Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Generalized linear-in-parameter models : theory and audio signal processing applications

This thesis presents a mathematically oriented perspective to some basic concepts of digital signal processing. A general framework for the development of alternative signal and system representations is attained by defining a generalized linear-in-parameter model (GLM) configuration. The GLM provides a direct view into the origins of many familiar methods in signal processing, implying a variety of generalizations, and it serves as a natural introduction to rational orthonormal model structures. In particular, the conventional division between finite impulse response (FIR) and infinite impulse response (IIR) filtering methods is reconsidered. The latter part of the thesis consists of audio oriented case studies, including loudspeaker equalization, musical instrument body modeling, and room response modeling. The proposed collection of IIR filter design techniques is submitted to challenging modeling tasks. The most important practical contribution of this thesis is the introduction of a procedure for the optimization of rational orthonormal filter structures, called the BU-method. More generally, the BU-method and its variants, including the (complex) warped extension, the (C)WBU-method, can be consider as entirely new IIR filter design strategies.reviewe

Multispectral texture synthesis

Synthesizing texture involves the ordering of pixels in a 2D arrangement so as to display certain known spatial correlations, generally as described by a sample texture. In an abstract sense, these pixels could be gray-scale values, RGB color values, or entire spectral curves. The focus of this work is to develop a practical synthesis framework that maintains this abstract view while synthesizing texture with high spectral dimension, effectively achieving spectral invariance. The principle idea is to use a single monochrome texture synthesis step to capture the spatial information in a multispectral texture. The first step is to use a global color space transform to condense the spatial information in a sample texture into a principle luminance channel. Then, a monochrome texture synthesis step generates the corresponding principle band in the synthetic texture. This spatial information is then used to condition the generation of spectral information. A number of variants of this general approach are introduced. The first uses a multiresolution transform to decompose the spatial information in the principle band into an equivalent scale/space representation. This information is encapsulated into a set of low order statistical constraints that are used to iteratively coerce white noise into the desired texture. The residual spectral information is then generated using a non-parametric Markov Ran dom field model (MRF). The remaining variants use a non-parametric MRF to generate the spatial and spectral components simultaneously. In this ap proach, multispectral texture is grown from a seed region by sampling from the set of nearest neighbors in the sample texture as identified by a template matching procedure in the principle band. The effectiveness of both algorithms is demonstrated on a number of texture examples ranging from greyscale to RGB textures, as well as 16, 22, 32 and 63 band spectral images. In addition to the standard visual test that predominates the literature, effort is made to quantify the accuracy of the synthesis using informative and effective metrics. These include first and second order statistical comparisons as well as statistical divergence tests

Efficient compression of motion compensated residuals

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Stability and Stabilization of the Wave Model.

The stability properties of 2-D systems are an important aspect of the design of acoustic, seismic, image and sonar signal processors. This research utilizes the Wave model format to transport 1-D stability techniques to the 2-D setting. The research studies stability through multistep growth bounds on the Wave state. The use of Lyapunov theory is also considered. The research considers also the problem of stabilizing a 2-D system using state and/or output information feedback to interior and/or boundary controls. Finally the problem of observer design for 2-D systems is considered, with the new stability criteria being used to assure observer/system convergence. New results based on symmetrizability are also discussed. The principal results are illustrated by a number of examples. The results are also interpreted in the context of other contemporary local state models

Efficient algorithms for arbitrary sample rate conversion with application to wave field synthesis

Arbitrary sample rate conversion (ASRC) is used in many fields of digital signal processing to alter the sampling rate of discrete-time signals by arbitrary, potentially time-varying ratios. This thesis investigates efficient algorithms for ASRC and proposes several improvements. First, closed-form descriptions for the modified Farrow structure and Lagrange interpolators are derived that are directly applicable to algorithm design and analysis. Second, efficient implementation structures for ASRC algorithms are investigated. Third, this thesis considers coefficient design methods that are optimal for a selectable error norm and optional design constraints. Finally, the performance of different algorithms is compared for several performance metrics. This enables the selection of ASRC algorithms that meet the requirements of an application with minimal complexity. Wave field synthesis (WFS), a high-quality spatial sound reproduction technique, is the main application considered in this work. For WFS, sophisticated ASRC algorithms improve the quality of moving sound sources. However, the improvements proposed in this thesis are not limited to WFS, but applicable to general-purpose ASRC problems.Verfahren zur unbeschränkten Abtastratenwandlung (arbitrary sample rate conversion,ASRC) ermöglichen die Änderung der Abtastrate zeitdiskreter Signale um beliebige, zeitvarianteVerhältnisse. ASRC wird in vielen Anwendungen digitaler Signalverarbeitung eingesetzt.In dieser Arbeit wird die Verwendung von ASRC-Verfahren in der Wellenfeldsynthese(WFS), einem Verfahren zur hochqualitativen, räumlich korrekten Audio-Wiedergabe, untersucht.Durch ASRC-Algorithmen kann die Wiedergabequalität bewegter Schallquellenin WFS deutlich verbessert werden. Durch die hohe Zahl der in einem WFS-Wiedergabesystembenötigten simultanen ASRC-Operationen ist eine direkte Anwendung hochwertigerAlgorithmen jedoch meist nicht möglich.Zur Lösung dieses Problems werden verschiedene Beiträge vorgestellt. Die Komplexitätder WFS-Signalverarbeitung wird durch eine geeignete Partitionierung der ASRC-Algorithmensignifikant reduziert, welche eine effiziente Wiederverwendung von Zwischenergebnissenermöglicht. Dies erlaubt den Einsatz hochqualitativer Algorithmen zur Abtastratenwandlungmit einer Komplexität, die mit der Anwendung einfacher konventioneller ASRCAlgorithmenvergleichbar ist. Dieses Partitionierungsschema stellt jedoch auch zusätzlicheAnforderungen an ASRC-Algorithmen und erfordert Abwägungen zwischen Performance-Maßen wie der algorithmischen Komplexität, Speicherbedarf oder -bandbreite.Zur Verbesserung von Algorithmen und Implementierungsstrukturen für ASRC werdenverschiedene Maßnahmen vorgeschlagen. Zum Einen werden geschlossene, analytischeBeschreibungen für den kontinuierlichen Frequenzgang verschiedener Klassen von ASRCStruktureneingeführt. Insbesondere für Lagrange-Interpolatoren, die modifizierte Farrow-Struktur sowie Kombinationen aus Überabtastung und zeitkontinuierlichen Resampling-Funktionen werden kompakte Darstellungen hergeleitet, die sowohl Aufschluss über dasVerhalten dieser Filter geben als auch eine direkte Verwendung in Design-Methoden ermöglichen.Einen zweiten Schwerpunkt bildet das Koeffizientendesign für diese Strukturen, insbesonderezum optimalen Entwurf bezüglich einer gewählten Fehlernorm und optionaler Entwurfsbedingungenund -restriktionen. Im Gegensatz zu bisherigen Ansätzen werden solcheoptimalen Entwurfsmethoden auch für mehrstufige ASRC-Strukturen, welche ganzzahligeÜberabtastung mit zeitkontinuierlichen Resampling-Funktionen verbinden, vorgestellt.Für diese Klasse von Strukturen wird eine Reihe angepasster Resampling-Funktionen vorgeschlagen,welche in Verbindung mit den entwickelten optimalen Entwurfsmethoden signifikanteQualitätssteigerungen ermöglichen.Die Vielzahl von ASRC-Strukturen sowie deren Design-Parameter bildet eine Hauptschwierigkeitbei der Auswahl eines für eine gegebene Anwendung geeigneten Verfahrens.Evaluation und Performance-Vergleiche bilden daher einen dritten Schwerpunkt. Dazu wirdzum Einen der Einfluss verschiedener Entwurfsparameter auf die erzielbare Qualität vonASRC-Algorithmen untersucht. Zum Anderen wird der benötigte Aufwand bezüglich verschiedenerPerformance-Metriken in Abhängigkeit von Design-Qualität dargestellt.Auf diese Weise sind die Ergebnisse dieser Arbeit nicht auf WFS beschränkt, sondernsind in einer Vielzahl von Anwendungen unbeschränkter Abtastratenwandlung nutzbar

Relationships between digital signal processing and control and estimation theory

Bibliography: leaves 83-97.NASA Grant NGL-22-009-124 and NSF Grant GK-41647.Alan S. Willsky

Least-Squares Image Resizing Using Finite Differences

We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree n. A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a. For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio. The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives