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

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 ScienceInternational Christian University, Tokyo, Japan

http://erkenntnis.icu.ac.jp/

Asian Workshop AWCGAIT2015 onClifford’s Geometric Algebra and Information Technology

26-28 Mar. 2015, Danang University of Science and Technology, Vietnam

Eckhard Hitzer International Christian University



Quote and Acknowledgments

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































Research under The Creative Peace License v0.3

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.




Research under The Creative Peace License v0.3

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.




CONTENTS

1 Clifford geometric algebra and calculusClifford geometric algebras over vector spaces Rp,qBasic Clifford geometric algebra based calculus

2 Overview of Clifford Fourier Transforms (CFT)General geometric (Clifford) FTSommen+Bülow Cl(0, n) CFT

3 Two-sided CFTsQuaternion Fourier transform (QFT)Generalizations of the QFT

4 One-sided CFTsSpinor and pseudoscalar Clifford FTsClifford Linear Canonical Transforms

5 Conclusion and referencesConclusionReferences and further information




Clifford geometric algebras over vector spaces Rp,q

Motivation 1: Complex Number Invariants [H. Li, 2008]Two complex numbers

x = x1 + ix2, y = y1 + iy2 (1)

have the productxy = x1y1 − x2y2 + i(x1y2 + x2y1). (2)

Product xy is not invariant under rotation in complex plain!

x→ xeiθ, y → yeiθ, xy → xye2iθ. (3)

But xy is invariant: xy → xe−iθeiθy = xy.

xy = (x1 − ix2)(y + iy2) = x1y1 + x2y2︸︷︷︸(symmetric) inner product

of vectors ~x · ~y

+ i (x1y2 − x2y1)︸︷︷︸~x, ~y parallelogram

area (anti-symmetric)

, (4)

i ℝ

ℝy1

y2

x1

x2 Area = x1y2− x2y1y

x





Motivation 2.1: Hamilton’s Quaternions∼= Cl(0, 2) ∼= Cl+(3, 0) ∼= Cl+(0, 3)

Gauss, Rodrigues and Hamilton’s 4D quaternion algebra H over R with 3imaginary units:

ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1. (5)

Quaternionq = qr + qii + qjj + qkk ∈ H, qr, qi, qj , qk ∈ R (6)

has quaternion conjugate (principle reverse in Cl+3,0, Cl(0, 2))

q = qr − qii− qjj − qkk, (7)

Leads to norm of q ∈ H

|q| =√qq =

√q2r + q2

i + q2j + q2

k, |pq| = |p||q|. (8)

Scalar part

Sc(q) = qr =1

2(q + q). (9)

Inner product (defines orthogonality !)

Sc(pq) = prqr + piqi + pjqj + pkqk ∈ R. (10)





Motivation 2.2: Pure Quaternions

Two pure quaternions

x = x1i + x2j + x3k, y = y1i + y2j + y3k, (11)

have xy product

xy = x1y1 + x2y2 + x3y3︸︷︷︸(symmetric) scalar part

+ (x3y2 − x2y3)i+ (x1y3 − x3y1)j + (x2y1 − x1y2)k︸︷︷︸(anti-symmetric) pure quaternion part

.

(12)

The symmetric part is the inner product of 3D vectors ~x · ~y .

The antisymmetric part is the outer product of 3D vectors ~x, ~y. Magnitude equalsarea of parallelogram spanned by ~x, ~y .

General outer product subject of Grassmann’s (1844) calculus of extension.

Inner and outer product in nD setting were unified by Clifford (1878) in hisassociative geometric algebras Cl(p, q) = Clp,q :

ab = a · b + a ∧ b, ∀a,b ∈ 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)





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)





Clifford Algebra: Multivectors, blades, reverse, scalar product

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)





Clifford Algebra: Modulus, blade subspace, pseudoscalar

The modulus |M | of a multivector M ∈ Clp,q is defined as

|M |2 = M ∗ M =∑A

M2A. (18)

This includes: magnitude, length, area, volume, hypervolume.Important for applications: n = 2 or 3(mod 4) pseudoscalar in = e1e2 . . . en

i2n = −1. (19)

Geometry: A blade B describes a vector subspace (example: e1, e2-plane)

VB = {x ∈ Rp,q |x∧B = 0}, e.g. B = e1 ∧e2 ⇔ VB = {x|x = x1e1 +x2e2}.

Its dual blade

B∗ = Bi−1n , e.g. B∗ = e1e2i

−13 = e3, Ve3 ⊥ e1e2,

describes the complimentary vector subspace V ⊥B .Pseudoscalar in central for n = p+ q = odd, [but not for n = even!]

inM = M in, ∀M ∈ Clp,q , n = odd.




Basic Clifford geometric algebra based calculus

Multivector functions

Multivector valued function f : Rp,q → Clp,q , p+ q = n, has 2n bladecomponents (fA : Rp,q → R)

f(x) =∑A

fA(x)eA. (20)

We define the inner product of Rp,q → Clp,q functions f, g by

(f, g) =

∫Rp,q

f(x)g(x)︸︷︷︸Clifford product

dnx, (21)

and the L2(Rp,q ;Clp,q)-norm

‖f‖2 = 〈(f, f)〉 , (22)

L2(Rp,q ;Clp,q) = {f : Rp,q → Clp,q | ‖f‖ <∞}. (23)





Square roots of −1 in Clifford algebras [21]i ∈ C in the transformation kernel eiφ = cosφ+ i sinφ is replaced by a

√−1 in

Cl(p, q). For example peseudoscalars i→ in ∈ Cl(n, 0), n = 2, 3(mod 4).What other

√−1 are there in Cl(p, q)? Answer: [21].

Example: In Cl(p, q), p+ q = 2, the√−1 are i = b1e1 + b2e2 + βe12,

with b1, b2, β ∈ R, and β2 = b21e22 + b22e

21 + e21e

22.

Manifolds of√−1 in: Cl(2, 0) (left), Cl(1, 1) (center), and Cl(0, 2) ∼= H (right).





Reference

[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.




Extensive overview of Clifford Fourier Transforms (CFT)

Clifford Fourier Transformations (CFT)

Clifford Algebra √−1

2-sided CFT

2-sided (1-sided)quaternion FT (QFT)

1-sided CFT

polar

Quaternion Fourier Mellin Transform

Clifford Fourier Mellin Transform pseudoscalar CFT

n=2,3(mod 4)

Spacetime FTMultivector Wavepackets

Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3

Spinorial CFTisomorphism

Cl(0,n) basis vector CFT

CommutativeCFT

Operator exponentialCFT

pola

r

Quaternion Fourier Stieltjes Transform

Clifford Wavelets

General geometric FT

Color image CFT

fract. QFTQuat. Lin.Can. Transf.

(fi) Cliff. Lin.Can. Transf.

Hyp.-compl. Lin.Can. Transf.

Quat. Domain

FT




Overview of Clifford Fourier Transforms (CFT)



2-sided CFT


1-sided CFT

polar


Clifford Fourier Mellin Transform

pseudoscalar CFTn=2,3(mod 4)


Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer



General geometric (Clifford) FT

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




Sommen+Bülow Cl(0, n) CFT




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer




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.








2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

Color image CFT



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.







2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

2-sided CFT


Main type: general two sided CFT [19]

One kernel factor on each side

Ff,g{h}(ω) =

∫Rp′,q′

e−fu(x,ω)h(x)e−gv(x,ω)dn′x, (26)

f, g two constant√−1 in Cl(p, q)

phase functions u, v : Rp′,q′ × Rp′,q′ → R

signal function h : Rp′,q′ → Cl(p, q)

often Rp′,q′ = Rp,q .

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 .



http://arxiv.org/abs/1306.2092


Quaternion Fourier transform (QFT)




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

2-sided (1-sided) quaternion FT (QFT) polar Clifford isomorphism



Quaternion Fourier transform (QFT) [14, 20]

Ff,g{h}(ω) =

∫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).





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





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.





Split of two-sided quaternion FT (QFT)

Fq{f}(ω) = f(ω) =

∫R2e−ix1ω1f(x) e−jx2ω2d2x. (28)

Linearity leads to

Fq{f}(ω) = Fq{f− + f+}(ω) = Fq{f−}(ω) + Fq{f+}(ω). (29)

Simple quasi-complex forms for QFT of f±

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 .






Discrete and fast QFT: Pei, Ding, Chang (2001) [29]

Discrete QFT

FDQFT {f}(ω) =

M−1∑x1=0

N−1∑x2=0

e−ix1ω1/Mf(x) e−jx2ω2/N , x1, x2 ∈ N. (31)

Fast QFT computation OK.

S.C. Pei, J.J. Ding, J.H. Chang, Efficient Implementation of Quaternion Fourier Transform, Convolution, andCorrelation by 2-D Complex FFT, IEEE Trans. on Sig. Proc., 49(11), p. 2783 (2001).





Application to image processing [3, 14]

Application: Image 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.





Geometric interpretation [20] of QFT integrand e−ix1ω1h(x) e−jx2ω2

Local rotation by phase angle−(x1ω1 + x2ω2) of h−(x) inthe two-dimensional q−plane, spanned by{i + j, 1− ij}.Local rotation by phase angle−(x1ω1 − x2ω2) of h+(x) inthe two-dimensional q+plane, spanned by{i− j, 1 + ij}.

q−-plane

q+-plane

i+j

i− j

1− ij

1+ ij

h+

h−

Geometry of ±-split of H in R4 picture.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.





Example [9]: Assigning colors RGB to qi, qj , qk, (qr = 0), simplex/perplex split(f = g = (i + j + k)/

√3 gray line).

Top left: Original. Top right: q−-part (luminance).Bottom left: Sum. Bottom right: q+-part (qi ↔ qj ) (chrominance).

T.A. Ell, S.J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process., 16(1)(2007), 22–35.





QFT of color image [9], f = g = gray line.

This representation uses polardecomposition of the QFT:F = ρF eθF nF .

Top left: Original.

Top right: Modulus ρF = |F|in log scale.

Bottom right: Phase angle θF(0=red, π/2 = green hue,π = cyan hue).

Bottom left: Pure part (axis)of nF RGB coded.

T.A. Ell, S.J. Sangwine, Hypercomplex Fouriertransforms of color images, IEEE Trans.Image Process., 16(1) (2007), 22–35.





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!





Applications of QFT (T. Bülow, thesis [3])

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.

C.J. Evans, S.J. Sangwine, T.A. Ell, Colour-Sensitive Edge Detection using Hypercomplex Filters, Proceedings ofEusipco 2000.

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).




QFT generalization




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

QFT) Quaternion Fourier Transform Clifford Fourier Mellin Transform


QFT generalization

Quaternion Fourier Mellin transform (QFMT) [22]

Polar coordinates in R2 lead to

FQM{h}(ν, k) =1

2π

∫ ∞0

∫ 2π

0r−fνh(r, θ)e−gkθdθdr/r, ∀(ν, k) ∈ R× Z,

(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.




QFT generalization

Comparison of complex FMT, Quat. FT and QFMT [22]




QFT generalization

The QFMT splits 2D image in parts with radial/angular symmetries [22]




QFT generalization

QFMT kernel: Radial/angular resolution tuning, scale invariance [22]




QFT generalization




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

Spacetime FT Multivector Wavepackets isomorphism


QFT generalization

Volume-time Fourier transform [14, 17]

The Cl(3, 1) volume-time subalgebra with basis {1, et, i3, ist} ∼= H and allows togeneralize the two-sided QFT to a volume-time Fourier transform

FV T {h}(ω) =

∫R3,1

e−et tωth(x)e−i3~x·~ωd4x, (33)

with x = tet + ~x ∈ R3,1, ~x = x1e1 + x2e2 + x3e3,ω = ωtet + ~ω ∈ R3,1, ~ω = ω1e1 + ω2e2 + ω3e3.

The split (52) with f = et, g = i3 = e∗t becomes the spacetime split of specialrelativity

h± =1

2(1± ethe

∗t ). (34)

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.




QFT generalization




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

Spacetime FT Multivector Wavepackets isomorphism


QFT generalization

Spacetime Fourier transform [14, 17]

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.







2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

1-sided CFT


One-sided Clifford FTs [23]

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



http://vixra.org/abs/1306.0130


Spinor and pseudoscalar Clifford FTs




2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

pseudoscalar Spinorial CFT



Spinor Clifford FT [1]

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/








2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

pseudoscalar CFT n=2,3(mod 4)



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].








2-sided CFT


1-sided CFT

polar





Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFT



hitzer

3D CFT i3 = e1e2e3



Pseodoscalar CFT and complex FT, Discrete CFT, Fast CFT

Case of pseudoscalar Cl3-CFT.

A multivector signal decomposition corresponds to 4 complex signals.

Clifford FT can then be implemented by 4 complex FTs.

This form of the pseudoscalar CFT leads to the discrete CFT using 4 discretecomplex FTs.

4 fast FTs (FFT) can be used to implement a fast pseudoscalar CFT.





Application: Vector Pattern Matching (J. Ebling, G. Scheuermann [7, 6])Fast Cl(3, 0) convolution in CFT Fourier domain of 3× 3× 3 = 33 (red), 53 (yellow),83 (green) rotation patterns with gas furnace chamber flow field.





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.




Clifford Linear Canonical Transforms

Extensive overview: CFT + Clifford Linear Canonical Transforms (LCT)+ QDFT



2-sided CFT


1-sided CFT

polar


Clifford Fourier Mellin Transform pseudoscalar CFT

n=2,3(mod 4)


Volume-time FT

Plane CFT i2 = e

1e

2

3D CFT i3 = e

1e

2e

3



CommutativeCFT


pola

r


Clifford Wavelets


Color image CFT

fract. QFTQuat. Lin.Can. Transf.

(fi) Cliff. Lin.Can. Transf.

Hyp.-compl. Lin.Can. Transf.

Quat. Domain

FT




Clifford Linear Canonical Transforms

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.




Conclusion

Conclusion

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).




References and further information

[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.





[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.





[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.






[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.



http://vixra.org/abs/1306.0130



[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



http://dx.doi.org/10.1016/j.sigpro.2008.04.012



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Documents

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