Wavelet and Multiscale Methods

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Combined Industry, Space and Earth Science Data Compression Workshop

The sixth annual Space and Earth Science Data Compression Workshop and the third annual Data Compression Industry Workshop were held as a single combined workshop. The workshop was held April 4, 1996 in Snowbird, Utah in conjunction with the 1996 IEEE Data Compression Conference, which was held at the same location March 31 - April 3, 1996. The Space and Earth Science Data Compression sessions seek to explore opportunities for data compression to enhance the collection, analysis, and retrieval of space and earth science data. Of particular interest is data compression research that is integrated into, or has the potential to be integrated into, a particular space or earth science data information system. Preference is given to data compression research that takes into account the scien- tist's data requirements, and the constraints imposed by the data collection, transmission, distribution and archival systems

Information Extraction and Modeling from Remote Sensing Images: Application to the Enhancement of Digital Elevation Models

To deal with high complexity data such as remote sensing images presenting metric resolution over large areas, an innovative, fast and robust image processing system is presented. The modeling of increasing level of information is used to extract, represent and link image features to semantic content. The potential of the proposed techniques is demonstrated with an application to enhance and regularize digital elevation models based on information collected from RS images

Anisotropic Harmonic Analysis and Integration of Remotely Sensed Data

This thesis develops the theory of discrete directional Gabor frames and several algorithms for the analysis of remotely sensed image data, based on constructions of harmonic analysis. The problems of image registration, image superresolution, and image fusion are separate but interconnected; a general approach using transform methods is the focus of this thesis. The methods of geometric multiresolution analysis are explored, particularly those related to the shearlet transform. Using shearlets, a novel method of image registration is developed that aligns images based on their shearlet features. Additionally, the anisotropic nature of the shearlet transform is deployed to smoothly superrsolve remotely-sensed image with edge features. Wavelet packets, a generalization of wavelets, are utilized for a flexible image fusion algorithm. The interplay between theoretical guarantees for these mathematical constructions, and their effectiveness for image processing is explored throughout

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Directional Haar Wavelet Systems in Image Processing

In der Dissertation werden Waveletmethoden betrachtet, die in der Lage sind, verschiedene Richtungen in digitalen Bildern zu erkennen. Die dabei verwendeten Haarwavelets sind stückweise konstante Funktionen mit kompaktem Träger im Zeitbereich. Zunächst konstruieren wir ein nicht-adaptives Waveletsystem, das aus gerichteten Haarwavelets auf Dreiecken besteht. Wir untersuchen Dreieckszerlegungen der reellen Ebene mit vier Richtungen, das daraus resultierende Waveletsystem generiert eine Orthonormalbasis des $L^2({\mathbb R}^2)$. Im nächsten Schritt erweitern wir den Ansatz, indem wir vier weitere Richtungen zulassen. Das nun entstehende Waveletsystem verliert seine Basiseigenschaft, wir erhalten einen Frame mit vierfacher Redundanz. Schließlich konstruieren wir ein noch allgemeineres Haarwaveletsystem, das aufgrund einer geeigneten Dreieckszerlegung beliebig viele Richtungen aufweist. Nachdem wir für alle Methoden den zugehörigen Filterbank-Algorithmus ausführlich beschrieben haben, wenden wir ihn zur Entstörung und zur Approximation von Bildern an. Im zweiten Teil der Arbeit führen wir eine neue adaptive Wavelettransformation ein, die sogenannte Tetrolet-Transformation. Die verwendeten Haarwavelets basieren auf einer adaptiven Tetromino-Zerlegung des diskreten Bildraumes und bilden eine Orthonormalbasis. Auch hier beschreiben wir den Filterbank-Algorithmus und illustrieren an vielen Beispielen die enorme Effizienz der Methode bei der Approximation von Bildern. Wir untersuchen die numerische Komplexität und die zusätzlichen Adaptivitätskosten, was zu kostenreduzierenden Modifikationen der Transformation führt. Das letzte Kapitel ist einer Postprocessing-Methode gewidmet, die die Schwächen von gerichteten Haarwavelet-Transformationen beheben soll. Durch die Verwendung von Haarwavelets liefern unsere Methoden eine stückweise konstante Approximation der Bilder. Durch Anwenden eines angepassten Postprocessing-Schemas können wir die Regularität des approximierten Bildes nachträglich erhöhen, ohne dabei die Bildkanten zu verwischen. Unsere Methode verwendet dazu eine anisotrope Totale-Variations-Minimierung. Die guten Ergebnisse in der Anwendung bestätigen die theoretischen Resultate, dass die Approximationsqualität der Bilder bei geeigneter Wahl der Parameter des Postprocessing-Verfahrens verbessert werden kann.We consider wavelet methods that are able to detect different directions in digital images. The constructed Haar wavelets are piecewise constant functions with compact support in the space domain. Firstly, we construct a non-adaptive wavelet system of directional Haar wavelets on triangles. We partition the real plane into triangles with four directions, the generated wavelet system forms an orthonormal basis of $L^2({\mathbb R}^2)$. In a next step we generalize this approach adding four further directions. The wavelet system looses its basis property, and we get a frame with fourfold redundancy. Finally, we construct an even more general Haar wavelet system that offers arbitrary many directions using a suitable triangle partition. After describing the corresponding filter bank algorithms for our methods we apply them to image denoising and approximation. In the second part we introduce a new adaptive wavelet transform, the so called tetrolet transform. The Haar wavelets base upon an adaptive tetromino partition of the discrete image space and form an orthonormal basis. We describe the filter bank, too, and illustrate the efficiency of the method for image approximation by many numerical examples. The analysis of the adaptivity costs leads to modified versions of the tetrolet transform. The last chapter is devoted to a postprocessing method which corrects the weakness of the directional Haar wavelet transforms. The usage of Haar wavelets in our methods leads to a piecewise constant image approximation. The application of the proposed postprocessing scheme subsequently improves the regularity of the approximated image without blurring the edges. In order to preserve edges we use an anisotropic total variation minimization. Good numerical experiments verify the theoretical results that the approximation quality can be improved by a certain choice of the parameter of the method