Asymptotics of wavelets and filters

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 126-131).by Jianhong (Jackie) Shen.Ph.D

Sampling systems matched to input processes and image classes

This dissertation investigates sampling and reconstruction of wide sense stationary (WSS) random processes from their sample random variables . In this context, two types of sampling systems are studied, namely, interpolation and approximation sampling systems. We aim to determine the properties of the filters in these systems that minimize the mean squared error between the input process and the process reconstructed from its samples. More specifically, for the interpolation sampling system we seek and obtain a closed form expression for an interpolation filter that is optimal in this sense. Likewise, for the approximation sampling system we derive a closed form expression for an optimal reconstruction filter given the statistics of the input process and the antialiasing filter. Using these expressions we show that Meyer-type scaling functions and wavelets arise naturally in the context of subsampled bandlimited processes. We also derive closed form expressions for the mean squared error incurred by both the sampling systems. Using the expression for mean squared error we show that for an approximation sampling system, minimum mean squared error is obtained when the antialiasing filter and the reconstruction filter are spectral factors of an ideal brickwall-type filter. Similar results are derived for the discrete-time equivalents of these sampling systems. Finally, we give examples of interpolation and approximation sampling filters and compare their performance with that of some standard filters. The implementation of these systems is based on a novel framework called the perfect reconstruction circular convolution (PRCC) filter bank framework. The results obtained for the one dimensional case are extended to the multidimensional case. Sampling a multidimensional random field or image class has a greater degree of freedom and the sampling lattice can be defined by a nonsingular matrix D. The aim is to find optimal filters in multidimensional sampling systems to reconstruct the input image class from its samples on a lattice defined by D. Closed form expressions for filters in multidimensional interpolation and approximation sampling systems are obtained as are expressions for the mean squared error incurred by each system. For the approximation sampling system it is proved that the antialiasing and reconstruction filters that minimize the mean squared error are spectral factors of an ideal brickwall-type filter whose support depends on the sampling matrix D. Finally. we give examples of filters in the interpolation and approximation sampling systems for an image class derived from a LANDSAT image and a quincunx sampling lattice. The performance of these filters is compared with that of some standard filters in the presence of a quantizer

Phase History Decomposition for Efficient Scatterer Classification in SAR Imagery

A new theory and algorithm for scatterer classification in SAR imagery is presented. The automated classification process is operationally efficient compared to existing image segmentation methods requiring human supervision. The algorithm reconstructs coarse resolution subimages from subdomains of the SAR phase history. It analyzes local peaks in the subimages to determine locations and geometric shapes of scatterers in the scene. Scatterer locations are indicated by the presence of a stable peak in all subimages for a given subaperture, while scatterer shapes are indicated by changes in pixel intensity. A new multi-peak model is developed from physical models of electromagnetic scattering to predict how pixel intensities behave for different scatterer shapes. The algorithm uses a least squares classifier to match observed pixel behavior to the model. Classification accuracy improves with increasing fractional bandwidth and is subject to the high-frequency and wide-aperture approximations of the multi-peak model. For superior computational efficiency, an integrated fast SAR imaging technique is developed to combine the coarse resolution subimages into a final SAR image having fine resolution. Finally, classification results are overlaid on the SAR image so that analysts can deduce the significance of the scatterer shape information within the image context

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

Custom optimization algorithms for efficient hardware implementation

The focus is on real-time optimal decision making with application in advanced control systems. These computationally intensive schemes, which involve the repeated solution of (convex) optimization problems within a sampling interval, require more efficient computational methods than currently available for extending their application to highly dynamical systems and setups with resource-constrained embedded computing platforms. A range of techniques are proposed to exploit synergies between digital hardware, numerical analysis and algorithm design. These techniques build on top of parameterisable hardware code generation tools that generate VHDL code describing custom computing architectures for interior-point methods and a range of first-order constrained optimization methods. Since memory limitations are often important in embedded implementations we develop a custom storage scheme for KKT matrices arising in interior-point methods for control, which reduces memory requirements significantly and prevents I/O bandwidth limitations from affecting the performance in our implementations. To take advantage of the trend towards parallel computing architectures and to exploit the special characteristics of our custom architectures we propose several high-level parallel optimal control schemes that can reduce computation time. A novel optimization formulation was devised for reducing the computational effort in solving certain problems independent of the computing platform used. In order to be able to solve optimization problems in fixed-point arithmetic, which is significantly more resource-efficient than floating-point, tailored linear algebra algorithms were developed for solving the linear systems that form the computational bottleneck in many optimization methods. These methods come with guarantees for reliable operation. We also provide finite-precision error analysis for fixed-point implementations of first-order methods that can be used to minimize the use of resources while meeting accuracy specifications. The suggested techniques are demonstrated on several practical examples, including a hardware-in-the-loop setup for optimization-based control of a large airliner.Open Acces

Pattern Recognition

Pattern recognition is a very wide research field. It involves factors as diverse as sensors, feature extraction, pattern classification, decision fusion, applications and others. The signals processed are commonly one, two or three dimensional, the processing is done in real- time or takes hours and days, some systems look for one narrow object class, others search huge databases for entries with at least a small amount of similarity. No single person can claim expertise across the whole field, which develops rapidly, updates its paradigms and comprehends several philosophical approaches. This book reflects this diversity by presenting a selection of recent developments within the area of pattern recognition and related fields. It covers theoretical advances in classification and feature extraction as well as application-oriented works. Authors of these 25 works present and advocate recent achievements of their research related to the field of pattern recognition

Wavelets and Subband Coding

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book

On the automatic detection of otolith features for fish species identification and their age estimation

This thesis deals with the automatic detection of features in signals, either extracted from photographs or captured by means of electronic sensors, and its possible application in the detection of morphological structures in fish otoliths so as to identify species and estimate their age at death. From a more biological perspective, otoliths, which are calcified structures located in the auditory system of all teleostean fish, constitute one of the main elements employed in the study and management of marine ecology. In this sense, the application of Fourier descriptors to otolith images, combined with component analysis, is habitually a first and a key step towards characterizing their morphology and identifying fish species. However, some of the main limitations arise from the poor interpretation that can be obtained with this representation and the use that is made of the coefficients, as generally they are selected manually for classification purposes, both in quantity and representativity. The automatic detection of irregularities in signals, and their interpretation, was first addressed in the so-called Best-Basis paradigm. In this sense, Saito's Local discriminant Bases algorithm (LDB) uses the Discrete Wavelet Packet Transform (DWPT) as the main descriptive tool for positioning the irregularities in the time-frequency space, and an energy-based discriminant measure to guide the automatic search of relevant features in this domain. Current density-based proposals have tried to overcome the limitations of the energy-based functions with relatively little success. However, other measure strategies more consistent with the true classification capability, and which can provide generalization while reducing the dimensionality of features, are yet to be developed. The proposal of this work focuses on a new framework for one-dimensional signals. An important conclusion extracted therein is that such generalization involves a mesure system of bounded values representing the density where no class overlaps. This determines severely the selection of features and the vector size that is needed for proper class identification, which must be implemented not only based on global discriminant values but also on the complementary information regarding the provision of samples in the domain. The new tools have been used in the biological study of different hake species, yielding good classification results. However, a major contribution lies on the further interpretation of features the tool performs, including the structure of irregularities, time-frequency position, extension support and degree of importance, which is highlighted automatically on the same images or signals. As for aging applications, a new demodulation strategy for compensating the nonlinear growth effect on the intensity profile has been developed. Although the method is, in principle, able to adapt automatically to the specific growth of individual specimens, preliminary results with LDB-based techniques suggest to study the effect of lighting conditions on the otoliths in order to design more reliable techniques for reducing image contrast variation. In the meantime, a new theoretic framework for otolith-based fish age estimation has been presented. This theory suggests that if the true fish growth curve is known, the regular periodicity of age structures in the demodulated profile is related to the radial length the original intensity profile is extracted from. Therefore, if this periodicity can be measured, it is possible to infer the exact fish age omitting feature extractors and classifiers. This could have important implications in the use of computational resources anc current aging approaches.El eje principal de esta tesis trata sobre la detección automática de singularidades en señales, tanto si se extraen de imágenes fotográ cas como si se capturan de sensores electrónicos, así como su posible aplicación en la detección de estructuras morfológicas en otolitos de peces para identi car especies, y realizar una estimación de la edad en el momento de su muerte. Desde una vertiente más biológica, los otolitos, que son estructuras calcáreas alojadas en el sistema auditivo de todos los peces teleósteos, constituyen uno de los elementos principales en el estudio y la gestión de la ecología marina. En este sentido, el uso combinado de descriptores de Fourier y el análisis de componentes es el primer paso y la clave para caracterizar su morfología e identi car especies marinas. Sin embargo, una de las limitaciones principales de este sistema de representación subyace en la interpretación limitada que se puede obtener de las irregularidades, así como el uso que se hace de los coe cientes en tareas de clasi cación que, por lo general, acostumbra a seleccionarse manualmente tanto por lo que respecta a la cantidad y a su importancia. La detección automática de irregularidades en señales, y su interpretación, se abordó por primera bajo el marco del Best-Basis paradigm. En este sentido, el algoritmo Local Discriminant Bases (LDB) de N. Saito utiliza la Transformada Wavelet Discreta (DWT) para describir el posicionamiento de características en el espacio tiempo-frecuencia, y una medida discriminante basada en la energía para guiar la búsqueda automática de características en dicho dominio. Propuestas recientes basadas en funciones de densidad han tratado de superar las limitaciones que presentaban las medidas de energía con un éxito relativo. No obstante, todavía están por desarrollar nuevas estrategias más consistentes con la capacidad real de clasi cación y que ofrezcan mayor generalización al reducir la dimensión de los datos de entrada. La propuesta de este trabajo se centra en un nuevo marco para señales unidimensionales. Una conclusión principal que se extrae es que dicha generalización pasa por un marco de medidas de valores acotados que re ejen la densidad donde las clases no se solapan. Esto condiciona severamente el proceso de selección de características y el tamaño del vector necesario para identi car las clases correctamente, que se ha de establecer no sólo en base a valores discriminantes globales sino también en la información complementaria sobre la disposición de las muestras en el dominio. Las nuevas herramientas han sido utilizadas en el estudio biológico de diferentes especies de merluza, donde se han conseguido buenos resultados de identi cación. No obstante, la contribución principal subyace en la interpretación que dicha herramienta hace de las características seleccionadas, y que incluye la estructura de las irregularidades, su posición temporal-frecuencial, extensión en el eje y grado de relevancia, el cual, se resalta automáticamente sobre la misma imagen o señal. Por lo que respecta a la determinación de la edad, se ha planteado una nueva estrategia de demodulación para compensar el efecto del crecimiento no lineal en los per les de intensidad. Inicialmente, aunque el método implementa un proceso de optimización capaz de adaptarse al crecimiento individual de cada pez automáticamente, resultados preliminares obtenidos con técnicas basadas en el LDB sugieren estudiar el efecto de las condiciones lumínicas sobre los otolitos con el n de diseñar algoritmos que reduzcan la variación del contraste de la imagen más ablemente. Mientras tanto, se ha planteado una nueva teoría para estimar la edad de los peces en base a otolitos. Esta teoría sugiere que si la curva de crecimiento real del pez se conoce, el período regular de los anillos en el per l demodulado está relacionado con la longitud total del radio donde se extrae el per l original. Por tanto, si dicha periodicidad es medible, es posible determinar la edad exacta sin necesidad de utilizar extractores de características o clasi cadores, lo cual tendría implicaciones importantes en el uso de recursos computacionales y en las técnicas actuales de estimación de la edad.L'eix principal d'aquesta tesi tracta sobre la detecció automàtica d'irregularitats en senyals, tant si s'extreuen de les imatges fotogrà ques com si es capturen de sensors electrònics, així com la seva possible aplicació en la detecció d'estructures morfològiques en otòlits de peixos per identi car espècies, i realitzar una estimació de l'edat en el moment de la seva mort. Des de la vesant més biològica, els otòlits, que son estructures calcàries que es troben en el sistema auditiu de tots els peixos teleostis, constitueixen un dels elements principals en l'estudi i la gestió de l'ecologia marina. En aquest sentit, l'ús combinat de descriptors de Fourier i l'anàlisi de components es el primer pas i la clau per caracteritzar la seva morfologia i identi car espècies marines. No obstant, una de les limitacions principals d'aquest sistema de representació consisteix en la interpretació limitada de les irregularitats que pot desenvolupar, així com l'ús que es realitza dels coe cients en tasques de classi cació, els quals, acostumen a ser seleccionats manualment tant pel que respecta a la quantitat com la seva importància. La detecció automàtica d'irregularitats en senyals, així com la seva interpretació, es va tractar per primera vegada sota el marc del Best-Basis paradigm. En aquest sentit, l'algorisme Local Discriminant Bases (LDB) de N. Saito es basa en la Transformada Wavelet Discreta (DWT) per descriure el posicionament de característiques dintre de l'espai temporal-freqüencial, i en una mesura discriminant basada en l'energia per guiar la cerca automàtica de característiques dintre d'aquest domini. Propostes més recents basades en funcions de densitat han tractat de superar les limitacions de les mesures d'energia amb un èxit relatiu. No obstant, encara s'han de desenvolupar noves estratègies que siguin més consistents amb la capacitat real de classi cació i ofereixin més generalització al reduir la dimensió de les dades d'entrada. La proposta d'aquest treball es centra en un nou marc per senyals unidimensionals. Una de las conclusions principals que s'extreu es que aquesta generalització passa per establir un marc de mesures acotades on els valors re ecteixin la densitat on cap classe es solapa. Això condiciona bastant el procés de selecció de característiques i la mida del vector necessari per identi car les classes correctament, que s'han d'establir no només en base a valors discriminants globals si no també en informació complementària sobre la disposició de les mostres en el domini. Les noves eines s'han utilitzat en diferents estudis d'espècies de lluç, on s'han obtingut bons resultats d'identi cació. No obstant, l'aportació principal consisteix en la interpretació que l'eina extreu de les característiques seleccionades, i que inclou l'estructura de les irregularitats, la seva posició temporal-freqüencial, extensió en l'eix i grau de rellevància, el qual, es ressalta automàticament sobre les mateixa imatge o senyal. En quan a l'àmbit de determinació de l'edat, s'ha plantejat una nova estratègia de demodulació de senyals per compensar l'efecte del creixement no lineal en els per ls d'intensitat. Tot i que inicialment aquesta tècnica desenvolupa un procés d'optimització capaç d'adaptar-se automàticament al creixement individual de cada peix, els resultats amb el LDB suggereixen estudiar l'efecte de les condicions lumíniques sobre els otòlits amb la nalitat de dissenyar algorismes que redueixin la variació del contrast de les imatges més ablement. Mentrestant s'ha plantejat una nova teoria per realitzar estimacions d'edat en peixos en base als otòlits. Aquesta teoria suggereix que si la corba de creixement és coneguda, el període regular dels anells en el per l d'intensitat demodulat està relacionat amb la longitud total de radi d'on s'agafa el per l original. Per tant, si la periodicitat es pot mesurar, es possible conèixer l'edat exacta del peix sense usar extractors de característiques o classi cadors, la qual cosa tindria implicacions importants en l'ús de recursos computacionals i en les tècniques actuals d'estimació de l'edat.Postprint (published version

Dynamical Properties of Quasi One-Dimensional Correlated Electron Systems

Correlated electrons in one spatial dimension have very unusual properties such as the dynamical separation of spin and charge degrees of freedom. Typically, dynamical correlation functions of these systems are investigated with field theoretical methods that are valid in the limit of vanishingly low energies. However, there are few reliable results at finite energies which can be directly compared with spectra of scattering experiments, such as angle-resolved photoemission. It has therefore been difficult to find direct spectroscopic evidence of spin-charge separation in experimental realizations of quasi one-dimensional electron systems. The Dynamical Density-Matrix Renormalization Group can accurately determine spectral properties of correlated one-dimensional lattice models for all energy scales and interaction strengths. We use this method to determine the frequency- and momentum dependent correlation functions of extended Hubbard Hamiltonians. We obtain results for various quantities including the optical conductivity, the one-particle spectral function, dynamical charge- and spin structure factors