Cliff. Alg. + Calc. CFT overview Two-sided CFTs One-sided CFTs Conclusion + Refs. Overview of Quaternion and Clifford Fourier Transformations Eckhard Hitzer Department of Material Science International Christian University, Tokyo, Japan Asian Workshop AWCGAIT2015 on Clifford’s Geometric Algebra and Information Technology 26-28 Mar. 2015, Danang University of Science and Technology, Vietnam Eckhard Hitzer International Christian University Overview of Quaternion and Clifford Fourier Transformations
67
Embed
Overview of Quaternion and Clifford Fourier Transformations · 2015-04-02 · Cliff. Alg. + Calc. CFT overview Two-sided CFTs One-sided CFTs Conclusion + Refs. Overview of Quaternion
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Jesus Christ on World Peace:"Peace I leave with you; my peace I give you.I do not give to you as the world gives.Do not let your hearts be troubled and do not be afraid."
Bible: Gospel of John, chp. 14, verse 27.
I thank my wife, my children, my parents.
R. Abłamowicz, M. Berthier, F. Brackx, R. Bujack, D. Eelbode, S. Georgiev, J.Helmstetter, B. Mawardi, J. Morais, S. Sangwine, G. Scheuermann, F. Sommen,W. Sproessig, G. Guerlebeck, T. Simos
M. T. Pham, K. Tachibana
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Jesus Christ on World Peace:"Peace I leave with you; my peace I give you.I do not give to you as the world gives.Do not let your hearts be troubled and do not be afraid."
Bible: Gospel of John, chp. 14, verse 27.
I thank my wife, my children, my parents.
R. Abłamowicz, M. Berthier, F. Brackx, R. Bujack, D. Eelbode, S. Georgiev, J.Helmstetter, B. Mawardi, J. Morais, S. Sangwine, G. Scheuermann, F. Sommen,W. Sproessig, G. Guerlebeck, T. Simos
M. T. Pham, K. Tachibana
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Jesus Christ on World Peace:"Peace I leave with you; my peace I give you.I do not give to you as the world gives.Do not let your hearts be troubled and do not be afraid."
Bible: Gospel of John, chp. 14, verse 27.
I thank my wife, my children, my parents.
R. Abłamowicz, M. Berthier, F. Brackx, R. Bujack, D. Eelbode, S. Georgiev, J.Helmstetter, B. Mawardi, J. Morais, S. Sangwine, G. Scheuermann, F. Sommen,W. Sproessig, G. Guerlebeck, T. Simos
M. T. Pham, K. Tachibana
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Jesus Christ on World Peace:"Peace I leave with you; my peace I give you.I do not give to you as the world gives.Do not let your hearts be troubled and do not be afraid."
Bible: Gospel of John, chp. 14, verse 27.
I thank my wife, my children, my parents.
R. Abłamowicz, M. Berthier, F. Brackx, R. Bujack, D. Eelbode, S. Georgiev, J.Helmstetter, B. Mawardi, J. Morais, S. Sangwine, G. Scheuermann, F. Sommen,W. Sproessig, G. Guerlebeck, T. Simos
M. T. Pham, K. Tachibana
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
1 The study and application of this research is only permitted for peaceful,non-offensive and non-criminal purposes. This permission includes passivedefensive technologies, like missile defense shields.
2 Any form of study, use and application of research, which is licensed under theCreative Peace License, for military purposes, with the explicit or implicit intent tocreate or contribute to military offensive technologies is strictly prohibited.
3 Civil technologies that pose excessive risks to human life are also excluded fromthis license.
4 Use for surveillance, surveillance of communications, their interception andcollection of personal data, including mass surveillance, interception andcollection, which infringes the right to privacy under international human rights lawand national regulations that comply with international human rights law, is strictlyprohibited.
5 Individuals, groups, teams, public and private entities engaging in any form in thestudy, use and application of research licensed under the Creative Peace License,thereby agree in a legally binding sense to strictly adhere to the terms of theCreative Peace License.
Dr. Eckhard S. M. Hitzer,
14 December 2011, Hiroshima, Japan. 30 August 2013, Osaka, Japan. 30 May 2014, Tokyo, Japan.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
1 The study and application of this research is only permitted for peaceful,non-offensive and non-criminal purposes. This permission includes passivedefensive technologies, like missile defense shields.
2 Any form of study, use and application of research, which is licensed under theCreative Peace License, for military purposes, with the explicit or implicit intent tocreate or contribute to military offensive technologies is strictly prohibited.
3 Civil technologies that pose excessive risks to human life are also excluded fromthis license.
4 Use for surveillance, surveillance of communications, their interception andcollection of personal data, including mass surveillance, interception andcollection, which infringes the right to privacy under international human rights lawand national regulations that comply with international human rights law, is strictlyprohibited.
5 Individuals, groups, teams, public and private entities engaging in any form in thestudy, use and application of research licensed under the Creative Peace License,thereby agree in a legally binding sense to strictly adhere to the terms of theCreative Peace License.
Dr. Eckhard S. M. Hitzer,
14 December 2011, Hiroshima, Japan. 30 August 2013, Osaka, Japan. 30 May 2014, Tokyo, Japan.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Clifford geometric algebras over vector spaces Rp,q
Example: Cl3 Clifford geometric algebra (GA) of Euclidean space R3
Orthonormal basis {e1, e2, e3} of 3-dim. Euclidean space R3 = R3,0 gives 8-dim.basis of Cl3 = Cl3,0
{1, e1, e2, e3︸ ︷︷ ︸vectors
, e2e3, e3e1, e1e2︸ ︷︷ ︸area bivectors
, i = e1e2e3︸ ︷︷ ︸volume trivector
}. (13)
i ... unit trivector, i.e. oriented volume of unit cube, i2 = −1.
Even grade subalgebra Cl+3 isomorphic to quaternions H (Hamilton), prominent invirtual reality, aerospace applications, crystal texture (orientation) analysis, etc.
{1, i, j,k}. (14)
Therefore elements of Cl+3 rotors (rotation operators), rotating all vectors andmultivectors of Cl3
x ∈ Cl3 : x→ R−1xR, R ∈ Cl+3 . (15)
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Clifford geometric algebras over vector spaces Rp,q
Example: Cl3 Clifford geometric algebra (GA) of Euclidean space R3
Orthonormal basis {e1, e2, e3} of 3-dim. Euclidean space R3 = R3,0 gives 8-dim.basis of Cl3 = Cl3,0
{1, e1, e2, e3︸ ︷︷ ︸vectors
, e2e3, e3e1, e1e2︸ ︷︷ ︸area bivectors
, i = e1e2e3︸ ︷︷ ︸volume trivector
}. (13)
i ... unit trivector, i.e. oriented volume of unit cube, i2 = −1.
Even grade subalgebra Cl+3 isomorphic to quaternions H (Hamilton), prominent invirtual reality, aerospace applications, crystal texture (orientation) analysis, etc.
{1, i, j,k}. (14)
Therefore elements of Cl+3 rotors (rotation operators), rotating all vectors andmultivectors of Cl3
x ∈ Cl3 : x→ R−1xR, R ∈ Cl+3 . (15)
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
A general element (multivector) M ∈ Clp,q represents a collection of subspacesof different dimensions (grades) k : 0 ≤ k ≤ n: scalars, vectors, bi-vectors, . . . ,pseudoscalars (Name: k-vector parts)
M =∑A
MAeA = 〈M〉+ 〈M〉1 + 〈M〉2 + . . .+ 〈M〉n, (16)
blade index A ∈ powerset P(1, 2, . . . , n), 2n coefficients MA ∈ R.
Principle reverse of M ∈ Clp,q replaces complex / quaternion conjugation.
The scalar product of two multivectors M, N ∈ Clp,q is defined as
M ∗ N = 〈MN〉 = 〈MN〉0 =∑A
MANA. (17)
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[21] E. Hitzer, et al, Square roots of −1 in real Clifford algebras, in: E. Hitzer, S.J.Sangwine (eds.), "Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27,Birkhauser, Basel, 2013, 123–153.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
General geometric (Clifford) Fourier transform [5]
Generalizations: in e−ixω : i ∈ C→√−1 ∈ Cl(p, q), −ixω → −s(x, ω) with
s(x, ω)2 < 0.
Incorporates most of previously known CFTs with the help of very general sets ofleft and right kernel factor products
FGFT {h}(ω) =
∫Rp′,q′
L(x, ω)h(x)R(x, ω)dn′x, L(x, ω) =
∏s∈FL
e−s(x,ω),
(24)with p′ + q′ = n′, FL = {s1(x, ω), . . . , sL(x, ω)} a set of mappingsRp′,q′ × Rp′,q′ → Ip,q into the manifold of real multiples of
√−1 in Cl(p, q).
R(x, ω) is defined similarly, and h : Rp′,q′ → Cl(p, q) is the multivector signalfunction.
R. Bujack, G. Scheuermann, E. H. A General Geom. Fourier Transf., in: E. Hitzer, S.J. Sangwine (eds.), "Quaternionand Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser, Basel, 2013, 155–176.
R. Bujack, E. H., G. Scheuermann, Demystification of the Geometric Fourier Transforms,In T. Simos, G. Psihoyios and C. Tsitouras (eds.), Numerical Analysis and Applied Mathematics ICNAAM 2013,AIP Conf. Proc. 1558, pp. 525–528 (2013). DOI: 10.1063/1.4825543, Preprint: http://vixra.org/abs/1310.0255
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Sommen+Bülow: Cl(0, n) basis vector Clifford FT [32, 4]
FSB{h}(ω) =
∫Rn
h(x)n∏k=1
e−2πxkωkek
︸ ︷︷ ︸R(x,ω)
dnx, (25)
where x, ω ∈ Rn with components xk, ωk, and {e1, . . . ek} is an orthonormal basis ofR0,n, e21 = . . . = e2k = −1, h : Rn → Cl(0, n).
For n = 2 and real signals h : R2 → R, this transform is identical to the quaternionicFT (see later).
F. Sommen, Hypercomplex Fourier and Laplace Transforms I, Illinois J. of Math., 26(2) (1982), 332–352.
T. Bülow, et al, Non-comm. Hypercomplex Fourier Transf. of Multidim. Signals, in G. Sommer (ed.), "Geom. Comp.with Cliff. Algebras", Springer 2001, 187–207.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Application: For generalized color image Fourier descriptors.Phase correlation of color images: analogous to shift g(ω) = f(ω)eiω·∆,get shift ∆ from color image CFT by score aggregation.
Fig. 9.2 of J. Mennesson, et al, Color Object Recognition Based on a Clifford Fourier Transf., in L. Dorst, J. Lasenby,"Guide to Geometric Algebra in Pract.", Springer 2011, 175–191.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
E. Hitzer, Two-sided Clifford Fourier transform with two square roots of−1 in Cl(p, q), Advances in Applied CliffordAlgebras, 2014, DOI: 10.1007/s00006-014-0441-9. First published in M. Berthier, L. Fuchs, C. Saint-Jean (eds.)electronic Proceedings of AGACSE 2012, La Rochelle, France, 2-4 July 2012. Preprint:http://arxiv.org/abs/1306.2092 .
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
∫R2e−fx1ω1h(x)e−gx2ω2d2x, h : R2 → H, f, g ∈ H : f2 = g2 = −1.
Variants: left or right kernel factors is dropped, or both are placed together at theright or left side, or rotor forms (L = R).
It was first described by Ernst, et al, [10, pp. 307-308] (with f = i, g = j) forspectral analysis in two-dimensional nuclear magnetic resonance, suggesting touse the QFT as a method to independently adjust phase angles with respect totwo frequency variables in two-dimensional spectroscopy.
Later Ell [8] independently formulated and explored the QFT for the analysis oflinear time-invariant systems of PDEs (next).
The QFT was further applied by Bülow, et al [3] for image, video and textureanalysis (see later).
Sangwine et al [31, 2]: color image analysis and analysis of non-stationaryimproper complex signals, vector image processing, and quaternion polar signalrepresentations (see later).
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
2D complex FT and QFT (Images: T. Bülow, thesis [3])
T. Bülow: Applications to 2D gray scale images. (Color images: Ell & Sangwine [9].)One component of each transform kernel for different frequency values u, v.
complex FT intrinsically 1D QFT intrinsically 2D
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Convenient ± split of quaternions [E.H.: AACA 2007, ICCA9]...
q = q+ + q−, q± =1
2(q ± iqj). (27)
=⇒ 2 orthogonal 2D planes in R4. For explicit forms of q±, see references (below).
Consequence: modulus identity |q|2 = |q−|2 + |q+|2.Generalization: i, j → any 2 pure unit quaternions f, g: f2 = g2 = −1.
NB: Even f = g makes perfect sense. Identical to simplex/perplex split of Ell &Sangwine.
E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. App. Cliff. Algs, 17(3)(2007), 497–517. DOI: 10.1007/s00006-007-0037-8
E. Hitzer, S. J. Sangwine, The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transf.,in: E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser,Basel, 2013, 15–39.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
The QFT of f± split parts of quaternion function f ∈ L2(R2,H) have simplequasi-complex forms
Fq{f±} =
∫R2f±e−j(x2ω2∓x1ω1)d2x =
∫R2e−i(x1ω1∓x2ω2)f±d
2x . (30)
Generalization: E. Hitzer, Two-sided Clifford Fourier transform with two square roots of−1 in Cl(p, q), Advances inApplied Clifford Algebras, 2014, DOI: 10.1007/s00006-014-0441-9. First published in M. Berthier, L. Fuchs, C.Saint-Jean (eds.) electronic Proceedings of AGACSE 2012, La Rochelle, France, 2-4 July 2012. Preprint:http://arxiv.org/abs/1306.2092 .
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Computation of image stretches, reflections and rotations in QFT spectrum of image.
Quaternionic Gaussian filter (GF)
Minimal uncertaintyTexture segmentation
T. Bülow, Hypercomplex Spectral Signal Repr. for the Proc. and Analysis of Images, PhD thesis, Univ. of Kiel,Germany, Inst. fuer Informatik und Prakt. Math., Aug. 1999.
E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, AACA, 17 (2007), 497–517.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Direct frequency domain filtering [9], f = g = gray line.
From left to right (vertical pairs):1) Original and modulus ρF . 2) Low-pass filtered.
3) Band-pass filtered. 4) High-pass filtered.T.A. Ell, S.J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process., 16(1)(2007), 22–35. See also Kogakuin/Tokyo 2015 lecture of S. Sangwine on YouTube!
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
The QFT allows superior two-dimensional texture segmentation.
The QFT leads to omni-directional disparity estimation (e.g. for video frames).
NB: In both cases intrinsic limitations of corresponding complex methods areovercome.
Colour-Sensitive Edge Detection using Hypercomplex (QFT) Filters.
Quaternion Wiener Deconvolution for Noise Robust Color Image Registration.
T. Bülow, Hypercomplex Spectral Signal Repr. for the Proc. and Analysis of Images, PhD thesis, Univ. of Kiel,Germany, Inst. fuer Informatik und Prakt. Math., Aug. 1999.
M. Pedone et al, Quaternion Wiener Deconvolution for Noise Robust Color Image Registration, IEEE SIGNALPROCESSING LETTERS, VOL. 22, NO. 9, pp. 1278 âAS 1282 (2015).
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
(32)with quaternion valued signal h : R2 → H such that |h| is summable over R∗+ × S1
under the measure dθdr/r, R∗ the multiplicative group of positive non-zeronumbers, and f, g ∈ H two
√−1.
FQM can characterize 2D shapes rotation, translation and scale invariant,possibly including object color and vector patterns encoded in the quaternioniccomponents of h.The QFMT can be generalized straightforward to Clifford FMT applied to signalsh : R2 → Cl(p, q), p+ q = 2 , with two
√−1: f, g ∈ Cl(p, q), p+ q = 2.
Case Cl(1, 1) important for hyperbolic geometry and two-dimensional specialrelativity.
E. Hitzer, Quaternionic Fourier-Mellin Transf., in T. Sugawa (ed.), Proc. of ICFIDCAA 2011, Hiroshima, Japan,Tohoku Univ. Press, Sendai (2013), ii, 123–131.E. Hitzer, Clifford Fourier-Mellin transform with two real square roots of−1 in Cl(p, q), p + q = 2, 9th ICNPAA2012, AIP Conf. Proc., 1493, (2012), 480–485.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
The volume-time Fourier transform can indeed be applied to multivector signalfunctions valued in the whole spacetime algebra h : R3,1 → Cl(3, 1) [14, 17]
FST {h}(ω) =
∫R3,1
e−et tωth(x)e−i3~x·~ωd4x. (35)
The volume-time FT includes the pseudoscalar spatial Clifford FT of Cl(3, 0):FPS{h}(~ω) =
∫R3 h(~x)e−i3~x·~ωd3x.
The split (34) applied to spacetime Fourier transform (35) leads to a multivectorwavepacket analysis
FST {h}(ω) =
∫R3,1
h+(x)e−i3(~x·~ω−tωt)︸ ︷︷ ︸right propagation
d4x +
∫R3,1
h−(x)e−i3(~x·~ω+tωt)︸ ︷︷ ︸left propagation
d4x,
(36)in terms of right and left propagating spacetime multivector wave packets.Application: stretches, reflections, rotations, acceleration, boost of space-timesignal in spectrum of SFT.
E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, AACA, 17 (2007), 497–517.E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier Transforms, AACA, 20(2) (2010), 271–284.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Relationship: One-sided CFTs are obtained by setting one of the phase functions u orv to zero in the two-sided CFT (26).
Definition (CFT with respect to one square root of −1)
Let f ∈ Cl(p, q), f2 = −1, be any square root of −1. The general one-sided CliffordFourier transform (CFT) of h ∈ L1(Rp,q ;Cl(p, q)), with respect to f is
Ff{h}(ω) =
∫Rp,q
h(x) e−fu(x,ω)dnx, (37)
where dnx = dx1 . . . dxn, x,ω ∈ Rp,q , and u : Rp,q × Rp,q → R.
E. Hitzer, The Clifford Fourier transform in real Clifford algebras, in E. H., K. Tachibana (eds.), "Session onGeometric Algebra and Applications, IKM 2012", Special Issue of Clifford Analysis, Clifford Algebras and theirApplications, Vol. 2, No. 3, pp. 227-240, (2013). First published in K. Guerlebeck, T. Lahmer and F. Werner (eds.),electronic Proc. of 19th International Conference on the Application of Computer Science and Mathematics inArchitecture and Civil Engineering, IKM 2012, Weimar, Germany, 04âAS06 July 2012. Preprint:http://vixra.org/abs/1306.0130
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
A recent discrete spinor CFT is used for edge and texture detection, where thesignal is represented as a spinor and the
√−1 is a local tangent bivector
B ∈ Cl(3, 0) to the image intensity surface (e3 is the intensity axis).
Can be applied to Gaussian filtering.
T. Batard, M. Berthier, Clifford Fourier Transf. and Spinor Repr. of Images, in: E. Hitzer, S.J. Sangwine (eds.),"Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser, Basel, 2013, 177–195.
B
http://www.dailymail.co.uk/
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
One-sided CFTs which use a single pseudoscalar [15]
Well studied and applied (in pseudoscalar, i2n = −1)
FPS{h}(ω) =
∫Rn
h(x)e−inx·ωdnx, in = e1e2 . . . en, n = 2, 3(mod 4), (38)
where h : Rn → Cl(n, 0), and {e1, e2, . . . , en} is the orthonormal basis of Rn.Historically the special case of (38), n = 3, was already introduced in 1990 [24] forthe processing of electromagnetic fields.This same transform (n = 3) was later applied [12] to two-dimensional imagesembedded in Cl(3, 0) to yield a two-dimensional analytic signal, and in imagestructure processing.Moreover, it (n = 3) was successfully applied to three-dimensional vector fieldprocessing in [7, 6] with vector signal convolution based on Clifford’s full geometricproduct of vectors.For embedding one-dimensional signals in R2, [12] considered in (38) the specialcase of n = 2, and in [7, 6] this was also applied to the processing oftwo-dimensional vector fields.Recent applications of (38) with n = 2, 3, to geographic information systems (GIS)and climate data can be found in [34, 33, 26].
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Clifford FFT in Geographic Information Systems (GIS)
Spatio-temporal raster and vector field data analysis.Linwang Yuan, et al, Geom. Alg. for Multidim.-Unified Geogr. Inf. System, AACA, 23 (2013), 497–518.Please read: Yuan Linwang, et al, Pattern Forced Geophys. Vec. Field Segm. based on Clifford FFT, Computer &Geoscience, Vol. 60 (2013), pp. 63–69.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Generalizing to Clifford Linear Canonical Transforms (LCT)
Real and complex linear canonical transforms parametrize a continuum oftransforms, which include the Fourier, fractional Fourier, Laplace, fractionalLaplace, Gauss-Weierstrass, Bargmann, Fresnel, and Lorentz transforms, as wellas scaling operations.Generalization to (finite-extension, complex) Clifford LCTs from the Cl(0, n) basisvector CFT of Buelow and Sommen.
K. Kou, J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonicaltransform in Clifford Analysis, Math. Meth. Appl. Sci., (2013).
Hypercomplex LCT related to pseudoscalar CFT in Cl(3, 0).
Y. Yang, K. Kou, Uncertainty principles for hypercomplex signals in the linear canoncial transform domains,Signal Proc., Vol. 95 (2014), pp. 67–75.
Quaternionic LCT related to 2-sided QFT and fractional QFT.
K. Kou, J-Y. Ou, J. Morais, On Uncertainty Principle for Quaternionic Linear Canonical Transform,Abs. and App. Anal., Vol. 2013, IC 72592, 14 pp.
Further new constructions of Clifford LCTs in PMAMCM 2014 proceedings(Santorini/Greece, 2014).Further new constructions of Quaternion Domain FT (QDFT) in Group30proceedings (Ghent/Belgium, 2014). Signal domain now H, not only R2.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
Clifford Fourier transforms can apply the manifolds of√−1 ∈ Cl(p, q) to create a
rich variety of new FTs.History of just over 30 years. Major steps: Cl(0, n) CFTs, then pseudoscalarCFTs, Quaternion FTs.In the 90ies especially applications in electromagnetic fiels/electronics andsignal/image processing dominated.This was followed by color image processing and most recently applications inGeographic Information Systems.This presentation could only feature a part of the approaches in CFT research,and only a part of the applications. Omitted: operator exponential CFT approach[H. De Bie, et al], CFT for conformal geom. algebra. Regarding applications, e.g.CFT Fourier descriptor representations of shape [B. Rosenhahn, et al] wasomitted.Note that there are further types of Clifford algebra/analysis related integraltransforms: Clifford wavelets, Clifford radon transforms, Clifford Hilbert transforms,... which we did not discuss.Generalization to Clifford Linear Canonical Transforms, and quaternion domain FT(QDFT).
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[1] T. Batard, M. Berthier, Clifford-Fourier Transf. and Spinor Repr. of Images, in: E. Hitzer,S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27,Birkhauser, Basel, 2013, 177–195.
[2] F. Brackx, et al, History of Quaternion and Clifford-Fourier Transf., in: E. Hitzer, S.J.Sangwine (eds.), "Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27,Birkhauser, Basel, 2013, xi–xxvii.
[3] T. Bülow, Hypercomplex Spectral Signal Repr. for the Proc. and Analysis of Images,PhD thesis, Univ. of Kiel, Germany, Inst. fuer Informatik und Prakt. Math., Aug. 1999.
[4] T. Bülow, et al, Non-comm. Hypercomplex Fourier Transf. of Multidim. Signals, in G.Sommer (ed.), "Geom. Comp. with Cliff. Algebras", Springer 2001, 187–207.
[5] R. Bujack, et al, A General Geom. Fourier Transf., in: E. Hitzer, S.J. Sangwine (eds.),"Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser, Basel,2013, 155–176.
[6] J. Ebling, G. Scheuermann, Clifford convolution and pattern matching on vector fields,In Proc. IEEE Vis., 3, IEEE Computer Society, Los Alamitos, 2003. 193–200,
[7] J. Ebling, G. Scheuermann, Cliff. Four. transf. on vector fields, IEEE Trans. on Vis. andComp. Graph., 11(4), (2005), 469–479.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[8] T. A. Ell, Quaternionic Fourier Transform for Analysis of Two-dimensional LinearTime-Invariant Partial Differential Systems. in Proceedings of the 32nd IEEEConference on Decision and Control, December 15-17, 2 (1993), 1830–1841.
[9] T.A. Ell, S.J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans.Image Process., 16(1) (2007), 22–35.
[10] R. R. Ernst, et al, Princ. of NMR in One and Two Dim., Int. Ser. of Monogr. on Chem.,Oxford Univ. Press, 1987.
[11] M. Felsberg, et al, Comm. Hypercomplex Fourier Transf. of Multidim. Signals, in G.Sommer (ed.), "Geom. Comp. with Cliff. Algebras", Springer 2001, 209–229.
[12] M. Felsberg, Low-Level Img. Proc. with the Struct. Multivec., PhD thesis, Univ. of Kiel,Inst. fuer Inf. & Prakt. Math., 2002.
[13] S. Georgiev, J. Morais, Bochner’s Theorems in the Framework of Quaternion Analysisin: E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transf. andWavelets", TIM 27, Birkhauser, Basel, 2013, 85–104.
[14] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations,AACA, 17 (2007), 497–517.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[15] E. Hitzer, B. Mawardi, Clifford Fourier Transf. on Multivector Fields and Unc. Princ. forDim. n = 2 (mod 4) and n = 3 (mod 4), P. Angles (ed.), AACA, 18(S3,4), (2008),715–736.
[16] E. Hitzer, Cliff. (Geom.) Alg. Wavel. Transf., in V. Skala, D. Hildenbrand (eds.), Proc.GraVisMa 2009, Plzen, 2009, 94–101.
[17] E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier Transforms, AACA,20(2) (2010), 271–284.
[18] E. Hitzer, Clifford Fourier-Mellin transform with two real square roots of −1 in Cl(p, q),p+ q = 2, 9th ICNPAA 2012, AIP Conf. Proc., 1493, (2012), 480–485.
[19] E. Hitzer, Two-sided Clifford Fourier transf. with two square roots of −1 in Cl(p, q),Advances in Applied Clifford Algebras, 2014, DOI: 10.1007/s00006-014-0441-9,http://arxiv.org/abs/1306.2092
[20] E. Hitzer, S. J. Sangwine, The Orthogonal 2D Planes Split of Quaternions andSteerable Quaternion Fourier Transf., in: E. Hitzer, S.J. Sangwine (eds.), "Quaternionand Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser, Basel, 2013, 15–39.
[21] E. Hitzer, et al, Square roots of −1 in real Clifford algebras, in: E. Hitzer, S.J. Sangwine(eds.), "Quaternion and Clifford Fourier Transf. and Wavelets", TIM 27, Birkhauser,Basel, 2013, 123–153.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[22] E. Hitzer, Quaternionic Fourier-Mellin Transf., in T. Sugawa (ed.), Proc. of ICFIDCAA2011, Hiroshima, Japan, Tohoku Univ. Press, Sendai (2013), ii, 123–131.
[23] E. Hitzer, The Clifford Fourier transform in real Clifford algebras, in E. H., K. Tachibana(eds.), "Session on Geometric Algebra and Applications, IKM 2012", Special Issue ofClifford Analysis, Clifford Algebras and their Applications, Vol. 2, No. 3, pp. 227-240,(2013). Preprint: http://vixra.org/abs/1306.0130 .
[24] B. Jancewicz. Trivector Fourier transf. and electromag. Field, J. of Math. Phys., 31(8),(1990), 1847–1852.
[25] H. Li, Invariant Algebras And Geometric Reasoning, World Scientific, Singapore, 2008.
[26] Linwang Yuan, et al, Geom. Alg. for Multidim.-Unified Geogr. Inf. System, AACA, 23(2013), 497–518.
[27] B. Mawardi, E. Hitzer, Clifford Algebra Cl(3, 0)-valued Wavelet Transf., Clifford WaveletUncertainty Inequality and Clifford Gabor Wavelets, Int. J. of Wavelets, Multiresolutionand Inf. Proc., 5(6) (2007), 997–1019.
[28] J. Mennesson, et al, Color Obj. Recogn. Based on a Clifford Fourier Transf., in L. Dorst,J. Lasenby, "Guide to Geom. Algebra in Pract.", Springer 2011, 175–191.
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations
[29] S.C. Pei, J.J. Ding, J.H. Chang, Efficient Implementation of Quaternion FourierTransform, Convolution, and Correlation by 2-D Complex FFT, IEEE Trans. on Sig.Proc., 49(11), p. 2783 (2001).
[30] B. Rosenhahn, G. Sommer Pose estimation of free-form objects, EuropeanConference on Computer Vision, Springer-Verlag, Berlin, 127, pp. 414–427, Prague,2004, edited by Pajdla, T.; Matas, J.
[31] S. J. Sangwine, Fourier transf. of color images using quat., or hyperc., numbers, El.Lett., 32(21) (1996), 1979–1980.
[32] F. Sommen, Hypercomplex Fourier and Laplace Transforms I, Illinois J. of Math., 26(2)(1982), 332–352.
[33] Yuan Linwang, et al, Pattern Forced Geophys. Vec. Field Segm. based on Clifford FFT,Computer & Geoscience, Vol. 60 (2013), pp. 63–69.
[34] Yu Zhaoyuan, et al, Clifford Algebra for Geophys. Vector Fields, To appear in NonlinearProcesses in Geophysics.
[35] G. Xu, X. Wang, X. Xu, Fractional quaternion Fourier transform, convolution andcorrelation, Signal Processing, 88(10), (2008), pp. 2511–2517,http://dx.doi.org/10.1016/j.sigpro.2008.04.012
Eckhard Hitzer International Christian University
Overview of Quaternion and Clifford Fourier Transformations