Adaptive lifting schemes with perfect reconstruction

In this paper, we propose a framework for constructing adaptive wavelet decompositions using the lifting scheme. A major requirement is that perfect reconstruction is possible without any overhead cost. In this paper we restrict ourselves to the update lifting stage. It is assumed that the update filter utilises local gradient information to adapt itself to the signal in the sense that smaller gradients `evoke' stronger update filters. As a result, sharp transitions in a signal will not be smoothed to the same extent as regions which are more homogeneous. The approach taken in this paper differs from other adaptive schemes found in the literature in the sense that that no bookkeeping is required in order to have perfect reconstruction

Lifting transforms on graphs and their application to video coding

Compact representations of data are very useful in many applications such as coding, denoising or feature extraction. “Classical” transforms such as Discrete Cosine Transforms (DCT) or Discrete Wavelets Transforms (DWT) provide sparse approximations of smooth signals, but lose efficiency when they are applied to signals with large discontinuities. In such cases, directional transforms, which are able to adapt their basis functions to the underlying signal structure, improve the performance of “classical” transforms. In this PhD Thesis we describe a general class of lifting transforms on graphs that can be seen as N-dimensional directional transforms. Graphs are constructed so that every node corresponds to a specific sample point of a discrete N-dimensional signal and links between nodes represent correlation between samples. Therefore, non-correlated samples (e.g., samples across a large discontinuity in the signal) should not be linked. We propose a lifting-based directional transform that can be applied to any undirected graph. In this transform, filtering operations are performed following highcorrelation directions (indicated by the links between nodes), thus avoiding filtering across large discontinuities that give rise to large high-pass coefficients in those locations. In this way, the transform efficiently exploits the correlation that exists between data on the graph, leading to a more compact representation. We mainly focus on the design and optimization of these lifting transforms on graphs, studying and discussing the three main steps required to obtain an invertible and critically sampled transform: (i) graph construction, (ii) design of “good” graph bipartitions, and (iii) filter design. We also explain how to extend the transform to J levels of decomposition, obtaining a multiresolution analysis of the original N-dimensional signal. The proposed transform has many desirable properties, such as perfect reconstruction, critically-sampled, easy generalization to N-dimensional domains, non-separable and one-dimensional filtering operations, localization in frequency and in the original domain, and the ability to choose any filtering direction. As an application, we develop a graph-based video encoder where the goal is to obtain a compact representation of the original video sequence. To this end, we first propose a graph-representation of the video sequence and then design a 3-dimensional (spatio-temporal) non-separable directional transform. This can be viewed as an extension of wavelet transform-based video encoders that operate in the spatial and in the temporal domains independently. Our transform yields better compaction ability (in terms of non-linear approximation) than a state of the art motion-compensated temporal filtering transform (which can be interpreted as a temporal wavelet transform) and a comparable hybrid Discrete Cosine Transform (DCT)-based video encoder (which is the basis of the latest video coding standards). In order to obtain a complete video encoder, the transform coefficients and the side information (needed to obtain an invertible scheme) should be entropy coded and sent to the decoder. Therefore, we also propose a coefficient-reordering method based on the information of the graph which allows to improve the compression ability of the entropy encoder. Furthermore, we design two different low-cost approaches which aim to reduce the extensive computational complexity of the proposed system without causing significant losses of compression performance. The proposed complete system leads to an efficient encoder which significantly outperforms a comparable hybrid DCT-based encoder in rate-distortion terms. Finally, we investigate how rate-distortion optimization can be applied to the proposed coding scheme.La representación compacta de señales resulta útil en diversas aplicaciones, tales como compresión, reducción de ruido, o extracción de características. Transformadas “clásicas” como la Transformada Discreta del Coseno (DCT) o la TransformadaWavelet Discreta (DWT) logran aproximaciones compactas de señales suaves, pero pierden su eficiencia al ser aplicadas sobre se˜nales que contienen grandes discontinuidades. En estos casos, las transformadas direccionales, capaces de adaptar sus funciones base a la estructura de la señal a analizar, mejoran la eficiencia de las transformadas “clásicas”. En esta tesis nos centramos en el diseño y optimización de transformadas “lifting” sobre grafos, las cuales pueden ser interpretadas como transformadas direccionales N-dimensionales. Los grafos son construidos demanera que cada nodo se corresponde con una muestra específica de una señal discreta N-dimensional, y los enlaces entre los nodos representan correlación entre muestras. Así, muestras no correlacionadas (por ejemplo, muestras que se encuentran a ambos lados de una discontinuidad) no deberían estar unidas. Sobre el grafo formado aplicaremos transformadas basadas en el esquema “lifting”, en las que las operaciones de filtrado se realizan siguiendo las direcciones indicadas por los enlaces entre nodos (direcciones de alta correlación). De esta manera, evitaremos filtrar cruzando a través de largas discontinuidades (lo que resultaría en coeficientes con alto valor en dichas discontinuidades), dando lugar a una transformada direccional que explota la correlación que existe entre las muestras de la señal en el grafo, obteniendo una representación compacta de dicha señal. En esta tesis nos centramos, principalmente, en investigar los tres principales pasos requeridos para obtener una transformada direccional basada en el esquema “lifting” aplicado en grafos: (i) la construcción del grafo, (ii) el diseño de biparticiones del grafo, y (iii) la definición de los filtros. El buen diseño de estos tres procesos determinará, entre otras cosas, la capacidad para compactar la energía de la transformada. También explicamos cómo extender este tipo de transformadas a J niveles de descomposición, obteniendo un análisis multi-resolución de la señal N-dimensional original. La transformada propuesta tiene muchas propiedades deseables, tales como reconstrucción perfecta, muestreo crítico, fácil generalización a dominios N-dimensionales, operaciones de filtrado no separables y unidimensionales, localización en frecuencia y en el dominio original, y capacidad de elegir cualquier dirección de filtrado. Como aplicación, desarrollamos un codificador de vídeo basado en grafos donde el objetivo es obtener una versión compacta de la señal de vídeo original. Para ello, primero proponemos una representación en grafos de la secuencia de vídeo y luego diseñamos transformadas no separables direccionales 3-dimensionales (espacio-tiempo). Nuestro codificador puede interpretarse como una extensión de los codificadores de vídeo basados en “wavelets”, los cuales operan independientemente (de forma separable) en el dominio espacial y en el temporal. La transformada propuesta consigue mejores resultados (en términos de aproximación no lineal) que un método del estado del arte basado en “wavelets” temporales compensadas en movimiento, y un codificador DCT comparable (base de los últimos estándares de codificación de vídeo). Para conseguir un codificador de vídeo completo, los coeficientes resultantes de la transformada y la información secundaria (necesaria para obtener un esquema invertible) deben ser codificados entrópicamente y enviados al decodificador. Por ello, también proponemos en esta tesis un método de reordenación de los coeficientes basado en la información del grafo que permite mejorar la capacidad de compresión del codificador entrópico. El esquema de codificación propuesto mejora significativamente la eficiencia de un codificador híbrido basado en DCT en términos de tasa-distorsión. Sin embargo, nuestro método tiene la desventaja de su gran complejidad computacional. Para tratar de paliar este problema, diseñamos dos algoritmos que tratan de reducir dicha complejidad sin que ello afecte en la capacidad de compresión. Finalmente, investigamos como realizar optimización tasa-distorsión sobre el codificador basado en grafos propuesto

MASCOT : metadata for advanced scalable video coding tools : final report

The goal of the MASCOT project was to develop new video coding schemes and tools that provide both an increased coding efficiency as well as extended scalability features compared to technology that was available at the beginning of the project. Towards that goal the following tools would be used: - metadata-based coding tools; - new spatiotemporal decompositions; - new prediction schemes. Although the initial goal was to develop one single codec architecture that was able to combine all new coding tools that were foreseen when the project was formulated, it became clear that this would limit the selection of the new tools. Therefore the consortium decided to develop two codec frameworks within the project, a standard hybrid DCT-based codec and a 3D wavelet-based codec, which together are able to accommodate all tools developed during the course of the project

PHM survey: implementation of signal processing methods for monitoring bearings and gearboxes

The reliability and safety of industrial equipments are one of the main objectives of companies to remain competitive in sectors that are more and more exigent in terms of cost and security. Thus, an unexpected shutdown can lead to physical injury as well as economic consequences. This paper aims to show the emergence of the Prognostics and Health Management (PHM) concept in the industry and to describe how it comes to complement the different maintenance strategies. It describes the benefits to be expected by the implementation of signal processing, diagnostic and prognostic methods in health-monitoring. More specifically, this paper provides a state of the art of existing signal processing techniques that can be used in the PHM strategy. This paper allows showing the diversity of possible techniques and choosing among them the one that will define a framework for industrials to monitor sensitive components like bearings and gearboxes

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

SUBDIVIDE AND CONQUER RESOLUTION

This contribution will be freewheeling in the domain of signal, image and surface processing and touch briefly upon some topics that have been close to the heart of people in our research group. A lot of the research of the last 20 years in this domain that has been carried out world wide is dealing with multiresolution. Multiresolution allows to represent a function (in the broadest sense) at different levels of detail. This was not only applied in signals and images but also when solving all kinds of complex numerical problems. Since wavelets came into play in the 1980's, this idea was applied and generalized by many researchers. Therefore we use this as the central idea throughout this text. Wavelets, subdivision and hierarchical bases are the appropriate tools to obtain these multiresolution effects. We shall introduce some of the concepts in a rather informal way and show that the same concepts will work in one, two and three dimensions. The applications in the three cases are however quite different, and thus one wants to achieve very different goals when dealing with signals, images or surfaces. Because completeness in our treatment is impossible, we have chosen to describe two case studies after introducing some concepts in signal processing. These case studies are still the subject of current research. The first one attempts to solve a problem in image processing: how to approximate an edge in an image efficiently by subdivision. The method is based on normal offsets. The second case is the use of Powell-Sabin splines to give a smooth multiresolution representation of a surface. In this context we also illustrate the general method of construction of a spline wavelet basis using a lifting scheme

Discrete Wavelet Transforms

The discrete wavelet transform (DWT) algorithms have a firm position in processing of signals in several areas of research and industry. As DWT provides both octave-scale frequency and spatial timing of the analyzed signal, it is constantly used to solve and treat more and more advanced problems. The present book: Discrete Wavelet Transforms: Algorithms and Applications reviews the recent progress in discrete wavelet transform algorithms and applications. The book covers a wide range of methods (e.g. lifting, shift invariance, multi-scale analysis) for constructing DWTs. The book chapters are organized into four major parts. Part I describes the progress in hardware implementations of the DWT algorithms. Applications include multitone modulation for ADSL and equalization techniques, a scalable architecture for FPGA-implementation, lifting based algorithm for VLSI implementation, comparison between DWT and FFT based OFDM and modified SPIHT codec. Part II addresses image processing algorithms such as multiresolution approach for edge detection, low bit rate image compression, low complexity implementation of CQF wavelets and compression of multi-component images. Part III focuses watermaking DWT algorithms. Finally, Part IV describes shift invariant DWTs, DC lossless property, DWT based analysis and estimation of colored noise and an application of the wavelet Galerkin method. The chapters of the present book consist of both tutorial and highly advanced material. Therefore, the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications

The confluence of Gaussian process emulation and wavelets

We discuss two thriving research areas, emulation (in the statistical sense) and wavelet analysis, and explore ways in which the two areas can complement each other to tackle problems that both areas face. The Gaussian process, which is the popular choice in emulation, is used due to its ability to be a surrogate for a function when we are only able to make a limited number of observations from the function. The Gaussian process, however, does not perform well when the underlying function contains a discontinuity. Wavelet analysis, on the other hand, is known for its ability to model and analyse functions that contain discontinuities. Wavelet analysis tends to require a large number of datapoints to be able to model functions accurately, tending to struggle when the amount of data is limited. As it appears that one area’s strength is the other area’s weakness, this thesis is aimed at exploring the possible overlaps between the two methods, and the ways in which they could benefit each other. Particular attention in the thesis is paid to the challenges that are faced when the function that we are attempting to model contains discontinuities, or, areas of space in which there is a sharp increase/decrease in the value of our observations. We develop methods to select the location of additional design points after we have observed the function at our original design points with the objective of better defining the location of the discontinuity. We also develop novel methods to model the unknown function that we believe contains discontinuities, and look to accurately find our uncertainty in this function