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Multidimensional Filter Banks and Multiscale Geometric Representations Full Text Available At Full text available at: http://dx.doi.org/10.1561/2000000012 Multidimensional Filter Banks and Multiscale Geometric Representations Full text available at: http://dx.doi.org/10.1561/2000000012 Multidimensional Filter Banks and Multiscale Geometric Representations Minh N. Do University of Illinois at Urbana-Champaign Urbana, IL 61801 USA [email protected] Yue M. Lu Harvard University Cambridge, MA 02138 USA [email protected] Boston { Delft Full text available at: http://dx.doi.org/10.1561/2000000012 Foundations and Trends R in Signal Processing Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is M. N. Do and Y. M. Lu, Multidimen- sional Filter Banks and Multiscale Geometric Representations, Foundations and Trends R in Signal Processing, vol 5, no 3, pp 157{264, 2011 ISBN: 978-1-60198-584-2 c 2012 M. N. Do and Y. M. Lu All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The `services' for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. Authorization does not extend to other kinds of copy- ing, such as that for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. In the rest of the world: Permission to pho- tocopy must be obtained from the copyright owner. Please apply to now Publishers Inc., PO Box 1024, Hanover, MA 02339, USA; Tel. +1-781-871-0245; www.nowpublishers.com; [email protected] now Publishers Inc. has an exclusive license to publish this material worldwide. Permission to use this content must be obtained from the copyright license holder. Please apply to now Publishers, PO Box 179, 2600 AD Delft, The Netherlands, www.nowpublishers.com; e-mail: [email protected] Full text available at: http://dx.doi.org/10.1561/2000000012 Foundations and Trends R in Signal Processing Volume 5 Issue 3, 2011 Editorial Board Editor-in-Chief: Robert M. Gray Dept of Electrical Engineering Stanford University 350 Serra Mall Stanford, CA 94305 USA [email protected] Editors Abeer Alwan (UCLA) Jelena Kovacevic (CMU) John Apostolopoulos (HP Labs) Jia Li (Pennsylvania State Pamela Cosman (UCSD) University) Michelle Effros (California Institute B.S. Manjunath (UCSB) of Technology) Urbashi Mitra (USC) Yonina Eldar (Technion) Thrasos Pappas (Northwestern Yariv Ephraim (George Mason University) University) Mihaela van der Shaar (UCLA) Sadaoki Furui (Tokyo Institute Michael Unser (EPFL) of Technology) P.P. Vaidyanathan (California Vivek Goyal (MIT) Institute of Technology) Sinan Gunturk (Courant Institute) Rabab Ward (University Christine Guillemot (IRISA) of British Columbia) Sheila Hemami (Cornell) Susie Wee (HP Labs) Lina Karam (Arizona State Clifford J. Weinstein (MIT Lincoln University) Laboratories) Nick Kingsbury (Cambridge Min Wu (University of Maryland) University) Josiane Zerubia (INRIA) Alex Kot (Nanyang Technical Pao-Chi CHang (National Central University) University) Full text available at: http://dx.doi.org/10.1561/2000000012 Editorial Scope Foundations and Trends R in Signal Processing will publish sur- vey and tutorial articles on the foundations, algorithms, methods, and applications of signal processing including the following topics: • Adaptive signal processing • Signal processing for • Audio signal processing communications • Biological and biomedical signal • Signal processing for security and processing forensic analysis, biometric signal processing • Complexity in signal processing • Signal quantization, sampling, • Digital and multirate signal analog-to-digital conversion, processing coding and compression • Distributed and network signal • Signal reconstruction, processing digital-to-analog conversion, • Image and video processing enhancement, decoding and • Linear and nonlinear filtering inverse problems • Multidimensional signal processing • Speech/audio/image/video • Multimodal signal processing compression • Multiresolution signal processing • Speech and spoken language processing • Nonlinear signal processing • Statistical/machine learning • Randomized algorithms in signal processing • Statistical signal processing • Sensor and multiple source signal { classification and detection processing, source separation { estimation and regression • Signal decompositions, subband { tree-structured methods and transform methods, sparse representations Information for Librarians Foundations and Trends R in Signal Processing, 2011, Volume 5, 4 issues. ISSN paper version 1932-8346. ISSN online version 1932-8354. Also available as a combined paper and online subscription. Full text available at: http://dx.doi.org/10.1561/2000000012 Foundations and Trends R in Signal Processing Vol. 5, No. 3 (2011) 157{264 c 2012 M. N. Do and Y. M. Lu DOI: 10.1561/2000000012 Multidimensional Filter Banks and Multiscale Geometric Representations Minh N. Do1 and Yue M. Lu2 1 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA, [email protected] 2 School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA, [email protected] Abstract Thanks to the explosive growth of sensing devices and capabilities, multidimensional (MD) signals | such as images, videos, multispectral images, light fields, and biomedical data volumes | have become ubiq- uitous. Multidimensional filter banks and the associated constructions provide a unified framework and an efficient computational tool in the formation, representation, and processing of these multidimensional data sets. In this survey we aim to provide a systematic development of the theory and constructions of multidimensional filter banks. We thoroughly review several tools that have been shown to be particularly effective in the design and analysis of multidimensional filter banks, including sampling lattices, multidimensional bases and frames, polyphase representations, Gr¨obner bases, mapping methods, frequency domain constructions, ladder structures and lifting schemes. We then focus on the construction of filter banks and signal represen- tations that can capture directional and geometric features, which are unique and key properties of many multidimensional signals. Next, Full text available at: http://dx.doi.org/10.1561/2000000012 we study the connection between iterated multidimensional filter banks in the discrete domain and the associated multiscale signal representations in the continuous domain through a directional mul- tiresolution analysis framework. Finally, we show several examples to demonstrate the power of multidimensional filter banks and geometric signal representations in applications. Full text available at: http://dx.doi.org/10.1561/2000000012 Contents 1 Introduction 1 2 Preliminaries: Multidimensional Signals, Transforms, and Filtering 7 2.1 Multidimensional Fourier and z-Transforms 7 2.2 Multidimensional Filters 9 3 Multidimensional Sampling 13 3.1 Sampling on Lattices 13 3.2 The Effect of Sampling in the Fourier Domain 15 3.3 Downsampling and Upsampling of Discrete Signals 18 3.4 Key Properties of Sampling Lattices 21 4 Multidimensional Filter Banks 25 4.1 Filter Banks: from Frequency Decomposition to Signal Representation 25 4.2 Basic Tools: Multirate Identities and Polyphase Representations 28 4.3 Perfect Reconstruction Filter Banks 34 4.4 Example: Two-Channel Filter Banks in 2D 37 ix Full text available at: http://dx.doi.org/10.1561/2000000012 5 Characterization and Design of Multidimensional Filter Banks 43 5.1 Characterizing MD Filter Banks Using Gr¨obnerBases 43 5.2 The Mapping-Based Design for MD Filter Banks 48 5.3 Designing Filter Banks in the Frequency Domain 57 6 Iterated and Directional Filter Banks 61 6.1 Directional Filter Banks 61 6.2 Directional Filter Banks in Higher Dimensions 65 7 Multiscale Geometric Representations 73 7.1 The Contourlet and Surfacelet Transforms 73 7.2 Multiresolution Directional Analysis 79 7.3 Other Multiscale Geometric Representations 83 8 Example Applications 89 8.1 Signal Decomposition and Nonlinear Approximation 89 8.2 Image and Video Denoising 92 8.3 Edge-Preserving Image Interpolation 96 8.4 Compressed Sensing 99 Conclusions 103 Acknowledgments 105 References 107 Full text available at: http://dx.doi.org/10.1561/2000000012 1 Introduction Multidimensional (MD) signals are information-carrying physical quan- tities that depend on several variables, each representing a unique dimension. For example, a video is a three-dimensional (3D) signal with two spatial dimensions (horizontal and vertical) and one temporal dimension. A particularly important and common class of MD signals contains visual information, ranging from general images and videos on the Web to special medical images
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