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Trends in Mathematics Eckhard Hitzer Stephen J. Sangwine Editors Quaternion and Clifford Fourier Transforms and Wavelets

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Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submis- sion Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any signiﬁcantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of ﬁles must be in one particular dialect of TEX and uniﬁed according to simple instructions available from Birkha¨user. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the ﬁnal version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961

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Quaternion and Clifford Fourier Transforms and Wavelets Eckhard Hitzer Stephen J. Sangwine Editors

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Editors Eckhard Hitzer Stephen J. Sangwine Department of Material Science School of Computer Science International Christian University and Electronic Engineering Tokyo, Japan University of Essex Colchester, United Kingdom ISBN 978-3-0348-0602-2 ISBN 978-3-0348-0603-9 (eBook) DOI 10.1007/978-3-0348-0603-9 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2013939066 Mathematics Subject Classification (2010): 11R52, 15A66, 42A38, 65T60, 42C40, 68U10, 94A08, 94A12 © Springer Basel 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.springer.com)

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii F. Brackx, E. Hitzer and S.J. Sangwine History of Quaternion and Cliﬀord–Fourier Transforms and Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Part I: Quaternions 1 T.A. Ell Quaternion Fourier Transform: Re-toolingImage and Sig nal Processing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 E. Hitzer and S.J. Sangwine The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations . . . . . . . . . . . . . . . . . . . . . . . 15 3 N. Le Bihan and S.J. Sangwine Quaternionic Spectral Analysis of Non-Stationary Improper Complex Sig nals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 E.U. Moya-S´anchez and E. Bayro-Corrochano Quaternionic Local Phase for Low-level Image Processing UsingAtomic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 S. Georgiev and J. Morais Bochner’s Theorems in the Framework of Quaternion Analysis . . . . . . 85 6 S. Georgiev, J. Morais, K.I. Kou and W. Spro¨ßig Bochner–Minlos Theorem and Quaternion Fourier Transform . . . . . . . . 105 Part II: Cliﬀord Algebra 7 E. Hitzer, J. Helmstetter and R. Abl̷amowicz Square Roots of −1 in Real Cliﬀord Alg ebras . . . . . . . . . . . . . . . . . . . . . . . 123 8 R. Bujack, G. Scheuermann and E. Hitzer A General Geometric Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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vi Contents 9 T. Batard and M. Berthier Cliﬀord–Fourier Transform and Spinor Representation of Imag es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10 P.R. Girard, R. Pujol, P. Clary sse, A. Marion, R. Goutte and P. Delachartre Analytic Video (2D + 𝑡) Signals Using Cliﬀord–Fourier Transforms in Multiquaternion Grassmann–Hamilton–Cliﬀord Algebras . . . . . . . . . 197 11 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 12 R. Soulard and P. Carr´e Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Imag e Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 13 S. Bernstein Seeingthe Invisible and Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 269 14 M. Bahri A Generalized Windowed Fourier Transform in Real Cliﬀord Algebra 𝐶ℓ0,𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 15 Y. Fu, U. Ka¨hler and P. Cerejeiras The Balian–Low Theorem for the Windowed Cliﬀord–Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 16 S. Li and T. Qian Sparse Representation of Sig nals in Hardy Space . . . . . . . . . . . . . . . . . . . . 321 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

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Quaternion and Cliﬀord–Fourier Transforms and Wavelets Trends in Mathematics, vii–x ⃝c 2013 Springer Basel Preface One hundred and seventy years ago (in 1843) W.R. Hamilton formally introduced the four-dimensional quaternions, perceivingthem as one of the major discoveries of his life. One year later, in 1844, H. Grassmann published the ﬁrst version of his Ausdehnungslehre, now known as Grassmann algebra, without any dimensional limitations. Circa thirty years later (in 1876) W.K. Cliﬀord supplemented the Grassmann product of vectors with an inner product, which fundamentally uniﬁed the precedingworks of Hamilton and Grassmann in the form of Cliﬀord’s geometric algebras or Cliﬀord algebras. A Cliﬀord algebra is a complete algebra of a vector space and all its subspaces, includingthe measurement of volumes and dihedral angles between any pair of subspaces. To work in higher dimensions with quaternion and Cliﬀord algebras allows us to systematically generalize known concepts of symmetry, phase, analytic signal and holomorphic function to higher dimensions. And as demonstrated in the cur- rent proceedings, it successfully generalizes Fourier and wavelet transformations to higher dimensions. This is interesting both for the development of analysis in higher dimensions, as well as for a broad range of applications in multi-dimensional signal, image and color image processing. Therefore a wide variety of readers from pure mathematicians, keen to learn about the latest developments in quaternion and Cliﬀord analysis, to physicists and engineers in search of dimensionally ap- propriate and eﬃcient tools in concrete applications, will ﬁnd many interesting contributions in this book. The contributions in this volume originated as papers in a session on Quater- nion and Cliﬀord–Fourier transforms and wavelets of the 9th International Con- ference on Cliﬀord Algebras and their Applications (ICCA9), which took place from 15th to 20th July 2011 at the Bauhaus-University in Weimar, Germany. The session was organized by the editors of this volume. After the conference we asked the contributors to prepare expanded versions of their works for this volume, and many of them agreed to participate. The ex- panded submissions were subjected to a further round of reviews (in addition to the original reviews for the ICCA9 itself) in order to ensure that each contribution was clearly presented and worthy of publication. We are very grateful to all those reviewers whose eﬀorts contributed signiﬁcantly to the quality of the ﬁnal chapters by askingthe authors to revise, clarify or to expand on points in their drafts.

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viii Preface The contributions have been edited to achieve as much uniformity in presen- tation and notation as can reasonably be achieved across the somewhat diﬀerent traditions that have arisen in the quaternion and Cliﬀord communities. We hope that this volume will contribute to a growing uniﬁcation of ideas across the ex- pandingﬁeld of hypercomplex Fourier transforms and wavelets. The book is divided into two parts: Chapters 1 to 6 deal exclusively with quaternions ℍ, while Chapters 7 to 16 mainly deal with Cliﬀord algebras 𝐶ℓ𝑝,𝑞, but sometimes include high-dimensional complex as well as quaternionic results ∼ in several subsections. This is natural, since complex numbers (ℂ = 𝐶ℓ0,1) and ∼ quaternions (ℍ = 𝐶ℓ0,2) are low-dimensional Cliﬀord algebras, and often appear ∼ + ∼ + as subalgebras, e.g., ℂ = 𝐶𝑙 , ℍ = 𝐶𝑙 , etc. The ﬁrst chapter was written 2,0 3,0 especially for this volume to provide some background on the history of the subject, and to show how the contributions that follow relate to each other and to prior work. We especially thank Fred Brackx (Ghent/Belgium) for agreeing to contribute to this chapter at a late stage in the preparation of the book. The quaternionic part begins with an exploration by Ell (Chapter 1) of the evolution of quaternion Fourier transform (QFT) deﬁnitions as a framework for problems in vector-image and vector-signal processing, ranging from NMR prob- lems to applications in colour image processing. Next, follows an investigation by Hitzer and Sangwine (Chapter 2) into a steerable quaternion algebra split, which leads to: a local phase rotation interpretation of the classical two-sided QFT, eﬃ- cient fast numerical implementations and the design of new steerable QFTs. Then Le Bihan and Sangwine (Chapter 3) perform a quaternionic spectral analysis of non-stationary improper complex signals with possible correlation of real and imaginary signal parts. With a one-dimensional QFT they introduce a hyperanalytic signal closely linked to the geometric features of improper com- plex signals. In the ﬁeld of low level image processing Moya-S´anchez and Bayro- Corrochano (Chapter 4) employ quaternionic atomic functions to enhance geo- metric image features and to analytically express image processing operations like low-pass, steerable and multiscale ﬁltering, derivatives, and local phase computa- tion. In the next two chapters on quaternion analysis Georgiev and Morais (Chap- ter 5) characterize a class of quaternion Bochner functions generated via a quater- nion Fourier–Stieltjes transform and generalize Bochner’s theorem to quaternion functions. In Chapter 6 Georgiev, Morais, Kou and Spr¨oßigstudy the asymptotic behavior of the QFT, apply the QFT to probability measures, includingpositive deﬁnite measures, and extend the classical Bochner–Minlos theorem to the frame- work of quaternion analysis. The Cliﬀord algebra part begins with Chapter 7 by Hitzer, Helmstetter and Ab̷lamowicz, who establish a detailed algebraic characterization of the continuous manifolds of (multivector) square roots of −1 in all real Cliﬀord algebras 𝐶ℓ𝑝,𝑞, includingas examples detailed computer generated tables of representative square roots of −1 in dimensions 𝑛 = 𝑝 + 𝑞 = 5, 7 with signature 𝑠 = 𝑝 − 𝑞 = 3(mod 4).

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Preface ix Their work is fundamental for any form of Cliﬀord–Fourier transform (CFT) using multivector square roots of −1 instead of the complex imaginary unit. Based on this Bujack, Scheuermann and Hitzer (Chapter 8) introduce a general (Cliﬀord) geometric Fourier transform covering most CFTs in the literature. They prove a range of standard properties and specify the necessary conditions in the transform design. A series of four chapters on image processing begins with Batard and Ber- thier’s (Chapter 9) on spinorial representation of images focusing on edge- and texture detection based on a special CFT for spinor ﬁelds, that takes into account the Riemannian geometry of the image surface. Then Girard, Pujol, Clarysse, Mar- ion, Goutte and Delachartre (Chapter 10) investigate analytic signals in Cliﬀord algebras of 𝑛-dimensional quadratic spaces, and especially for three-dimensional ∼ video (2D + 𝑇) signals in (complex) biquaternions (= 𝐶ℓ3,0). Generalizingfrom the right-sided QFT to a rotor CFT in 𝐶ℓ3,0, which allows a complex fast Fourier transform (FFT) decomposition, they investigate the corresponding analytic video signal including its generalized six biquaternionic phases. Next, Bernstein, Bou- chot, Reinhardt and Heise (Chapter 11) undertake a mathematical overview of gen- eralizations of analytic signals to higher-dimensional complex and Cliﬀord analysis together with applications (and comparisons) for artiﬁcial and real-world image samples. Soulard and Carr´e (Chapter 12) deﬁne a novel colour monogenic wavelet transform, leading to a non-marginal multiresolution colour geometric analysis of images. They show a ﬁrst application through the deﬁnition of a full colour image denoisingscheme based on statistical modelingof coeﬃcients. Motivated by applications in optical coherence tomography, Bernstein (Chap- ter 13) studies inverse scatteringfor Dirac operators with scalar, vector and quater- nionic potentials, by writingMaxwell’s equations as Dirac equations in Cliﬀord algebra (i.e., complex biquaternions). For that she considers factorizations of the Helmholtz equation and related fundamental solutions; standard- and Faddeev’s Green functions. In Chapter 14 Bahri introduces a windowed CFT for signal functions 𝑓 : 𝑛 ℝ → 𝐶ℓ0,𝑛, and investigates some of its properties. For a diﬀerent type of win- 𝑛 dowed CFT for signal functions 𝑓 : ℝ → 𝐶ℓ𝑛,0, 𝑛 = 2, 3(mod 4), Fu, K¨ahler and Cerejeiras establish in Chapter 15 a Balian–Low theorem, a strongform of Heisen- berg’s classical uncertainty principle. They make essential use of Cliﬀord frames and the Cliﬀord–Zak transform. Finally, Li and Qian (Chapter 16) employ a compressed sensingtechnique in order to introduce a new kind of sparse representation of signals in a Hardy space dictionary (of elementary wave forms) over a unit disk, together with examples illustratingthe new algorithm. We thank all the authors for their enthusiastic participation in the project and their enormous patience with the review and editingprocess. We further thank the organizer of the ICCA9 conference K. Guerlebeck and his dedicated team for

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