Many geophysical inverse problems are ill-posed and have to be regularized. The most often used solution methods for solving ill-posed problems are based on the use of quadratic regularization that results in smooth solutions. Solutions of this type are not to be suitable when the model parameter is piecewise continuous blocky and edges are desired in the regularized solution. To avoid the smoothing of edges, which are very important attributes of an image, an edge-preserving regularization (non-quadratic regularization) term has to be employed. Total Variation (TV) regularization is one of the most effective regularization techniques for allowing sharp edges and the existence of discontinuities in the solutions. The edge-preserving regularization based on the TV method for small-scale geophysical inverse problems to the problem of estimating the acoustic velocity perturbation from a multi-source-receiver geophysical experiment is studied. The acoustic velocity perturbation is assumed to be piecewise continuous and blocky. The problem is based on linearization acoustic modeling using the framework of the single-scattering Born approximation from a known constant background medium. To solve this non-linear and ill-posed problem, an iterative scheme based on the conjugate gradient method is employed. The TV regularization method provides us with the opportunity to recover more useful information of velocity profiles from the measured seismic data. Though it requires more effort in implementing the TV term to control the smoothing and regularization parameter, the algorithm possesses the strong ability of marking the discontinuities and ensures their preservation from over-smoothing. / Geophysics

Images may be degraded by some random error which is called noise. Noise may occur during image capture, transmission or processing and its elimination is achieved by smoothing filters. Linear smoothing filters blur the edges and edges are important characteristics of images and must be preserved. Various edge preserving smoothing filters are proposed in the literature. In this thesis, most common smoothing and edge preserving smoothing filters are discussed and a new method is proposed by modifying Ambrosio Tortorelli approximation of Mumford Shah Model. New method takes into edge direction consistency account and produces sharper results at comparable scales.

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The self-organizing map (SOM) is an unsupervised learning algorithm which has been successfully applied to various applications. One of advantages of SOM is it maintains an incremental property to handle data on the fly. In the last several decades, there have been variants of SOM used in many application domains. In this dissertation, two new SOM algorithms are developed for image quantization and compression. The first algorithm is a sample-size adaptive SOM algorithm that can be used for color quantization of images to adapt to the variations of network parameters and training sample size. The sweep size of neighborhood function is modulated by the size of the training data. In addition, the minimax distortion principle which is modulated by training sample size is used to search the winning neuron. Based on the sample-size adaptive self-organizing map, we use the sampling ratio of training data, rather than the conventional weight change between adjacent sweeps, as a stop criterion. As a result, it can significantly speed up the learning process. Experimental results show that the proposed sample-size adaptive SOM achieves much better PSNR quality, and smaller PSNR variation under various combinations of network parameters and image size. The second algorithm is a novel classified SOM method for edge preserving quantization of images using an adaptive subcodebook and weighted learning rate. The subcodebook sizes of two classes are automatically adjusted in training iterations based on modified partial distortions that can be estimated incrementally. The proposed weighted learning rate updates the neuron efficiently no matter of how large the weighting factor is. Experimental results show that the proposed classified SOM method achieves better quality of reconstructed edge blocks and more spread out codebook and incurs a significantly less computational cost as compared to the competing methods.

A non-linear edge-preserving solution to the least-squares migration problem with sparseness & illumination constraints is proposed. The applied formalism explores Curvelets as basis functions. By virtue of their sparseness and locality, Curvelets not only reduce the dimensionality of the imaging problem but they also naturally lead to a dense preconditioning that almost diagonalizes the normal/Hessian operator. This almost diagonalization allows us to recast the imaging problem into a ’simple’ denoising problem. As such, we are in the position to use non-linear estimators based on thresholding. These estimators exploit the sparseness and locality of Curvelets and allow us to compute a first estimate for the reflectivity, which approximates the least-squares solution of the seismic inverse scattering problem. Given this estimate, we impose sparseness and additional amplitude corrections by solving a constrained optimization problem. This optimization problem is initialized and constrained by the thresholded image and is designed to remove remaining imaging artifacts and imperfections in the estimation and reconstruction.

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On suppose que S=Sg,n est un surface connexe orientable de type topologique fini, de genre g≥3 et n≥0 épointements. Dans les chapitres 1 et 2 on décrit l'ensemble principal d'une surface et prouve que en utilisant expansions rigides itérés, on peut créer suites croissantes d'ensembles finis qui sa réunion est le complexe des courbes de la surface C(S). Dans le 3ème chapitre on introduit l'ensemble rigide X(S) de Aramayona et Leininger et l'utilise pour montrer que la suite des chapitres précédents est eventuellement une suite d'ensembles rigides. On utilise cela pour prouver que si Si=Sgi,ni pour i=1,2 sont surfaces telles que k(S1)≥k(S2) et g1≥3, toute application qui préserve les arêtes de C(S1) dans C(S2) est induite par un homéomorphisme. Ceci est utilisé pour montrer un résultat similaire pour les homomorphismes de sous-groupes de Mod*(S1) dans Mod*(S2). Dans le 4ème chapitre on utilise les résultats précédents pour prouver que l'unique façon d'obtenir une application qui préserve les arêtes et qui est alternante du graphe de Hatcher-Thurston de S1, HT(S1), dans soi de S2, HT(S2) est en utilisant un homéomorphisme de S1 et puis piquer la surface n fois pour obtenir S2. Ceci implique que toute application qui préserve les arêtes et qui est alternante de HT(S) dans soi même et aussi tous les automorphismes de HT(S), sont induits par homéomorphismes. Dans le 5ème chapitre on montre que toute application super-injective du graphe des courbes qui ne sépare pas et courbes extérieures de S1, NO(S1), dans soi de S2, NO(S2), est induite par un homéomorphisme. Finalement, dans les conclusions on discute la signifiance des résultats et les façons possibles d'étendre leur. / Suppose S = Sg,n is an orientable connected surface of finite topological type, with genus g ≥ 3 and n ≥ 0 punctures. In the first two chapters we describe the principal set of a surface, and prove that through iterated rigid expansions we can create an increasing sequence of finite sets whose union in the curve complex of the surface C(S). In the third chapter we introduced Aramayona and Leininger's finite rigid set X(S) and use it to prove that the increasing sequence of the previous two chapters becomes an increasing sequence of finite rigid sets after, at most, the fifth iterated rigid expansion. We use this to prove that given S1 = Sg1,n1 and S2 = Sg2,n2 surfaces such that k(S1) ≥ k(S2) and g1 ≥ 3, any edge-preserving map from C(S1) to C(S2) is induced by a homeomorphism from S1 to S2. This is later used to prove a similar statement using homomorphisms from certain subgroups of Mod*(S1) to Mod*(S2). In the fourth chapter we use the previous results to prove that the only way to obtain an edge-preserving and alternating map from the Hatcher-Thurston graph of S1 = Sg,0, HT(S1), to the Hatcher-Thurston graph of S2 = Sg,n, HT(S2), is using a homeomorphism of S1 and then make n punctures to the surface to obtain S2. As a consequence, any edge-preserving and alternating self-map of HT(S) as well as any automorphism is induced by a homeomorphism. In the fifth chapter we prove that any superinjective map from the nonseparating and outer curve graph of S1, NO(S1), to that of S2, NO(S2), is induced by a homeomorphism assuming the same conditions as in the previous chapters. Finally, in the conclusions we discuss the meaning of these results and possible ways to expand them.