Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Tutorial on Fourier and Wavelet Transformations in Geometric Algebra 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 2 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 2 f0; 1; 2; 3; 12; 23; 31; 123;:::; 12 : : : ng, MA 2 R. Reverse of M 2 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 2 Gp;q is defined as M ∗ Ne = hMNei = hMNei0 (3) P For M; Ne 2 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 jMj of a multivector M 2 Gn = Gn;0 is defined as n 2 X 2 jMj = 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 = fx 2 R jx ^ B = 0g: Its dual blade ∗ −1 B = Bin ? describes the complimentary vector subspace VB . pseudoscalar in commutes for n = 3(mod 4) in M = M in; 8M 2 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 = jf(x)j2dnx: (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 2 R ) f(x + a) − f(x) a · rf(x) = lim ; (9) !0 NB: a · r scalar. Vector derivative r can be expanded as n X @ r = e @ ;@ = e · r = ; (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 · rf1 = a rf1 = 3; (n = 3) 2 f2 = x a · rf2 = 2a · x rf2 = 2x f = jxj a·x 3 a · rf3 = jxj rf3 = x=jxj f = x · hAi , 0≤k≤n 4 k a · rf4 = a · hAik rf4 = khAik −1 2 f5 = log r a · rf = a·r rf5 = r = r=r : 5 r2 r = x − x0 r = jrj 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 f1 = x a · rf1 = a rf1 = 3; (n = 3) 2 f2 = x a · rf2 = 2a · x rf2 = 2x f = jxj a·x 3 a · rf3 = jxj rf3 = x=jxj f = x · hAi , 0≤k≤n 4 k a · rf4 = a · hAik rf4 = khAik −1 2 f5 = log r a · rf = a·r rf5 = r = r=r : 5 r2 r = x − x0 r = jrj 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 f1 = x a · rf1 = a rf1 = 3; (n = 3) 2 f2 = x a · rf2 = 2a · x rf2 = 2x f = jxj a·x 3 a · rf3 = jxj rf3 = x=jxj f = x · hAi , 0≤k≤n 4 k a · rf4 = a · hAik rf4 = khAik −1 2 f5 = log r a · rf = a·r rf5 = r = r=r : 5 r2 r = x − x0 r = jrj 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 f1 = x a · rf1 = a rf1 = 3; (n = 3) 2 f2 = x a · rf2 = 2a · x rf2 = 2x f = jxj a·x 3 a · rf3 = jxj rf3 = x=jxj f = x · hAi , 0≤k≤n 4 k a · rf4 = a · hAik rf4 = khAik −1 2 f5 = log r a · rf = a·r rf5 = r = r=r : 5 r2 r = x − x0 r = jrj 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) rf = ra (a · rf) (11) Sum rules, product rules exist. Modification by non-commutativity: r_ fg_ 6= fr_ g_ n _ _ _ _ _ X r(fg) = (rf)g + rfg_ = (rf)g + ekf(@kg): (12) k=1 Vector differential / derivative of f(x) = g(λ(x)); λ(x) 2 R : @g @g a · rf = fa · rλ(x)g ; rf = (rλ) : (13) @λ @λ @f Example: For a = ek (1 ≤ k ≤ n) ek · rf = @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 2 L (R) \ L (R), the Fourier transform of f is the function Fffg: R ! C = G0;1 Z Fffg(!) = f(x) e−i!x dx; (14) R i imaginary unit: i2 = −1. Inverse FT 1 Z f(x) = F −1[Fffg(!)] = Fffg(!) ei!x d!: (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 2 L (R ; Gn) Definition (with pseudoscalars in dims.2,3,6,7,10,11, ...) n The GA Fourier transform Fffg: R ! Gn = Cln;0; n =2 ; 3(mod 4) is given by Z Fffg(!) = f(x) e−in!·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[Fffg(!)] = Fffg(!) ein!·x dn!: (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 −i!x FCffg(!) = f(x) e dx (18) R Clifford Geometric Algebra (GA) FT in G3 (i ! i3) Z −i3~!·~x 3 F 3 ffg(~!) = f(~x) e d ~x (19) G 3 R GA FT in Gn;0; G0;n0 (i ! in) Z −in!·x n F n ffg(!) = f(x) e d x (20) G n R QFT in H (i ! i; j) Z F ffg(u) = f^(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 FST Affg(u) = f(u) = e f(x) e d ~xdt (22) 3;1 R E.