This report is a master thesis, written by an engineering physics and electrical engineering student at Luleå University of Technology.The desires of this project was to remove dislocations from wrapped phase maps using sparse reconstructive methods. Dislocations is an error that can appear in phase maps due to improper filtering or inadequate sampling. Dislocations makes it impossible to correctly unwrap the phasemap.The report contains a mathematical description of a sparse reconstructive method. The sparse reconstructive method is based on KSVDbox which was created by R. Rubinstein and is free for download and use. The KSVDbox is a MATLAB implementation of a dictionary learning algorithm called K-SVD with Orthogonal Matching Pursuit and a sparse reconstructive algorithm. A guide for adapting the toolbox for inpainting is included, with a couple of examples on natural images which supports the suggested adaptation. For experimental purposes a set of simulated wrapped phase maps with and without disloca-tions were created. These simulated phase maps are based on work by P. Picart. The MATLAB implementation that was used to generate these test images can be found in the appendix of this report such that they can easily be generated by anyone who has the interest to do so. Finally the report leads to an outline of five different experiments that was designed to test the KSVDbox for the processing of dislocations. Each one of these experiments uses a different dictionary. These experiments are due to inpainting with, 1. A dictionary based on Discrete Cosine Transform. 2. An adaptive dictionary, where the dictionary learning algorithm has been shown what thearea in the phase map that was damaged by dislocations should look like. 3. An adaptive dictionary, where the dictionary learning algorithm has been allowed to trainon the phase map that with damages. This is done such that areas with dislocations areignored. 4. An adaptive dictionary, where training is done on a separate image that has been designedto contain general phase patterns. 5. An adaptive dictionary, that results from concatenating the dictionaries used in experiment 3 and 4. The first three experiments are complimented with experiments done on a natural image for comparison purposes.The results show that sparse reconstructive methods, when using the scheme used in this work, is unsuitable for processing of dislocations in phase maps. This is most likely because the reconstructive method has difficulties in acquiring a high contrast reconstruction and there is nothing in the algorithm that causes the inpainting from any direction to match with the inpainting from other directions.

Ingeniero Civil Eléctrico / La identificación de husos sigma se realiza manualmente por expertos en la medicina del sueño. El proceso consiste en inspeccionar el electroencefalograma (EEG) de los registros polisomnográficos y marcar los intervalos en los que se observan los patrones. Este proceso es bastante tedioso y complicado, especialmente considerando que se buscan patrones de onda que no suelen durar más de algunos segundos en registros de aproximadamente 8 horas. Para aliviar el trabajo de los expertos se han desarrollado sistemas automáticos de detección de husos sigma capaces de identificar estos patrones en el EEG. En esta memoria se propone un nuevo método de detección automático de husos sigma en que se entrenan las formas de onda de un diccionario, usando un algoritmo de aprendizaje supervisado, para que éstas sean representativas de los husos sigma. Posteriormente, se utiliza un modelo de descomposición de señal para descomponer la señal de un canal del EEG en un número finito de componentes representados por la convolución entre las formas de onda del diccionario aprendido y un conjunto de trenes de pulsos que indican los intervalos de la señal donde se identifican patrones de onda semejantes a las formas de onda del diccionario aprendido. Los intervalos de la señal que son descompuestos por el modelo de descomposición, son consideradas como las detecciones del método, debido a que estos intervalos presentan una alta correlacción con las formas de onda representativas de los husos sigma que componen el diccionario aprendido. En el desarrollo de este método se utilizó un único registro polisom- nográfico de un niño de 10 años, con el cual se formaron los conjuntos de entrenamiento y de prueba usando fragmentos del registro en la etapa de sueño N2. El método obtuvo resultados preliminares satisfactorios que verifican su capacidad para detector husos sigma en la etapa de sueño N2 de un registro polisomnográfico, con una tasa de verdaderos positivos promedio de 85,080 % y una tasa de falsos positivos promedio de 14,995 %. El método de detección de husos sigma propuesto ofrece una metodología novedosa que no utiliza los usuales métodos espectrales para analizar el EEG. Además con este proceso se obtiene un diccionario con formas de onda representativas de los husos sigma que se puede utilizar para estudiar y caracterizar los husos sigma detectados por el método.

This diploma thesis discusses methods of dictionary learning to inpaint missing sections in the audio signal. There was theoretically analyzed and practically used algorithms K-SVD and INK-SVD for dictionary learning. These dictionaries have been applied to the reconstruction of audio signals using OMP (Orthogonal Matching Pursuit). Furthermore, there was proposed an algorithm for selecting the stationary segments and their subsequent use as training data for K-SVD and INK-SVD. In the practical part of thesis have been observed efficiency with training set selection from whole signal compared with algorithm for stationary segmentation used. The influence of mutual coherence on the quality of reconstruction with incoherent dictionary was also studied. With created scripts for multiple testing in Matlab, there was performed comparison of these methods on genre distinct songs.

Les décompositions parcimonieuses décrivent un signal comme une combinaison d'un petit nombre de formes de base, appelées atomes. Le dictionnaire d'atomes, crucial pour l'efficacité de la décomposition, peut résulter d'un choix a priori (ondelettes, Gabor, ...) qui fixe la structure du dictionnaire, ou d'un apprentissage à partir d'exemples représentatifs du signal. Nous proposons ici un cadre hybride combinant des contraites structurelles et une approche par apprentissage. Les dictionnaires ainsi structurés apportent une meilleure adaptation aux propriétés du signal et permettent de traiter des volumes importants de données. Nous exposons les concepts et les outils qui étayent cette approche, notamment l'adaptation des algorithmes Matching Pursuit et K-SVD à des dictionnaires d'atomes constitués de motifs linéairement déformables, via une propriété d'adjonction. Nous présentons également des résultats de séparation de signaux monocanaux dans le cadre proposé.

Introduction of this thesis is dedicated to the description of basic concepts and algorithms for image processing using sparse representation. Furthermore there is mentioned neural network model called Restricted Boltzmann machine, which is in the practical part of the thesis subject of behaving observation in the task of determining the optimal block size for extrapolation using K-SVD algorithm