Preface Wavelets are a generic name for a collection of self similar localized waveforms suitable for signal and image processing The first set of such functions that constituted an orthonormal basis for L^(R) was introduced in 1910 by Haar However the Haar functions not have good localization in the combined time-frequency space and, therefore, in many cases not satisfy the properties required in signal and image processing and analysis The problem of how to construct functions that are well localized in both time and frequency was confronted by communication engineers dealing with the analysis of speech in the 1920s and 1930s About half a century ago Gabor introduced the optimally localized function, obtained by windowing a complex exponential with a Gaussian window The main advantage of this localized waveform is in achieving the lowest bound on the joint entropy, defined as the product of effective temporal or spatial extent and frequency bandwidth However, the Gabor elementary functions, which span L^(R), are not orthogonal The subject of representation in combined spaces refers to wavelettype and Gabor-type expansions Such expansions are more suitable for the analysis and processing of natural signals and images than expansion by the traditional application of Fourier series, polynomials, and other functions of infinite support, since the nonstationarity of natural signals calls for localization in both time (or spatial variables in the case of images) and frequency (or scale) in their representation While global transforms such as the Fourier transform, which is the most widely used in engineering, describe the spectrum of the entire signal as a whole, the wavelet-type and Gabor-type transforms allow for extraction of the local signatures of the signal as they vary in time, or along the spatial coordinates in the case of images By correlating signals with appropriately chosen wavelets, certain analysis tasks such as feature extraction, signal compression, and recognition can be facilitated The ability of wavelets to localize signals in time, or spatial variables in the case of images, allows for a multiresolution approach in signal processing In fact, since the wavelet transform is defined by either its basic time-scale, position-scale, or decomposition structure, ix X Preface it naturally lends itself to multiresolution analysis Yet, a great deal of freedom is left for the exact choice of the transform's kernel and various parameters Thus, the wavelet approach provides us with a wide range of powerful tools for signal processing and analysis These are described in this volume The general interrelated topics involving multiscale analysis, wavelet and Gabor analysis, can all be viewed as enhancing the traditional Fourier analysis by enabling an adaptation of combined time and frequency localization procedures to various tasks The simple and basic transition from the global Fourier transform to the localized (windowed) Fourier analysis, consists of segmenting the signal into windows of fixed length, each of which is expanded by a Fast Fourier Transform (FFT) or Discrete Cosine Transform (DCT) This type of procedure corresponds to spectrograms, to Gabor transform, as well as to localized trigonometric transforms A dual version of this procedure corresponds to filtering the signal, or windowing its Fourier transform, usually referred to as wavelet, wavelet packets, or subband coding transforms Wavelet analysis and more generally adapted waveform analysis has provided a simple comprehensive mathematical and algorithmic infrastructure for the localized signal processing tools, as well as many new tools which evolved as a result of the cross-fertilization of ideas originated in many fields, such as the Calderon-Zygmund theory in mathematics, multiscale ideas from geophysical seismic prospecting, mathematical physics of coherent states and wave packets, pyramid structures in image processing, band and subband filtering in signal processing, music, numerical analysis, etc In this volume we don't intend to elaborate on the origin of these ideas, but rather on the current state of this elaborate toolkit and the relative advantages it brings to the scene While to some extent most of the qualitative analytical aspects of wavelet analysis, and of the windowed Fourier transform, have been well understood by mathematicians for at least 30 years, the recent explosion of activity and algorithms is due to the discovery of the orthogonal wavelets by Stromberg and Meyer, and the connection to Quadrature Mirror Filter (QMF) by Mallat and Daubechies More fundamental yet is our better understanding of structures permitting construction of a multitude of orthogonal and nonorthogonal expansions customized to tasks at hand, and enabling the introduction of fast computational methods and realtime processing The role and usefulness of redundancy in providing stability in signal representation, as opposed to efficiency, has also become clear by means of the application of frame analysis and the Zak transform Some of the main tasks that can be accomplished by the application of wavelet-based tools are related to feature extraction and efficient description of large data sets for processing and computations This is the Preface xi point where, instead of using algebraic or analytic formulas, functions or measured data are described efficiently by adapted waveforms which are, in turn, described algorithmically and designed specifically to optimize various tasks Perhaps the most natural analogy to the new modes of analysis (or signal transcription) is provided by musical scores and orchestration; an overlay of time frequency analysis The musical score is somewhat more general and abstract than the alphabet and corresponds roughly to a description of a piece of music by specifying which notes are being played, i.e., the note's characteristic pitch, amplitude, duration, and location in time While traditional windowed Fourier analysis considers a Fourier representation of the signal in each window of space (or time), wavelets, wavelet packets, and their variants provide a description in which notes of different duration (or resolution) are superimposed For images, this corresponds to an overlay of patterns of different size and scale This multiscale representation allows for a better separation of textures and structures, and of decomposition of the textures into their basic elements The complementary procedure introduces a new approach to speech, music, and image synthesis, yet to be further explored Most of the chapters in this book are based on the lectures delivered at the Neaman Workshop on Signal and Image Representation in Combined Spaces, held at Technion Additional chapters were contributed by invitees who could not attend the workshop The material presented in this volume brings together a rich variety of ideas that blend most aspects of analysis mentioned above These papers can be clustered into affinity groups as follows: Variations on the windowed Fourier transform and its applications, relating Fourier analysis to analysis on the Heisenberg group, are provided in the following group of papers: M An, A Bordzik, I Gertner, and R Tolimieri: "Weyl-Heisenberg System and the Finite Zak Transform;" M Bastiaans: "Gabor's Expansion and the Zak Transform for Continuous-Time and Discrete-Time Signals;" W Schempp: "Non-Commutative Affine Geometry and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets;" M Zibulski and Y Y Zeevi: "The Generalized Gabor Scheme and Its Application in Signal and Image Representation." Constructions of special waveforms suitable for specific tasks are given in: J S Byrnes: "A Low Complexity Energy Spreading Transform Coder;" A Coheu and N Dyn: "Nonstationary Subdivision Schemes, Multiresolution Analysis and Wavelet Packets." The use of redundant representations in reconstruction and enhancement is provided in: J J Benedetto: "Noise Reduction in Terms of the Theory of Frames;" Z Cvetkovic and M Vetterli: "Overcomplete Expansions and Robustness;" F Bergeaud and S Mallat: "Matching Pursuit of Images." xii Preface Applications of efBcient numerical compression as a tool for fast numerical analysis are described in: A Averbuch, G Beylkin, R Coifman, and M Israeli: "Multiscale Inversion of Elliptic Operators;" A Harten: "Multiresolution Representation of Cell-Averaged Data: A Promotional Review." Approximation properties of various waveforms in diflFerent contexts are described in the following series of papers: A J E M Janssen: "A Density Theorem for Time-Continuous Filter Banks;" V E Katsnelson: "Sampling and Interpolation for Functions with Multi-Band Spectrum: The Mean Periodic Continuation Method;" M A Kon and L A Raphael: "Characterizing Convergence Rates for Multiresolution Approximations;" C Chui and Chun Li: "Characterization of Smoothness via Functional Wavelet Transforms;" R Lenz and J Svanberg: "Group Theoretical Transforms, Statistical Properties of Image Spaces and Image Coding;" J Prestin and K Selig: "Interpolatory and Orthonormal Trigonometric Wavelets;" B Rubin: "On Calderon's Reproducing Formula;" and "Continuous Wavelet Transforms on a Sphere;" V A Zheludev: "Periodic Splines, Harmonic Analysis and Wavelets." Acknowledgments The Neaman Workshop was organized under the auspices of The Israel Academy of Sciences and Humanites and co-sponsored by The Neaman Institute for Advanced Studies in Science and Technology; The Institute of Advanced Studies in Mathematics; The Institute of Theoretical Physics; and The Ollendorff Center of the Department of Electrical Engineering, Technion—Israel Institute of Technology Several people helped in the preparation of this manuscript We wish to thank in particular Ms Lesley Price for her editorial assistance and word-processing of the manuscripts provided by the authors, Ms Margaret Chui for her editing and overall guidance in the preparation of the book, and Ms Katy Tynan of Academic Press for her communications assistance Haifa, Israel New Haven, Connecticut June 1997 Yehoshua Y Zeevi Ronald Coifman Contributors Numbers in parentheses indicate where the authors^ contributions begin M A N (1), Prometheus Inc., 52 Ashford Street, AUston, MA 02134 [myoung@ccs.neu.edu] (341), School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel [amir@math.tau.ac.il] A M I R AVERBUCH MARTIN JOHN J J BASTIAANS (23), Technische Universiteit Eindhoven, Faculteit Elektrotechniek, Postbus 513, 5600 MB Eindhoven, Netherlands [M.J.Bastiaans@ele.tue.nl] B E N E D E T T O (259), Department of Mathematics, Maryland, College Park, Maryland 29742 [jjb@math.umd.edu] University of FRANgois BERGEAUD (285), Ecole Centrale Paris, Applied Mathematics Laboratory, Grande Voie des Vignes, F-92290 Chatenay-Malabry, France [francois@mas.ecp.fr] (341), Program in Applied Mathematics, of Colorado at Boulder, Boulder, CO 80309-0526 [beylkin@julia.colorado.edu] GREGORY BEYLKIN A BRODZIK (1), Rome Laboratory/EROI, University Hanscom AFB, MA 01731- 2909 J S (167), Prometheus Inc., 52 Ashford St., Allston, MA 02134 [jbyrnes@cs.umb.edu] BYRNES xni xiv Contributors (395), Center for Approximation Theory, Texas A&;M University, College Station, TX 77843 [cchui@tamu.edu] CHARLES K CHUI (189), Laboratoire d^AnalyseNumerique, UniversitePierre et Marie Curie, Place Jussieu, 75005 Paris, France [cohen@ann j ussieu fr] ALBERT COHEN (341), Department of Mathematics, P.O Box 208283, Yale University New Haven, CT 06520-8283 [coifman@j ules mat h yale edu] RONALD COIFMAN ZORAN CVETKOVIC (301), Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720-1772 [zor an@eecs berkeley.edu] NiRA DYN (189), School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel [niradyn@math.tau.ac.il] I (1), Computer Science Department, The City College of New York, Convent Avenue at 138th Street, New York, NY 10031 [csicg@csfaculty.engr.ccny.cuny.edu] GERTNER AMI HARTEN (361), School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, 69978 Israel MosHE (341), Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa 32000, Israel [isr aeli@ cs t echnion ac il] ISRAELI A J E M JANSSEN (513), Philips Research Laboratories, WL-01, 5656 AA Eindhoven, The Netherlands (525), Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel [katze@wisdom.weizmann.ac.il] V I C T O R E KATSNELSON MARK A K O N (415), Department of Mathematics, Boston Boston, MA 02215 University, (553), Department of Electrical Engineering, University, S-58183 Linkoping, Sweden [reiner@isy.liu.se] Linkoping R E I N E R LENZ CHUN L I (395), Institute of Mathematics, Academia Sinica, Beijing 100080, China Contributors xv (285), Ecole Polytechnique, CMAP, 91128 Palaiseau Cedex, France [mallat@cmapx.polytechnique.fr] STEPHANE MALLAT JURGEN PRESTIN (201), FB Mathematik, Universitat Rostock, Rostock, Germany [prestin@mathematik.uni-rostock.d400.de] LOUISE A RAPHAEL (415), Department of Mathematics, versity, Washington, DC 20059 D-18051 Howard Uni- BORIS RUBIN (439, 457), Department of Mathematics, Hebrew University of Jerusalem, Givat Ram 91904, Jerusalem, Israel [boris@math.huji.ac.il] (71), Lehrstuhl fuer Mathematik I, University of Siegen, D-57068 Siegen, Germany [schempp@mathematik.uni-siegen.d400.de] WALTER SCHEMPP (201), FB Mathematik, Universitat Rostock, D-18051 Rostock, Germany [selig@mathematik.uni-rostock.d400.de] KATHI SELIG (553), Department of Electrical Engineering, Linkoping University, S-58183 Linkoping, Sweden [svan@isy.liu.se] JONAS SVANBERG R TOLIMIERI (1), Electrical Engineering Department, The City College of New York, Convent Avenue at 138th Street, New York, NY 10031 (301), Department d'Electricite EPFL, CH'1015 Lausanne, Switzerland [martin.vetterli@de.epfl.ch] MARTIN V E T T E R L I Y ZEEVI (121), Department of Electrical Engineering, TechnionIsrael Institute of Technology, Haifa 32000, Israel [zeevi@ee technion ac il] YEHOSHUA (477), School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel [zhel@math.tau.ac.il] VALERY A ZHELUDEV (121), Multimedia Department, IBM Science and Technology, MATAM, Haifa 31905, Israel [meir z @ vnet ibm com] M E I R ZIBULSKI Weyl-Heisenberg Systems and t h e Finite Zak Transform M An, A Brodzik, I Gertner, and R Tolimieri Abstract Previously, a theoretical foundation for designing algorithms for computing Weyl-Heisenberg (W-H) coefficients at critical sampling was established by applying the finite Zak transform This theory established clear and easily computable conditions for the existence of W-H expansion and for stability of computations The main computational task in the resulting algorithm was a 2-D finite Fourier transform In this work we extend the applicability of the approach to rationally over-sampled W-H systems by developing a deeper understanding of the relationship established by the finite Zak transform between linear algebra properties of W-H systems and function theory in Zak space This relationship will impact on questions of existence, parameterization, and computation of W-H expansions Implementation results on single RISC processor of i860 and the PARAGON parallel multiprocessor system are given The algorithms described in this paper possess highly parallel structure and are especially suited in a distributed memory, parallel-processing environment Timing results show that real-time computation of W-H expansions is realizable §1 Introduction During t h e last four years powerful new methods have been introduced for analyzing Wigner transforms of discrete and periodic signals [10, 11, 13] based on finite W-H expansions [2, 5, 6, 12] A recent work [10] adapted these m e t h o d s t o gain control over t h e cross-term interference problem [9] by constructing signal systems in time frequency space for expanding Wigner trg,nsforms from W-H systems based on Gaussian-like signals T h e computational feasibility of t h e method in [10] depends strongly on t h e availability of eflScient and stable algorithms for computing W-H expansion coefficients Since W-H systems are not orthogonal, standard Hilbert space inner-product methods not generally apply Moreover, Signal and Image Representation in Combined Spaces Y Y Zeevi and R R Coifman (Eds.), PP- 3-21 Copyright © 9 by Academic Press All rights of reproduction in any form reserved ISBN 0-12-777830-6 O M An et al since critically sampled W-H systems may not form a basis, over-sampling in time frequency is necessary for the existence of arbitrary signal expansions In fact, this is usually the case for systems based on the Gaussian In [10, 11, 12, 13, 15], the concept of biorthogonals was applied to the problem of W-H coefficient computation In [15], the Zak transform provided the framework for computing biorthogonals for rationally over-sampled WH systems forming frames A similar approach for critically and integer over-sampled W-H systems can be found in [3, 4] The goal in this work is somewhat different in that major emphasis is placed on describing linear spans of W-H systems that are not necessarily complete and on establishing, in a form suitable for RISC and parallel processing, algorithms for computing W-H coefficients of signals in such linear spans For the most part, our approach extends on that developed in [3, 14] and frame theory, although an important part in [15] plays no role in this work However, as in these previous works [7, 8], the finite Zak transform will be established as a fundamental and powerful tool for studying critically sampled and rationally over-sampled W-H systems and for designing algorithms for computing W-H coefficients for discrete and periodic signals The role of the finite Zak transform is analogous to that played by the Fourier transform in replacing complex convolution computations by simple pointwise multiplication In this new setting, properties of W-H systems, such as their spanning space and dimension, can be determined by simple operations on functions in Zak space This relationship will impact on questions of existence, parameterization, and computation of W-H expansions In the over-sampled case, both integer and rational over-sampling are investigated Implementation results on single RISC processor of i860 and the PARAGON parallel multiprocessor system are given for sample sizes both of powers of and mixed sizes with factors 2, 3, 4, 5, 6, 7, 8, and The algorithms described in this paper possess highly parallel structure and are especially suited in a distributed memory, parallel-processing environment Timing results on single i860 processor and on 4- and 8-node computing systems show that real-time computation of W-H expansions is realizable In Section 2, the basic preliminaries will be established Algorithms will be described in Section for critically sampled W-H systems, in Section for integer over-sampled systems, and in Section for rationally oversampled systems Implementation results will be given in Sections 6, 7, and Weyl-Heisenberg Systems and the Finite Zak Transform §2 2.1 Preliminaries Weyl-Heisenberg systems Choose an integer A > A discrete function / ( a ) , a e Z is called NT periodic if f{a + N) = f{a), aeZ Denote by L{N) the Hilbert space of all AT-periodic functions with inner product N-l a=0 For g G L{N) and < m, n < iV define gm,n ^ H^) by QmAc^) = 9{a + m ) e - - - / ^ , ae Z (2.1.1) The functions in the set {gm,n : < m, n < N} are called Weyl-Heisenberg wavelets having generator g Suppose N = KM with positive integers K and M The collection of N functions {QkM^mK :0