Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets

Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets

by Ali N. Akansu

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The uniqueness of this book is that it covers such important aspects of modern signal processing as block transforms from subband filter banks and wavelet transforms from a common unifying standpoint, thus demonstrating the commonality among these decomposition techniques. In addition, it covers such "hot" areas as signal compression and coding, including particular…  See more details below


The uniqueness of this book is that it covers such important aspects of modern signal processing as block transforms from subband filter banks and wavelet transforms from a common unifying standpoint, thus demonstrating the commonality among these decomposition techniques. In addition, it covers such "hot" areas as signal compression and coding, including particular decomposition techniques and tables listing coefficients of subband and wavelet filters and other important properties.
The field of this book (Electrical Engineering/Computer Science) is currently booming, which is, of course, evident from the sales of the previous edition. Since the first edition came out there has been much development, especially as far as the applications. Thus, the second edition addresses new developments in applications-related chapters, especially in chapter 4 "Filterbrook Families: Design and Performance," which is greatly expanded.

* Unified and coherent treatment of orthogonal transforms, subbands, and wavelets
* Coverage of emerging applications of orthogonal transforms in digital communications and multimedia
* Duality between analysis and synthesis filter banks for spectral decomposition and synthesis and analysis transmultiplexer structures
* Time-frequency focus on orthogonal decomposition techniques with applications to FDMA, TDMA, and CDMA

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Editorial Reviews

This second edition describes objective performance criteria to measure signal compression and coding and the time-frequency properties of signals and decompositions techniques, with an overall focus on orthonormal decompositions. The chapters cover orthogonal transforms; the theory of perfect reconstruction, orthonormal two-band and M-band filter; specific filter banks and their objective performance; joint time-frequency properties of signals and the localization features of decomposition tools; basic theory of orthonormal and biorthogonal wavelet transfers and their connection to the orthonormal dyadic subband tree; and recent applications in image coding and communications applications. Graduate students and researchers who know linear system theory and Fourier analysis and are acquainted with linear algebra, random signals and processes and digital signal processing will be the text's audience. Annotation c. Book News, Inc., Portland, OR (

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Chapter 1: Introduction

1.1 Introduction

In the first edition of this book, published in 1992, we stated our goals as threefold:

(1) To present orthonormal signal decomposition techniques-transforms, subbands, and wavelets-from a unified framework and point of view.
(2) To develop the interrelationships among decomposition methods in both time and frequency domains and to define common features.
(3) To evaluate and critique proposed decomposition strategies from a compression coding standpoint using measures appropriate to image processing.

The emphasis then was signal coding in an analysis/synthesis structure or codec. As the field matured and new insights were gained, we expanded our vistas to communications systems and other applications where objectives other than compression are vital - as for example, interference excision in CDMA spread spectrum systems. We can also represent certain communications systems such as TDMA, FDMA, and CDMA as synthesis/ analysis structures, i.e., the conceptual dual of the compression codec. This duality enables one to view all these systems from one unified framework.

The Fourier transform and its extensions have historically been the prime vehicle for signal analysis and representation. Since the early 1970s, block transforms with real basis functions, particularly the discrete cosine transform (DCT), have been studied extensively for transform coding applications. The availability of simple fast transform algorithms and good signal coding performance made the DCT the standard signal decomposition technique, particularly for image and video. The international standard image-video coding algorithms, i.e., CCITTH.261, JPEG, and MPEG, all employ DCT-based transform coding.

Since the recent research activities in signal decomposition are basically driven by visual signal processing and coding applications, the properties of the human visual system (HVS) are examined and incorporated in the signal decomposition step. It has been reported that the HVS inherently performs multiresolution signal processing. This finding triggered significant interest in multiresolution signal decomposition and its mathematical foundations in multirate signal processing theory. The multiresolution signal analysis concept also fits a wide spectrum of visual signal processing and visual communications applications. Lower, i.e., coarser, resolution versions of an image frame or video sequence are often sufficient in many instances. Progressive improvement of the signal quality in visual applications, from coarse to finer resolution, has many uses in computer vision, visual communications, and related fields.

The recognition that multiresolution signal decomposition is a by-product of multirate subband filter banks generated significant interest in the design of better performing filter banks for visual signal processing applications.

The wavelet transform with a capability for variable time-frequency resolution has been promoted as an elegant multiresolution signal processing tool. It was shown that this decomposition technique is strongly linked to subband decomposition. This linkage stimulated additional interest in subband filter banks, since they serve as the only vehicle for fast orthonormal wavelet transform algorithms and wavelet transform basis design.

1.2 Why Signal Decomposition?

The uneven distribution of signal energy in the frequency domain has made signal decomposition an important practical problem. Rate-distortion theory shows that the uneven spectral nature of real-world signals can provide the basis for source compression techniques. The basic concept here is to divide the signal spectrum into its subspectra or subbands, and then to treat those subspectra individually for the purpose at hand. From a signal coding standpoint, it can be appreciated that subspectra with more energy content deserve higher priority or weight for further processing. For example, a slowly varying signal will have predominantly low-frequency components. Therefore, the low-pass subbands contain most of its total energy. If one discards the high-pass analysis subbands and reconstructs the signal, it is expected that very little or negligible reconstruction error occurs after this analysis-synthesis operation.

The decomposition of the signal spectrum into subbands provides the mathematical basis for two important and desirable features in signal analysis and processing. First, the monitoring of signal energy components within the subbands or subspectra is possible. The subband signals can then be ranked and processed independently. A common use of this feature is in the spectral shaping of quantization noise in signal coding applications. By bit allocation we can allow different levels of quantization error in different subbands. Second, the subband decomposition of the signal spectrum leads naturally to multiresolution signal decomposition via multirate signal processing in accordance with the Nyquist sampling theorem.

Apart from coding/compression considerations, signal decomposition into subbands permits us to investigate the subbands for contraband signals, such as bandlimited or single tone interference. We have also learned to think more globally to the point of signal decomposition in a composite time-frequency domain, rather than in frequency subbands as such. This expansive way of thinking leads naturally to the concept of wavelet packets (subband trees), and to the block transform packets introduced in this text.

1.3 Decompositions: Transforms, Subbands, and Wavelets

The signal decomposition (and reconstruction) techniques developed in this book have three salient characteristics:

(1) Orthonormality. As we shall see, the block transforms will be square unitary matrices, i.e., the rows of the transformation matrix will be orthogonal to each other; the subband filter banks will be paraunitary, a special kind of orthonormality, and the wavelets will be orthonormal.
(2) Perfect reconstruction (PR). This means that, in the absence of encoding, quantization, and transmission errors, the reconstructed signal can be reassembled perfectly at the receiver.
(3) Critical sampling. This implies that the signal is subsampled at a minimum possible rate consistent with the applicable Nyquist theorem. From a practical standpoint, this means that if the original signal has a data rate of fs samples or pixels per second, the sum of the transmission rates out of all the subbands is also fs.

The aforementioned are the prime ingredients of the decomposition techniques. However, we also briefly present a few other decomposition methods for contrast or historical perspective. The oversampled Laplacian pyramid, biorthogonal filter banks, and non-PR filter banks are examples of these, which we introduce fordidactic value...

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