genetic programming - di erential evolutionary (GP-DE) algorithm is proposed to optimize not only the morphology chain but also the utilized structuring elements. GP as an outer layer optimizer is responsible to optimize the morphology chain while the di erential evolutionary (DE) as an inner layer optimizer optimizes the structure elements. Similarly in the testing phase, the obtained morphology chain is applied on test images. In term of utilized test images, the two test cases have been employed : synthesis and music note images. The results indicate that the proposed method is able to locate the object of interest. For the music note images, the proposed approach is able to extract the head notes, sta s, and vertical lines correctly. The training phase is iii time consuming, but it is acceptable; because one time training is required to obtain an optimal chain for a speci c image processing task.

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The derivation of nonlinear dispersive PDE, such as the nonlinear Schrödinger (NLS) or nonlinear Hartree equations, from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE. In this thesis we present two types of results related to obtaining NLS via the GP hierarchy. In the first part of the thesis, we derive a NLS with a linear combination of power type nonlinearities in R[superscript d] for d = 1, 2. In the second part of the thesis, we focus on considering solutions to the cubic GP hierarchy and we prove unconditional uniqueness of low regularity solutions to the cubic GP hierarchy in R[superscript d] with d ≥ 1: the regularity of solution in our result coincides with known regularity of solutions to the cubic NLS for which unconditional uniqueness holds. / text

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