Tutorial on Fourier and Wavelet Transformations in Geometric Algebra

Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

E. Hitzer

Department of Applied Physics University of Fukui Japan

17. August 2008, AGACSE 3 Hotel Kloster Nimbschen, Grimma, Leipzig, Germany

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Acknowledgements

I do believe in God’s creative power in having brought it all into being in the ﬁrst place, I ﬁnd that studying the natural world is an opportunity to observe the majesty, the elegance, the intricacy of God’s creation.

Francis Collins, Director of the US National Human Genome Research Institute, in Time Magazine, 5 Nov. 2006. I thank my wife, my children, my parents. B. Mawardi, G. Sommer, D. Hildenbrand, H. Li, R. Ablamowicz Organizers of AGACSE 3, Leipzig 2008.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

CONTENTS

1 Geometric Calculus Multivector Functions

2 Clifford GA Fourier Transformations (FT) Deﬁnition and properties of GA FTs Applications of GA FTs, discrete and fast versions

3 Quaternion Fourier Transformations (QFT) Basic facts about Quaternions and deﬁnition of QFT Properties of QFT and applications

4 Spacetime Fourier Transformation (SFT) GA of spacetime and deﬁnition of SFT Applications of SFT

5 Clifford GA Wavelets GA wavelet basics and GA wavelet transformation GA wavelets and uncertainty limits, example of GA Gabor wavelets

6 Conclusion

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Multivectors, blades, reverse, scalar product

Multivector M ∈ Gp,q = Clp,q, p + q = n, has k-vector parts (0 ≤ k ≤ n) scalar, vector, bi-vector, . . . , pseudoscalar X M = MAeA = hMi + hMi1 + hMi2 + ... + hMin, (1) A

blade index A ∈ {0, 1, 2, 3, 12, 23, 31, 123,..., 12 . . . n}, MA ∈ R. Reverse of M ∈ Gp,q n X k(k−1) Mf = (−1) 2 hMik. (2) k=0 replaces complex conjugation/quaternion conjugation

The scalar product of two multivectors M, Ne ∈ Gp,q is deﬁned as

M ∗ Ne = hMNei = hMNei0 (3) P For M, Ne ∈ Gn = Gn,0 we get M ∗ Ne = A MANA.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Modulus, blade subspace, pseudoscalar

The modulus |M| of a multivector M ∈ Gn = Gn,0 is deﬁned as n 2 X 2 |M| = M ∗ Mf = MA. (4) A=1

For n = 2(mod 4), n = 3(mod 4) pseudoscalar in = e1e2 ... en

2 in = −1. (5) A blade B describes a vector subspace

p,q VB = {x ∈ R |x ∧ B = 0}. Its dual blade ∗ −1 B = Bin ⊥ describes the complimentary vector subspace VB . pseudoscalar in commutes for n = 3(mod 4)

in M = M in, ∀M ∈ Gn,0.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Multivector functions

p,q n Multivector valued function f : R → Gp,q = Clp,q, p + q = n, has 2 blade components n X f(x) = fA(x)eA. (6) A=1 n We deﬁne the inner product of R → Cln,0 functions f, g by Z Z n X n (f, g) = f(x)gg(x) d x = eAeB fA(x)gB (x) d x, (7) n f n R A,B R

2 n and the L (R ; Cln,0)-norm Z kfk2 = h(f, f)i = |f(x)|2dnx. (8) n R

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Vector Differential & Vector Derivative

p,q Vector differential of f deﬁned (any const. a ∈ R ) f(x + a) − f(x) a · ∇f(x) = lim , (9) →0 NB: a · ∇ scalar. Vector derivative ∇ can be expanded as

n X ∂ ∇ = e ∂ , ∂ = e · ∇ = , (10) k k k k ∂x k=1 k

both coordinate independent!

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Geometric Calculus Examples

MV Functions V-Differentials V-Derivatives

f1 = x a · ∇f1 = a ∇f1 = 3, (n = 3) 2 f2 = x a · ∇f2 = 2a · x ∇f2 = 2x f = |x| a·x 3 a · ∇f3 = |x| ∇f3 = x/|x| f = x · hAi , 0≤k≤n 4 k a · ∇f4 = a · hAik ∇f4 = khAik −1 2 f5 = log r a · ∇f = a·r ∇f5 = r = r/r . 5 r2 r = x − x0 r = |r|

References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Geometric Calculus Examples

MV Functions V-Differentials V-Derivatives

References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Geometric Calculus Examples

MV Functions V-Differentials V-Derivatives

References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Geometric Calculus Examples

MV Functions V-Differentials V-Derivatives

References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Multivector Functions Rules for Vector Differential & Derivative

Derivative from differential (regard x = const., a = variable)

∇f = ∇a (a · ∇f) (11)

Sum rules, product rules exist. Modiﬁcation by non-commutativity: ∇˙ fg˙ 6= f∇˙ g˙

n ˙ ˙ ˙ ˙ ˙ X ∇(fg) = (∇f)g + ∇fg˙ = (∇f)g + ekf(∂kg). (12) k=1

Vector differential / derivative of f(x) = g(λ(x)), λ(x) ∈ R : ∂g ∂g a · ∇f = {a · ∇λ(x)} , ∇f = (∇λ) . (13) ∂λ ∂λ ∂f Example: For a = ek (1 ≤ k ≤ n) ek · ∇f = ∂kf = (∂kλ) ∂λ .

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Classical scalar complex Fourier Transformations

Deﬁnition (1D Cassical FT) 2 1 For an integrable function f ∈ L (R) ∩ L (R), the Fourier transform of f is the function F{f}: R → C = G0,1 Z F{f}(ω) = f(x) e−iωx dx, (14) R i imaginary unit: i2 = −1.

Inverse FT 1 Z f(x) = F −1[F{f}(ω)] = F{f}(ω) eiωx dω. (15) 2π R

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Examples of continuous and discrete 1D FT [Images: Wikipedia]

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Geometric Algebra Fourier Transformation (GA FT)

complex → geometric

complex unit i → Geometric roots of −1, e.g. pseudoscalars in, n = 2, 3(mod 4). 2 n complex f → multivector function f ∈ L (R , Gn)

Deﬁnition (with pseudoscalars in dims.2,3,6,7,10,11, ...)

n The GA Fourier transform F{f}: R → Gn = Cln,0, n =2 , 3(mod 4) is given by Z F{f}(ω) = f(x) e−inω·x dnx, (16) n R n for multivector functions f: R → Gn. NB: Also possible for Cl0,n, n =1 , 2(mod 4). Possible applications: dimension speciﬁc transformations for actual data and signals, with desired subspace structure.

Inversion of these GA FTs 1 Z f(x) = F −1[F{f}(ω)] = F{f}(ω) einω·x dnω. (17) n 3 (2π) R

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Overview of complex, Clifford, quaternion, spacetime FTs

Complex Fourier Transformation (FT) Z −iωx FC{f}(ω) = f(x) e dx (18) R Clifford Geometric Algebra (GA) FT in G3 (i → i3) Z −i3~ω·~x 3 F 3 {f}(~ω) = f(~x) e d ~x (19) G 3 R GA FT in Gn,0, G0,n0 (i → in) Z −inω·x n F n {f}(ω) = f(x) e d x (20) G n R QFT in H (i → i, j) Z F {f}(u) = fˆ(u) = e−ixuf(x) e−jyvd2x (21) H 2 R SFT in spacetime algebra G3,1 (i → et, j → i3) Z −et ts −i3~x·~u 3 FSTA{f}(u) = f(u) = e f(x) e d ~xdt (22) 3,1 R

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs

Properties of the GA FT in Gn, n = 3 (mod 4)

General properties: Linearity, shift, modulation (ω-shift), scaling, convolution, Plancherel theorem and Parseval theorem 1 kfk = kF{f}k. (23) (2π)n/2

2 n n Table: Unique properties of GA FT Multiv. Funct. f ∈ L (R , Gn), a ∈ R , m ∈ N. Property Multiv. Funct. GA FT m m m Vec. diff. (a · ∇) f(x) in (a · ω) F{f}(ω) m m m (a · x) f(x) in (a · ∇ω) F{f}(ω) m m m Moments of x x f(x) in ∇ω F{f}(ω) m m m f(x)x in F{f}(ω) ∇ω m m m Vec. deriv. ∇ f(x) in ω F{f}(ω) m m m dual moments f(x)∇ in F{f}(ω) ω

NB: Standard complex FT only gives coordinate moments and partial derivative properties.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs

Properties of the GA FT in Gn, n = 2 (mod 4)

Property Multiv. Funct. CFT Left lin. αf(x)+β g(x) αF{f}(ω)+ βF{g}(ω) x-Shift f(x − a) F{f}(ω) e−inω·a i ω ·x ω-Shift f(x) e n 0 F{f}(ω − ω0) 1 ω Scaling f(ax) an F{f}( a ) Convolution (f?g)(x) F{f}(−ω) F{godd}(ω) +F{f}(ω) F{geven}(ω) m m m Vec. diff. (a · ∇) f(x)(a · ω) F{f}(ω) in m m m (a · x) f(x)(a · ∇ω) F{f}(ω) in m m m Moments of x x f(x) ∇ω F{f}(ω) in m m m m f(x) x F{f}((−1) ω) ∇ω in m m m Vec. deriv. ∇ f(x) ω F{f}(ω) in m m m m f(x)∇ F{f}((−1) ω) ω in R n 1 R n Plancherel n f1(x)f^2(x) d x n n F{f1}(ω)F{^f2}(ω) d ω R (2π) R sc. Parseval kfk 1 kF{f}k (2π)n/2

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Applications

Sampling Representation of shape Uncertainty LSI filters (smoothing, edge detection) Signal analysis Image processing Fast (multi)vector pattern matching Visual flow analysis (Multi)vector field analysis

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Relation with complex FT, Discrete GA FT, Fast GA FT

Example of multivector signal decomposition

f = [f0 + f123i3] + [f1 + f23i3]e1 + [f2 + f31i3]e2 + [f3 + f12i3]e3 (24)

corresponds to 4 complex signals. Implementation of GA FT by 4 complex FTs

FG3 {f} = F[f0 + f123i3] + F[f1 + f23i3]e1 + F[f2 + f31i3]e2 + F[f3 + f12i3]e3 (25) This form of the GA FT leads to the discrete GA FT using 4 discrete complex FTs. 4 fast FTs (FFT) can be used to implement fast GA FT.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Application: Fourier Descriptor Repr. of Shape (B. Rosenhahn et al [6])

e e e f l l = 1, 2, 3 Set of 3D sample points is projected along 1, 2, 3: n1,n2 , . 0 Use of rotors for 2D discrete rotor FT (−N1,2 ≤ n1,2, k1,2 ≤ N1,2, N = 2N + 1)

k1,n1 2π k2,n2 R1,l = exp( 0 k1n1eli3),R2,l = ... (26) N1 f l Sample points n1,n2 give surface phase vectors 1 X pl = f l R^k1,n1 R^k2,n2 e (27) k1,k2 N 0 N 0 n1,n2 1,l 2,l l 1 2 n1,n2 Surface estimation 3 3 X X X F (Φ , Φ ) = f l(Φ , Φ )e = Rk1,Φ1 Rk2,Φ2 pl R^k1,Φ1 R^k2,Φ2 (28) 1 2 1 2 l 1,l 2,l k1,k2 1,l 2,l l=1 l=1 k1,k2

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Application: Fourier Descriptor Repr. of Shape (B. Rosenhahn et al [6])

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs GA FT of Convolution

Convolution The GA FT of the convolution of f(x) and g(x) Z (f?g)(x) = f(y)g(x − y) d3y, (29) 3 R equals for n = 3(mod 4) the product of the GA FTs of f(x) and g(x):

F{f?g}(ω) = F{f}(ω)F{g}(ω). (30)

Convolution for n = 2(mod 4)

For n = 2(mod 4) we get due to non-commutativity of the pseudoscalar in ∈ Gn

in M 6= M in, for M ∈ Gn. (31) that F{f ? g}(ω) = F{f}(−ω)F{godd}(ω) + F{f}(ω)F{geven}(ω) (32)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs GA FT of Correlation

Correlation The GA FT of the correlation of f(x) and g(x) Z (f g)(x) = f(y)g(x + y) d3y, (33) F 3 R equals for n = 3(mod 4) the product of the GA FTs of f 0(x) = f(−x) and g(x):

0 F{fFg}(ω) = F{f }(ω)F{g}(ω). (34)

Correlation for n = 2(mod 4)

For n = 2(mod 4) we get due to non-commutativity of the pseudoscalar in ∈ Gn

in M 6= M in, for M ∈ Gn. (35) that 0 0 F{fFg}(ω) = F{f }(−ω)F{godd}(ω) + F{f }(ω)F{geven}(ω) (36)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Application: Vector Pattern Analysis (J. Ebling, G. Scheuermann [7, 8]) 3 × 3 × 3 rotation pattern and discrete GA FT black: vector components, red: bivector components (shown by normal vectors)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Definition and properties of GA FTs Application: Vector Pattern Matching (J. Ebling, G. Scheuermann [7, 8]) Convolution of 3 × 3 × 3 = 33 (red), 53 (yellow), 83 (green) rotation patterns with gas furnace chamber flow field.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Uncertainty Principle in Physics (Image: Wikipedia)

Optical Measurement Heisenberg Uncertainty Princ. Standard deviations ∆x, ∆p of position x, momentum p measurements

∆x∆p ≥ ~ , 2 f . . . wave function.

f −→ ∆x,

F{f} −→ ∆p.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Deﬁnition and properties of GA FTs Application of GAFT: Uncertainty Principles

Directional Uncertainty Principle [Hitzer&Mawardi, Proc. ICCA7]

2 n Multivect. funct. f ∈ L (R , Gn), n = 2, 3 (mod 4) n GA FT F{f}(ω). arbitrary const. vectors a, b ∈ R 1 Z 1 Z (a · b)2 (a · x)2|f(x)|2 dnx (b · ω)2 |F{f}(ω)|2dnω ≥ 2 n n 2 n kfk R (2π) kfk R 4

Equality (minimal bound) for optimal Gaussian multivector functions

−k x2 f(x) = C0 e , (37)

C0 ∈ Gn arbitrary const. multivector, 0 < k ∈ R.

Uncertainty principle (direction independent) 2 n For f ∈ L (R , Gn), n = 2, 3 (mod 4) we obtain from the dir. UP that 1 Z 1 Z n x2 |f(x)|2 dnx ω2 |F{f}(ω)|2dnω ≥ . (38) 2 n n 2 n kfk R (2π) kfk R 4

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Basic facts about Quaternions and deﬁnition of QFT Quaternions

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

ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1. (39) Quaternion q = qr + qii + qj j + qkk ∈ H, qr, qi, qj , qk ∈ R (40) + has quaternion conjugate (reversion in Cl3,0)

q˜ = qr − qii − qj j − qkk, (41)

Leads to norm of q ∈ H q p 2 2 2 2 |q| = qq˜ = qr + qi + qj + qk, |pq| = |p||q|. (42)

Scalar part 1 Sc(q) = qr = (q +q ˜). (43) 2

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Basic facts about Quaternions and deﬁnition of QFT ± Split of quaternions [Hitzer, AACA 2007]

Convenient split of quaternions

1 q = q+ + q−, q± = (q ± iqj). (44) 2

Explicitly in real components qr, qi, qj , qk ∈ R using (39): 1 ± k 1 ± k q± = {qr ± q + i(qi ∓ qj )} = {qr ± q + j(qj ∓ qi)}. (45) k 2 2 k

Consequence: modulus identity

2 2 2 |q| = |q−| + |q+| . (46)

Property: Scalar part of mixed split product Given two quaternions p, q and applying the ± split we get zero for the scalar part of the mixed products Sc(p+q−) = 0, Sc(p−q+) = 0. (47)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Basic facts about Quaternions and deﬁnition of QFT

Deﬁnition of quaternion FT (QFT) Z −ix ω −jx ω 2 Fq{f}(ω) = fˆ(ω) = e 1 1 f(x) e 2 2 d x. (48) 2 R Linearity leads to

Fq{f}(ω) = Fq{f− + f+}(ω) = Fq{f−}(ω) + Fq{f+}(ω). (49)

Remark: Other variations exist (see later).

Simple complex forms for QFT of f±

2 2 The QFT of f± split parts of quaternion function f ∈ L (R , H) have simple complex forms Z Z −j(x ω ∓x ω ) 2 −i(x ω ∓x ω ) 2 fˆ± = f±e 2 2 1 1 d x = e 1 1 2 2 f±d x . (50) 2 2 R R Free of coordinates: Z Z −j x·ω 2 −j x·(Ue ω) 2 fˆ− = f−e d x, fˆ+ = f+e 1 d x, (51) 2 2 R R

the reﬂection Ue1 ω changes component ω1 → −ω1.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Basic facts about Quaternions and deﬁnition of QFT 2D complex FT and QFT (Images: T. Bülow, thesis [4])

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Applications of QFT, discrete and fast versions

partial differential systems colour image processing ﬁltering disparity estimation (two images differ by local translations) texture segmentation wide ranging higher dimensional generalizations

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Discrete and fast QFT: Pei, Ding, Chang (2001) [13]

Discrete QFT

M−1 N−1 X X −ix1ω1/M −jx2ω2/N FDQF T {f}(ω) = e f(x) e . (52) x1=0 x2=0

Fast QFT The simple complex forms Z Z −j x·ω 2 −j x·(Ue ω) 2 fˆ = fˆ− + fˆ+, = f−e d x, + f+e 1 d x, (53) 2 2 R R show that the QFT can be implemented with 2 complex M × N 2D discrete FTs. This requires 2MN log2 MN (54) real number multiplications.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Application to image processing

Application: Image transformations

2 2 2 The QFT of a quaternion function f ∈ L (R ; H) with a GL(R ) transformation A (stretches, reﬂections, rotations) of its vector argument x is

−1 ˆ −1 ˆ −1 f\(Ax)(u) = | det A | { f−(A u) + f+(Ue1 A Ue1 u) } , (55)

where A−1 is the adjoint of the inverse of A.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Properties of QFT

Standard properties: Linearity, shift, modulation, dilation, Plancherel theorem and Parseval theorem (signal energy) 1 kfk = kFq{f}k. (56) 2π

2 2 2 Table: Special QFT properties. Quat. Funct. f ∈ L (R ; H), with x, u ∈ R , m, n ∈ N0. Property Quat. Funct. QFT ∂m+n m ˆ n Part. deriv. ∂xm∂yn f(x)(iu) f(u)(jv) m n m ∂m+n ˆ n Moments of x, y x y f(x) i ∂um∂vn f(u) j Powers∗ of i, j imf(x) jn imfˆ(u) jn

∗ Easy with correct order!

± split and QFT commute: Fq{f±} = Fq{f}±.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Properties of QFT

Integration of parts 2 With the vector differential a · ∇ = a1∂1 + a2∂2, with arbitrary constant a ∈ R , 2 2 g, h ∈ L (R , H) Z Z a·x=∞ Z g(x)[a·∇h(x)]d2x = g(x)h(x)dx − [a·∇g(x)]h(x)d2x. (57) R2 R a·x=−∞ R2

Modulus identities 2 2 2 2 Due to |q| = |q−| + |q+| we get for f : R → H the following identities 2 2 2 |f(x)| = |f−(x)| + |f+(x)| , (58)

2 2 2 |Fq{f}(ω)| = |Fq{f−}(ω)| + |Fq{f+}(ω)| . (59)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Properties of QFT

QFT of vector differentials 2 Using the split f = f− + f+ we get the QFTs of the split parts. Let b ∈ R be an arbitrary constant vector.

Fq{b · ∇f−}(ω) = b · ωFq{f−}(ω) j = i b · ωFq{f−}(ω), (60)

Fq{b · ∇f+}(ω) = (b ·Ue1 ω) Fq{f+}(ω) j = i (b ·Ue2 ω) Fq{f+}(ω), (61)

Fq{(Ue1 b · ∇)f+}(ω) = b · ω Fq{f+}(ω) j, (62)

Fq{(Ue2 b · ∇)f+}(ω) = i b · ω Fq{f+}(ω). (63)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications

Gaussian quaternion ﬁlter (GF)

h(x) = g(x) ei2πx1ω01 ej2πx2ω02

2D Gaussian amplitude:

2 2 1 x1 x2 1 − 2 2 + 2 σ1 σ2 g(x) = 3 e . (2π) 2 σ1σ2

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Application of QFT to texture segmentation (T. Bülow, thesis [4])

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Application of QFT to disparity estimation (T. Bülow, thesis [4])

QFT method overcomes directional limitations of complex methods.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Properties of QFT and applications Uncertainty Principles for QFT (componentwise and directional)

Right sided quaternion Fourier transform (QFT) in H Z −ix ω −jx ω 2 Fr{f}(ω) = f(x) e 1 1 e 2 2 d x (64) 2 R Componentwise (k = 1, 2) uncertainty [Mawardi&Hitzer, 2008] Z Z 1 2 2 2 1 2 2 2 1 xk|f(x)| d x ωk|f(ω)| d ω ≥ . (65) 2 2 2 2 2 kfk R (2π) kfk R 4 Equality holds if and only if f is a Gaussian quaternion function.

Directional QFT Uncertainty Principle [Hitzer, 2008] 2 For two arbitrary constant vectors a, b ∈ R (selecting two directions), and 2 2 1/2 2 2 0 f ∈ L (R , H), |x| f ∈ L (R , H), b = −b1e1 + b2e2 we obtain Z Z 2 2 2 2 2 2 (a · x) |f(x)| d x (b · ω) |Fq{f}(ω)| d ω 2 2 R R 2 (2π) 2 4 0 2 4 ≥ (a · b) kf−k + (a · b ) kf+k , (66) 4

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA of spacetime and definition of SFT Motion in time: Video sequences, flow fields, ...

GA of spacetime (STA)

2 2 2 {et, e1, e2, e3}, e1 = e2 = e3 = 1. (67) 2 2 2 et = −1, i3 = e1e2e3, i3 = −1, ist = ete1e2e3, ist = −1, (68)

with time vector et, 3D space volume i3, hyper volume of spacetime ist.

Isomorphism: Volume-time subalgebra to Quaternions

{1, et, i3, ist} ←→ {1, i, j, k} (69)

NB: Does not work for G2,1 or G4,1, but works for conformal model of spacetime G5,2.

∗ −1 Split → Spacetime Split (et = etist = −etist = −eteti3 = i3)

1 ∗ f± = (f ± etfe ) (70) 2 t

Time direction et determines complimentary 3D space i3 as well!

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA of spacetime and deﬁnition of SFT

QFT → Spacetime Fourier transform (SFT): f → FSFT {f}

Z −et ts −i3~x·~u 4 FSFT {f}(u) = f(u) = e f(x) e d x . (71) 3,1 R with 3,1 3 spacetime vectors x = tet + ~x ∈ R , ~x = xe1 + ye2 + ze3 ∈ R spacetime volume d4x = dtdxdydz 3,1 3 spacetime frequency vectors u = set + ~u ∈ R , ~u = ue1 + ve2 + we3 ∈ R 3,1 maps 16D spacetime algebra functions f : R → G3,1 = Cl3,1 3,1 to 16D spacetime spectrum functions f : R → Cl3,1 R −i ~x·~u 3 Part f(x) e 3 d ~x ... GA FT of G3. Reversion replaced by principal involution (reversion and et → −et).

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Applications of SFT Application: Multivector Wave Packets in Space and Time

SFT = Sum of Right/Left Propagating Multivector Wave Packets

Z Z −i3( ~x·~u− ts ) 4 −i3( ~x·~u+ ts ) 4 f = f + + f − = f+ e d x + f− e d x. (72) 3,1 3,1 R R

Directional 4D Spacetime Uncertainty for Multivector Wave Packets 3,1 For two arbitrary constant spacetime vectors a, b ∈ R (selecting two directions), and 2 3,1 1/2 2 3,1 f ∈ L (R , G3,1), |x| f ∈ L (R , G3,1) we obtain Z Z 2 2 4 2 2 4 (att − ~a · ~x) |f(x)| d x (btωt −~b · ~ω) |F{f}(ω)| d ω 3,1 3,1 R R 4 (2π) h 2 4 2 4i ≥ (atbt − ~a ·~b) kf−k + (atbt + ~a ·~b) kf+k , 4

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Applications of SFT Spacetime signal transformations

Application: Space & Time Signal Transformations

2 3,1 3,1 The SFT of a spacetime function f ∈ L (R ; G3,1) with a GL(R ) transformation A (stretches, reﬂections, rotations, acceleration, boost) of its spacetime vector argument x is

−1 −1 −1 FSFT {f(Ax)}(u) = | det A | { f +(Uet A Uet u) + f −(A u) } , (73)

where A−1 is the adjoint of the inverse of A.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelet basics and GA wavelet transformation GA Wavelet Basics [Mawardi&Hitzer, 2007]

Transformation group SIM(3) We apply 3D (elements of G = SIM(3)) translations, scaling and rotations to a mother 3 wavelet ψ : R → G3 = Cl3,0 1 x − b ψ(x) 7−→ ψ (x) = ψ(r−1( )) (74) a,θ,b a3/2 θ a

Wavelet admissibility (condition on ψ)

Z e ψb(ω)ψb(ω) 3 Cψ = d ω. (75) 3 3 R |ω| 3 is an invertible multivector constant and ﬁnite at a.e. ω ∈ R , GA FT ψb = F{ψ}.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelet basics and GA wavelet transformation GA Wavelet Transformation

Deﬁnition (GA wavelet transformation)

2 3 2 Tψ : L (R ; Cl3,0) → L (G; Cl3,0) Z 3 f → Tψf(a, θ, b) = f(x)ψa,^, (x) d x, (76) 3 θ b R 2 3 ψ ∈ L (R ; Cl3,0) ... GA mother wavelet, f ... image, multivector ﬁeld, data, etc. Properties: Locality (different from global FTs), left linearity; translation, dilation, rotation covariance. Inversion of GA wavelet transform

2 3 −1 Any f ∈ L (R ; Cl3,0) can be decomposed as (Cψ ... admissibility) Z −1 3 f(x) = Tψf(a, b, θ) ψa,θ,b Cψ dµd b, (77) G

with left Haar measure on G = SIM(3): dλ(a, θ, b) = dµ(a, θ)d3b

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelet basics and GA wavelet transformation Applications (expected) of GA wavelets

multi-dimensional image/signal processing JPEG local spherical harmonics (lighting) geological exploration seismology local multivector pattern matching and the like

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelet basics and GA wavelet transformation Notation: Inner products and norms

We deﬁne the inner product of f, g : G → Cl3,0 by Z (f, g) = f(a, θ, b)g(^a, θ, b) dλ(a, θ, b), (78) G 2 and the L (G; Cl3,0)-norm Z kfk2 = h(f, f)i = |f(a, θ, b)|2dλ. (79) G

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelets and uncertainty limits, example of GA Gabor wavelets GA Wavelet Uncertainty

Generalized GA wavelet uncertainty principle

Let ψ be a Clifford algebra wavelet that satisﬁes the admissibility condition. Then for 2 3 every f ∈ L (R ; Cl3,0), the following inequality holds

2 ˆ ˆ kbTψf(a, θ, b)k 2 Cψ ∗ (ωff, ωff) 2 3 L (G;Cl3,0) L (R ;Cl3,0) 3(2π)3 h i2 ≥ C ∗ (f, f) 2 3 . (80) 4 ψ L (R ;Cl3,0)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelets and uncertainty limits, example of GA Gabor wavelets Uncertainty principle for scalar admissibility constant

For scalar admissibility constant Cψ we get

Uncertainty principle for GA wavelet

Let ψ be a Clifford algebra wavelet with scalar admissibility constant. Then for every 2 3 f ∈ L (R ; Cl3,0), the following inequality holds

3 2 ˆ 2 (2π) 4 kb Tψf(a, θ, b)k 2 kωfk 2 3 ≥ 3Cψ kfk 2 3 . (81) L (G;Cl3,0) L (R ;Cl3,0) 4 L (R ;Cl3,0) NB1: This shows indeed, that the previous inequality (80) represents a multivector generalization of the uncertainty principle of inequality (81) for Clifford wavelets with scalar admissibility constant. NB2: Compare with (direction independent) uncertainty principle for GA FT.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

GA wavelets and uncertainty limits, example of GA Gabor wavelets GA Gabor Wavelets

Example: GA Gabor Wavelets   c i ω ·x − 1 (σ2u2+σ2u2+σ2w2) ψ (x) = g(x) e 3 0 − e 2 1 0 2 0 3 0   | {z } constant

2 2 2 1 x1 x2 x3 1 − 2 2 + 2 + 2 σ1 σ2 σ3 3D Gaussian: g(x) = 3 e . (2π) 2 σ1σ2σ3

Uncertainty principle for GA Gabor wavelet

c 2 3 Let ψ be an admissible GA Gabor wavelet. Assume f ∈ L (R ; Cl3,0), then the following inequality holds

3 2 ˆ 2 (2π) 4 kb Tψc f(a, θ, b)k 2 kωfk 2 3 ≥ 3Cψc kfk . (82) L (G;Cl3,0) L (R ;Cl3,0) 4

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

Conclusion

Geometric calculus allows to construct a variety of GA Fourier transformations. Especially quaternion Fourier transformations are well studied. We studied the space time Fourier transformation giving rise to a multivector wave packet decomposition. Discrete and fast versions exist. Uncertainty limits established (principle bounds to accuracy). A diverse range of applications exists. For the future the application of local GA wavelets may overcome the limitations of global GA Fourier transformations.

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1999. E. Hitzer, Multivector Differential Calculus, AACA, 12(2), pp. 135–182 (2002).

T. A. Ell, Quat.-Four. Transf. for Analysis of Two-dim. Linear Time-Invariant Partial Diff. Systems, Proc. 32nd IEEE Conf. on Decision and Control, Dec. 1993, pp. 1830-1841. T. Bülow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images, Ph. D. thesis, University of Kiel (1999). T. Bülow, M. Felsberg and G. Sommer, Non-commutative Hypercomplex Four. Transf. of Multidim. Signals, in G. Sommer (ed.), Geom. Comp. with Cliff. Alg., Theor. Found. and Appl. in Comp. Vis. and Rob., Springer, (2001), 187–207. Rosenhahn B. and Sommer G., Pose estim. of free-form obj., Proc. of the Eur. Conf. on Comp. Vis. ECCV ’04, Part I, T. Pajdla and J. Matas (Eds.), Springer, Berlin, LNCS 3021, 414–427. http://www.mpi-inf.mpg.de/~rosenhahn/publications/eccv04F.pdf

J. Ebling, G. Scheuermann, Clifford Fourier Transform on Vector Fields, IEEE Trans. on Vis. and Comp. Graph., 11 (4), (July/August 2005). J. Ebling, G. Scheuermann, Clifford Convolution And Pattern Matching On Vector Fields, Proc 14th IEEE Vis. 2003 (VIS’03), p.26, (October 22-24, 2003)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

B. Mawardi, E. Hitzer. Clifford Fourier Transform and Uncertainty Principle for the Clifford Geometric Algebra Cl3,0, Adv. App. Cliff. Alg., 16(1), pp. 41-61, 2006. E. Hitzer, B. Mawardi, Cliff. FT on Multivector Fields and Unc. Princ. for Dimensions n = 2 (mod 4) and n = 3 (mod 4), Proc. of ICCA7, Toulouse, 2005. B. Mawardi, E. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion FT , in press, CAMWA, 2008. E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. in App. Cliff. Alg., 17, pp. 497–517, 2007. S.C. Pei, J.J. Ding, J.H. Chang, Efﬁcient Implementation of Quaternion Fourier Transform, Convolution, and Correlation by 2-D Complex FFT , IEEE TRANS. on SIG. PROC., Vol. 49, No. 11, p. 2783 (2001).

B. Mawardi, E. Hitzer, Cliff. Alg. Cl3,0-valued Wavelet Transf., Cliff. Wavelet Uncertainty Inequality and Cliff. Gabor Wavelets, IJWMIP, 2007. F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Vol. 76 of Research Notes in Mathematics, Pitman Advanced Publishing Program, 1982. C. Li, A. McIntosh and T. Qian, Cliff. Algs., Fourier Transf. and Sing. Conv. Operators On Lipschitz Surfaces, Rev. Mat. Iberoam., 10 (3), (1994), 665–695. Other publications: http://sinai.mech.fukui-u.ac.jp/gcj/pubs.html

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion

News about Clifford Geometric Algebra

GA-Net Electronic newsletter: http://sinai.mech.fukui-u.ac.jp/GA-Net/index.html

GA-Net Updates (blog) Immediate access to latest GA news: http://gaupdate.wordpress.com/

Soli Deo Gloria

J. S. Bach (Leipzig)

E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations