Denoising with patch-based principal component analysis

One important task in image processing is noise reduction, which requires to recover image information by removing noise without loss of local structures. In recent decades patch-based denoising techniques proved to have a better performance than pixel-based ones, since a spatial neighbourhood can represent high correlations between nearby pixels and improve the results of similarity measurements. This bachelor thesis deals with denoising strategies with patch-based principal component analysis. The main focus lies on learning a new basis on which the representation of an image has the best denoising effect. The first attempt is to perform principal component analysis on a global scale, which obtains a basis that reflects the major variance of an image. The second attempt is to learn bases respectively over patches in a local window, so that more image details can be preserved. In addition, local pixel grouping is introduced to find similar patches in a local window. Due to the importance of sufficient samples in the principal component analysis transform, the third attempt is to search for more similar patches in the whole image by using a vantage point tree for space partitioning. In the part of implementation, parameter selection and time complexity are discussed. The denoising performance of different approaches is evaluated in terms of both PSNR value and visual quality.Eine der wichtigen Aufgaben in der Bildverarbeitung ist die Entrauschung, die erfordert Bildinformationen ohne Verlust lokaler Strukturen wiederzuherstellen. In den letzten Jahrzehnten hat es sich herausgestellt, dass Patch-basierte Verfahren eine bessere Leistung bei der Bildentrauschung haben als Pixel-basierte Verfahren. Der Grund liegt darin, dass eine räumliche Nachbarschaft die Korrelationen zwischen benachbarten Pixels repräsentiert und die Ergebnisse des Ähnlichkeitsmaß verbessern. In dieser Bachelorarbeit geht es um Entrauschungsstrategien mit der Patch-basierten Hauptkomponentenanalyse. Der Schwerpunkt liegt im Lernen einer neuen Basis, auf welcher die Representation eines Bildes den besten Entrauschungseffekt hat. Der erste Versuch ist, die Hauptkomponentenanalyse global durchzuführen und eine Basis zu erhalten, welche die Hauptvarianz eines Bildes reflektiert. Der zweite Versuch ist, mehrere Basen jeweils über Patches in einem lokalen Fenster zu lernen, um mehr Details zu behalten. Außerdem wird Local Pixel Grouping benutzt um ähnliche Patches in einem lokalen Fenster zu suchen. Die Hauptkomponentenanalyse ist wichtig dass genügend Samples vorhanden sind, daher werden im dritten Versuch weitere ähnliche Patches innerhalb des ganzen Bildes mithilfe von einem Vantage Point Baum gesucht. Im Teil der Implementierung wird über die Auswahl der Parameter und die Zeitkomplexität diskutiert. Die Entrauschungsleistung von unterschiedlichen Verfahren wird nach dem PSNR-Wert und der visuellen Qualität evaluiert

A study on Diophantine equations via cluster theory

In this paper, we mainly answer a Lampe's question\cite{lampe} about the solutions of a Diophantine equation, that is, we give a discriminant to determine which solutions of the Diophantine equation are in the orbit of the initial solution $(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon)$ under the actions of the group $G$ which defined by mutations of a cluster algebra. In order to this, using a rational map $\varphi$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $\varphi(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon) = (3,4,4)$ under the actions of the group $\widetilde{G}$, and the set $S(3,4,4)$ is shown to be the orbit of $(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon)$ under the actions of a subgroup of $G$. Then the discriminant is proved as the main conclusion.Comment: 18 pages, 5 figure

Magnetic Frustration and Iron-Vacancy Ordering in Iron-Chalcogenide

We show that the magnetic and vacancy orders in the 122 $(A_{1-y}Fe_{2-x}Se_2)$ iron-chalcogenides can be naturally derived from the $J_1-J_2-J_3$ model with $J_1$ being the ferromagnetic (FM) nearest neighbor exchange coupling and $J_{2}, J_3$ being the antiferromagnetic (AFM) next and third nearest neighbor ones respectively, previously proposed to describe the magnetism in the 11(FeTe/Se) systems. In the 11 systems, the magnetic exchange couplings are extremely frustrated in the ordered bi-collinear antiferromagnetic state so that the magnetic transition temperature is low. In the 122 systems, the formation of iron vacancy order reduces the magnetic frustration and significantly increases the magnetic transition temperature and the ordered magnetic moment. The pattern of the 245 iron-vacancy order ($\sqrt{5}\times \sqrt{5}$) observed in experiments is correlated to the maximum reduction of magnetic frustration. The nature of the iron-vacancy ordering may hence be electronically driven. We explore other possible vacancy patterns and magnetic orders associated with them. We also calculate the spin wave excitations and their novel features to test our model.Comment: Figures are modified and more discussion is adde