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
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.
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
Multivector Functions Multivectors, blades, reverse, scalar product Multivector M G p,q = Cl p,q, p + q = n, has k-vector parts (0 k n) scalar, vector, bi-vector,..., pseudoscalar M = A M A e A = M + M 1 + M 2 +... + M n, (1) blade index A {0, 1, 2, 3, 12, 23, 31, 123,..., 12... n}, M A R. Reverse of M G p,q M = n k=0 replaces complex conjugation/quaternion conjugation ( 1) k(k 1) 2 M k. (2) The scalar product of two multivectors M, Ñ Gp,q is defined as M Ñ = MÑ = MÑ 0 (3) For M, Ñ Gn = G n,0 we get M Ñ = A M AN A.
Multivector Functions Modulus, blade subspace, pseudoscalar The modulus M of a multivector M G n = G n,0 is defined as M 2 = M M = n MA 2. (4) A=1 For n = 2(mod 4), n = 3(mod 4) pseudoscalar i n = e 1 e 2... e n A blade B describes a vector subspace Its dual blade i 2 n = 1. (5) V B = {x R p,q x B = 0}. B = Bi 1 n describes the complimentary vector subspace V B. pseudoscalar i n commutes for n = 3(mod 4) i n M = M i n, M G n,0.
Multivector Functions Multivector functions Multivector valued function f : R p,q G p,q = Cl p,q, p + q = n, has 2 n blade components n f(x) = f A (x)e A. (6) A=1 We define the inner product of R n Cl n,0 functions f, g by (f, g) = f(x) g(x) d n x = e A ẽ B f A (x)g B (x) d R n A,B R n x, (7) n and the L 2 (R n ; Cl n,0 )-norm f 2 = (f, f) = f(x) 2 d n x. (8) R n
Multivector Functions Vector Differential & Vector Derivative Vector differential of f defined (any const. a R p,q ) NB: a scalar. Vector derivative can be expanded as f(x + ɛa) f(x) a f(x) = lim, (9) ɛ 0 ɛ = n k=1 both coordinate independent! e k k, k = e k = x k, (10)
Multivector Functions Geometric Calculus Examples MV Functions f 1 = x f 2 = x 2 f 3 = x f 4 = x A k, 0 k n f 5 = log r r = x x 0 r = r V-Differentials a f 1 = a a f 2 = 2a x a f 3 = a x x a f 4 = a A k a f 5 = a r r 2 V-Derivatives f 1 = 3, (n = 3) f 2 = 2x f 3 = x/ x f 4 = k A k f 5 = r 1 = r/r 2. References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.
Multivector Functions Geometric Calculus Examples MV Functions f 1 = x f 2 = x 2 f 3 = x f 4 = x A k, 0 k n f 5 = log r r = x x 0 r = r V-Differentials a f 1 = a a f 2 = 2a x a f 3 = a x x a f 4 = a A k a f 5 = a r r 2 V-Derivatives f 1 = 3, (n = 3) f 2 = 2x f 3 = x/ x f 4 = k A k f 5 = r 1 = r/r 2. References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.
Multivector Functions Geometric Calculus Examples MV Functions f 1 = x f 2 = x 2 f 3 = x f 4 = x A k, 0 k n f 5 = log r r = x x 0 r = r V-Differentials a f 1 = a a f 2 = 2a x a f 3 = a x x a f 4 = a A k a f 5 = a r r 2 V-Derivatives f 1 = 3, (n = 3) f 2 = 2x f 3 = x/ x f 4 = k A k f 5 = r 1 = r/r 2. References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.
Multivector Functions Geometric Calculus Examples MV Functions f 1 = x f 2 = x 2 f 3 = x f 4 = x A k, 0 k n f 5 = log r r = x x 0 r = r V-Differentials a f 1 = a a f 2 = 2a x a f 3 = a x x a f 4 = a A k a f 5 = a r r 2 V-Derivatives f 1 = 3, (n = 3) f 2 = 2x f 3 = x/ x f 4 = k A k f 5 = r 1 = r/r 2. References 1) Hestenes & Sobczyk, Clifford Algebra to Geometric Calculus, 1984. 2) Hitzer, Vector Differential Calculus, 2002.
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: fġ f ġ (fg) = ( f)g + fġ = ( f)g + n e k f( k g). (12) k=1 Vector differential / derivative of f(x) = g(λ(x)), λ(x) R : a f = {a λ(x)} g λ, Example: For a = e k (1 k n) g f = ( λ) λ. (13) e k f = k f = ( k λ) f λ.
Classical scalar complex Fourier Transformations Definition (1D Cassical FT) For an integrable function f L 2 (R) L 1 (R), the Fourier transform of f is the function F{f}: R C = G 0,1 F{f}(ω) = f(x) e iωx dx, (14) R i imaginary unit: i 2 = 1. Inverse FT f(x) = F 1 [F{f}(ω)] = 1 F{f}(ω) e iωx dω. (15) 2π R
Examples of continuous and discrete 1D FT [Images: Wikipedia]
Definition and properties of GA FTs Geometric Algebra Fourier Transformation (GA FT) complex geometric complex unit i Geometric roots of 1, e.g. pseudoscalars i n, n = 2, 3(mod 4). complex f multivector function f L 2 (R n, G n) Definition (with pseudoscalars in dims. 2,3,6,7,10,11,...) The GA Fourier transform F{f}: R n G n = Cl n,0, n = 2, 3(mod 4) is given by F{f}(ω) = f(x) e inω x d n x, (16) R n for multivector functions f: R n G n. NB: Also possible for Cl 0,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 f(x) = F 1 [F{f}(ω)] = 1 (2π) n R 3 F{f}(ω) e inω x d n ω. (17)
Definition and properties of GA FTs Overview of complex, Clifford, quaternion, spacetime FTs Complex Fourier Transformation (FT) F C {f}(ω) = f(x) e iωx dx (18) R Clifford Geometric Algebra (GA) FT in G 3 (i i 3 ) F G3 {f}( ω) = f( x) e i3 ω x d 3 x (19) R 3 GA FT in G n,0, G 0,n (i i n) F Gn {f}(ω) = f(x) e inω x d n x (20) R n QFT in H (i i, j) F H {f}(u) = ˆf(u) = e ixu f(x) e jyv d 2 x (21) R 2 SFT in spacetime algebra G 3,1 (i e t, j i 3 ) F ST A {f}(u) = f(u) = e e t ts f(x) e i3 x u d 3 xdt (22) R 3,1
Definition and properties of GA FTs Properties of the GA FT in G n, n = 3 (mod 4) General properties: Linearity, shift, modulation (ω-shift), scaling, convolution, Plancherel theorem and Parseval theorem f = 1 F{f}. (23) (2π) n/2 Table: Unique properties of GA FT Multiv. Funct. f L 2 (R n, G n), a R n, m N. Property Multiv. Funct. GA FT Vec. diff. (a ) m f(x) (a ω) m F{f}(ω) i m n Moments of x (a x) m f(x) i m n (a ω)m F{f}(ω) x m f(x) i m n m ω F{f}(ω) Vec. deriv. f(x)x m m f(x) n i m n F{f}(ω) m ω i m ω m F{f}(ω) dual moments f(x) m i m n F{f}(ω) ωm NB: Standard complex FT only gives coordinate moments and partial derivative properties.
Definition and properties of GA FTs Properties of the GA FT in G n, 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 ω-shift f(x) e inω 0 x F{f}(ω ω 0 ) 1 Scaling f(ax) a n F{f}( ω a ) Convolution (f g)(x) F{f}( ω) F{g odd }(ω) +F{f}(ω) F{g even}(ω) Vec. diff. (a ) m f(x) (a ω) m F{f}(ω) i m n (a x) m f(x) (a ω) m F{f}(ω) i m n Moments of x x m f(x) m ω F{f}(ω) im n Vec. deriv. f(x) x m m f(x) F{f}(( 1) m ω) m ω i m n ω m F{f}(ω) i m n f(x) m F{f}(( 1) m ω) ω m i m n Plancherel R n f 1 (x) f 2 (x) d n 1 x (2π) n R n F{f 1 }(ω) F{f2 }(ω) d n ω 1 sc. Parseval f (2π) n/2 F{f}
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
Definition and properties of GA FTs Relation with complex FT, Discrete GA FT, Fast GA FT Example of multivector signal decomposition f = [f 0 + f 123 i 3 ] + [f 1 + f 23 i 3 ]e 1 + [f 2 + f 31 i 3 ]e 2 + [f 3 + f 12 i 3 ]e 3 (24) corresponds to 4 complex signals. Implementation of GA FT by 4 complex FTs F G3 {f} = F[f 0 + f 123 i 3 ] + F[f 1 + f 23 i 3 ]e 1 + F[f 2 + f 31 i 3 ]e 2 + F[f 3 + f 12 i 3 ]e 3 (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.
Definition and properties of GA FTs Application: Fourier Descriptor Repr. of Shape (B. Rosenhahn et al [6]) Set of 3D sample points is projected along e 1, e 2, e 3 : f l n 1,n 2, l = 1, 2, 3. Use of rotors for 2D discrete rotor FT ( N 1,2 n 1,2, k 1,2 N 1,2, N = 2N + 1) R k 1,n 1 1,l = exp( 2π N 1 k 1 n 1 e l i 3 ), R k 2,n 2 2,l =... (26) Sample points f l n 1,n 2 give surface phase vectors p l k 1,k 2 = 1 N 1 N 2 Surface estimation 3 F (Φ 1, Φ 2 ) = f l (Φ 1, Φ 2 )e l = l=1 n 1,n2 f l n 1,n 2 Rk 1,n 1 1,l 2,l e l (27) R k 2,n 2 3 R k1,φ1 1,l R k 2,Φ 2 2,l p l k 1,k Rk 1,Φ 1 2 1,l R k 2,Φ 2 2,l (28) l=1 k 1,k 2
Definition and properties of GA FTs Application: Fourier Descriptor Repr. of Shape (B. Rosenhahn et al [6])
Definition and properties of GA FTs GA FT of Convolution Convolution The GA FT of the convolution of f(x) and g(x) (f g)(x) = f(y)g(x y) d 3 y, (29) R 3 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 i n G n i n M M i n, for M G n. (31) that F{f g}(ω) = F{f}( ω)f{g odd }(ω) + F{f}(ω)F{g even}(ω) (32)
Definition and properties of GA FTs GA FT of Correlation Correlation The GA FT of the correlation of f(x) and g(x) (f g)(x) = f(y)g(x + y) d 3 y, (33) R 3 equals for n = 3(mod 4) the product of the GA FTs of f (x) = f( x) and g(x): F{f g}(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 i n G n i n M M i n, for M G n. (35) that F{f g}(ω) = F{f }( ω)f{g odd }(ω) + F{f }(ω)f{g even}(ω) (36)
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)
Definition and properties of GA FTs Application: Vector Pattern Matching (J. Ebling, G. Scheuermann [7, 8]) Convolution of 3 3 3 = 3 3 (red), 5 3 (yellow), 8 3 (green) rotation patterns with gas furnace chamber flow field.
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.
Definition and properties of GA FTs Application of GAFT: Uncertainty Principles Directional Uncertainty Principle [Hitzer&Mawardi, Proc. ICCA7] Multivect. funct. f L 2 (R n, G n), n = 2, 3 (mod 4) GA FT F{f}(ω). arbitrary const. vectors a, b R n (a x) 2 f(x) 2 d n x R n 1 f 2 1 Rn (2π) n f 2 (b ω) 2 F{f}(ω) 2 d n ω Equality (minimal bound) for optimal Gaussian multivector functions C 0 G n arbitrary const. multivector, 0 < k R. Uncertainty principle (direction independent) (a b)2 4 f(x) = C 0 e k x2, (37) For f L 2 (R n, G n), n = 2, 3 (mod 4) we obtain from the dir. UP that 1 f 2 x 2 f(x) 2 d n 1 x R n (2π) n f 2 ω 2 F{f}(ω) 2 d n ω n R n 4. (38)
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, i 2 = j 2 = k 2 = ijk = 1. (39) Quaternion q = q r + q i i + q j j + q k k H, q r, q i, q j, q k R (40) has quaternion conjugate (reversion in Cl + 3,0 ) q = q r q i i q j j q k k, (41) Leads to norm of q H q = q q = qr 2 + q2 i + q2 j + q2 k, pq = p q. (42) Scalar part Sc(q) = q r = 1 (q + q). (43) 2
Basic facts about Quaternions and definition of QFT ± Split of quaternions [Hitzer, AACA 2007] Convenient split of quaternions q = q + + q, q ± = 1 (q ± iqj). (44) 2 Explicitly in real components q r, q i, q j, q k R using (39): q ± = {q r ± q k + i(q i q j )} 1 ± k 2 Consequence: modulus identity Property: Scalar part of mixed split product = 1 ± k {q r ± q k + j(q j q i )}. (45) 2 q 2 = q 2 + q + 2. (46) 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)
Basic facts about Quaternions and definition of QFT Definition of quaternion FT (QFT) F q{f}(ω) = ˆf(ω) = e ix 1ω 1 f(x) e jx 2ω 2 d 2 x. R 2 (48) Linearity leads to F q{f}(ω) = F q{f + f + }(ω) = F q{f }(ω) + F q{f + }(ω). (49) Remark: Other variations exist (see later). Simple complex forms for QFT of f ± The QFT of f ± split parts of quaternion function f L 2 (R 2, H) have simple complex forms ˆf ± = f ± e j(x 2ω 2 x 1 ω 1 ) d 2 x = R 2 e i(x 1ω 1 x 2 ω 2 ) f ± d 2 x. R 2 (50) Free of coordinates: ˆf = f e j x ω d 2 x, R 2 ˆf+ = f + e j x (U e ω) 1 d 2 x, R 2 (51) the reflection Ue 1 ω changes component ω 1 ω 1.
Basic facts about Quaternions and definition of QFT 2D complex FT and QFT (Images: T. Bülow, thesis [4])
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
Properties of QFT and applications Discrete and fast QFT: Pei, Ding, Chang (2001) [13] Discrete QFT F DQF T {f}(ω) = M 1 N 1 x 1 =0 x 2 =0 e ix 1ω 1 /M f(x) e jx 2ω 2 /N. (52) Fast QFT The simple complex forms ˆf = ˆf + ˆf +, = f e j x ω d 2 x, + f + e j x (U e ω) 1 d 2 x, (53) R 2 R 2 show that the QFT can be implemented with 2 complex M N 2D discrete FTs. This requires 2MN log 2 MN (54) real number multiplications.
Properties of QFT and applications Application to image processing Application: Image transformations The QFT of a quaternion function f L 2 (R 2 ; H) with a GL(R 2 ) transformation A (stretches, reflections, rotations) of its vector argument x is f(ax)(u) = det A 1 { ˆf (A 1 u) + ˆf + (Ue 1 A 1 Ue 1 u) }, (55) where A 1 is the adjoint of the inverse of A.
Properties of QFT and applications Properties of QFT Standard properties: Linearity, shift, modulation, dilation, Plancherel theorem and Parseval theorem (signal energy) f = 1 Fq{f}. (56) 2π Table: Special QFT properties. Quat. Funct. f L 2 (R 2 ; H), with x, u R 2, m, n N 0. Easy with correct order! Property Quat. Funct. QFT m+n Part. deriv. x m y n f(x) (iu) m ˆf(u)(jv) n Moments of x, y x m y n f(x) i m m+n u m v n j n Powers of i, j i m f(x) j n i m ˆf(u) j n ± split and QFT commute: F q{f ± } = F q{f} ±.
Properties of QFT and applications Properties of QFT Integration of parts With the vector differential a = a 1 1 + a 2 2, with arbitrary constant a R 2, g, h L 2 (R 2, H) [ g(x)[a h(x)]d 2 x = R 2 g(x)h(x)dx R ] a x= a x= R 2 [a g(x)]h(x)d 2 x. (57) Modulus identities Due to q 2 = q 2 + q + 2 we get for f : R 2 H the following identities f(x) 2 = f (x) 2 + f + (x) 2, (58) F q{f}(ω) 2 = F q{f }(ω) 2 + F q{f + }(ω) 2. (59)
Properties of QFT and applications Properties of QFT QFT of vector differentials Using the split f = f + f + we get the QFTs of the split parts. Let b R 2 be an arbitrary constant vector. F q{b f }(ω) = b ωf q{f }(ω) j = i b ωf q{f }(ω), (60) F q{b f + }(ω) = (b Ue 1 ω) F q{f + }(ω) j = i (b Ue 2 ω) F q{f + }(ω), (61) F q{(ue 1 b )f + }(ω) = b ω F q{f + }(ω) j, (62) F q{(ue 2 b )f + }(ω) = i b ω F q{f + }(ω). (63)
Properties of QFT and applications Gaussian quaternion filter (GF) h(x) = g(x) e i2πx 1ω 01 e j2πx 2ω 02 2D Gaussian amplitude: g(x) = ( 1 e 1 x 2 ) 1 2 σ 2 + x2 2 (2π) 3 1 σ 2 2. 2 σ 1 σ 2
Properties of QFT and applications Application of QFT to texture segmentation (T. Bülow, thesis [4])
Properties of QFT and applications Application of QFT to disparity estimation (T. Bülow, thesis [4]) QFT method overcomes directional limitations of complex methods.
Properties of QFT and applications Uncertainty Principles for QFT (componentwise and directional) Right sided quaternion Fourier transform (QFT) in H F r{f}(ω) = f(x) e ix 1ω 1 e jx 2ω 2 d 2 x (64) R 2 Componentwise (k = 1, 2) uncertainty [Mawardi&Hitzer, 2008] 1 f 2 x 2 R 2 k f(x) 2 d 2 1 x (2π) 2 f 2 ω 2 R 2 k f(ω) 2 d 2 ω 1 4. (65) Equality holds if and only if f is a Gaussian quaternion function. Directional QFT Uncertainty Principle [Hitzer, 2008] For two arbitrary constant vectors a, b R 2 (selecting two directions), and f L 2 (R 2, H), x 1/2 f L 2 (R 2, H), b = b 1 e 1 + b 2 e 2 we obtain (a x) 2 f(x) 2 d 2 x (b ω) 2 F q{f}(ω) 2 d 2 ω R 2 R 2 (2π)2 4 [ (a b) 2 f 4 + (a b ) 2 f + 4], (66)
GA of spacetime and definition of SFT Motion in time: Video sequences, flow fields,... GA of spacetime (STA) {e t, e 1, e 2, e 3 }, e 2 1 = e 2 2 = e 2 3 = 1. (67) e 2 t = 1, i 3 = e 1 e 2 e 3, i 2 3 = 1, ist = ete 1e 2 e 3, i 2 st = 1, (68) with time vector e t, 3D space volume i 3, hyper volume of spacetime i st. Isomorphism: Volume-time subalgebra to Quaternions {1, e t, i 3, i st} {1, i, j, k} (69) NB: Does not work for G 2,1 or G 4,1, but works for conformal model of spacetime G 5,2. Split Spacetime Split (e t = eti 1 st = e ti st = e te ti 3 = i 3 ) f ± = 1 2 (f ± etfe t ) (70) Time direction e t determines complimentary 3D space i 3 as well!
GA of spacetime and definition of SFT QFT Spacetime Fourier transform (SFT): f F SF T {f} with F SF T {f}(u) = f(u) = e e t ts f(x) e i3 x u d 4 x. (71) R 3,1 spacetime vectors x = te t + x R 3,1, x = xe 1 + ye 2 + ze 3 R 3 spacetime volume d 4 x = dtdxdydz spacetime frequency vectors u = se t + u R 3,1, u = ue 1 + ve 2 + we 3 R 3 maps 16D spacetime algebra functions f : R 3,1 G 3,1 = Cl 3,1 to 16D spacetime spectrum functions f : R 3,1 Cl 3,1 Part f(x) e i 3 x u d 3 x... GA FT of G 3. Reversion replaced by principal involution (reversion and e t e t).
Applications of SFT Application: Multivector Wave Packets in Space and Time SFT = Sum of Right/Left Propagating Multivector Wave Packets f = f + + f = f + e i 3( x u ts ) d 4 x + f e i 3( x u+ ts ) d 4 x. (72) R 3,1 R 3,1 Directional 4D Spacetime Uncertainty for Multivector Wave Packets For two arbitrary constant spacetime vectors a, b R 3,1 (selecting two directions), and f L 2 (R 3,1, G 3,1 ), x 1/2 f L 2 (R 3,1, G 3,1 ) we obtain (a tt a x) 2 f(x) 2 d 4 x (b tω t b ω) 2 F{f}(ω) 2 d 4 ω R 3,1 R 3,1 (2π)4 4 [ (a tb t a b) 2 f 4 + (a tb t + a b) 2 f + 4],
Applications of SFT Spacetime signal transformations Application: Space & Time Signal Transformations The SFT of a spacetime function f L 2 (R 3,1 ; G 3,1 ) with a GL(R 3,1 ) transformation A (stretches, reflections, rotations, acceleration, boost) of its spacetime vector argument x is F SF T {f(ax)}(u) = det A 1 { f + (Ue t A 1 Ue t u) + f (A 1 u) }, (73) where A 1 is the adjoint of the inverse of A.
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 wavelet ψ : R 3 G 3 = Cl 3,0 Wavelet admissibility (condition on ψ) ψ(x) ψ a,θ,b (x) = 1 ψ(r 1 a3/2 θ ( x b )) (74) a ψ(ω) C ψ = ψ(ω) R 3 ω 3 d 3 ω. (75) is an invertible multivector constant and finite at a.e. ω R 3, GA FT ψ = F{ψ}.
GA wavelet basics and GA wavelet transformation GA Wavelet Transformation Definition (GA wavelet transformation) T ψ : L 2 (R 3 ; Cl 3,0 ) L 2 (G; Cl 3,0 ) f T ψ f(a, θ, b) = f(x) ψa,θ,b (x) d 3 x, (76) R 3 ψ L 2 (R 3 ; Cl 3,0 )... 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 Any f L 2 (R 3 ; Cl 3,0 ) can be decomposed as (C 1 ψ... admissibility) f(x) = T ψ f(a, b, θ) ψ a,θ,b C 1 ψ dµd3 b, (77) G with left Haar measure on G = SIM(3): dλ(a, θ, b) = dµ(a, θ)d 3 b
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
GA wavelet basics and GA wavelet transformation Notation: Inner products and norms We define the inner product of f, g : G Cl 3,0 by (f, g) = f(a, θ, b) g(a, θ, b) dλ(a, θ, b), (78) G and the L 2 (G; Cl 3,0 )-norm f 2 = (f, f) = f(a, θ, b) 2 dλ. (79) G
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 every f L 2 (R 3 ; Cl 3,0 ), the following inequality holds bt ψ f(a, θ, b) 2 L 2 (G;Cl 3,0 ) C ψ ( ω ˆf, ω ˆf) L 2 (R 3 ;Cl 3,0 ) 3(2π)3 4 [ C ψ (f, f) L 2 (R 3 ;Cl 3,0 )] 2. (80)
GA wavelets and uncertainty limits, example of GA Gabor wavelets Uncertainty principle for scalar admissibility constant For scalar admissibility constant C ψ we get Uncertainty principle for GA wavelet Let ψ be a Clifford algebra wavelet with scalar admissibility constant. Then for every f L 2 (R 3 ; Cl 3,0 ), the following inequality holds b T ψ f(a, θ, b) 2 L 2 (G;Cl 3,0 ) ω ˆf 2 L 2 (R 3 ;Cl 3,0 ) 3C (2π) 3 ψ f 4 4 L 2 (R 3 ;Cl 3,0 ). (81) 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.
GA wavelets and uncertainty limits, example of GA Gabor wavelets GA Gabor Wavelets Example: GA Gabor Wavelets ψ c (x) = g(x) e i 3ω 0 x e 2 1 (σ2 1 u2 0 +σ2 2 u2 0 +σ2 3 w2 0 }{{} ) constant 3D Gaussian: g(x) = ( 1 e 1 x 2 1 2 σ (2π) 3 1 2 2 σ 1 σ 2 σ 3 ) + x2 2 σ 2 2 + x2 3 σ 3 2. Uncertainty principle for GA Gabor wavelet Let ψ c be an admissible GA Gabor wavelet. Assume f L 2 (R 3 ; Cl 3,0 ), then the following inequality holds b T ψ c f(a, θ, b) 2 L 2 (G;Cl 3,0 ) ω ˆf 2 L 2 (R 3 ;Cl 3,0 ) 3C (2π) 3 ψ c f 4. (82) 4
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.
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)
B. Mawardi, E. Hitzer. Clifford Fourier Transform and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0, 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. Cl 3,0 -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
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