Wavelets and Image Compression Augusta State University April, 27, Joe Lakey. Department of Mathematical Sciences. New Mexico State University

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1 Wavelets and Image Compression Augusta State University April, 27, 6 Joe Lakey Department of Mathematical Sciences New Mexico State University 1

2 Signals and Images Goal Reduce image complexity with little loss in information Goal Fast algorithms for decomposition/reconstruction Goal Uniform success over a large class of signals/images Problem How to define information vs. content 2

3 Function spaces: How do we measure magnitudes, errors, etc? L 2 (R) : f, g = f(x) g(x) dx; f 2 = f, f 1/2 l 2 (Z) : c, d = k c k d k ; c l 2 = c, c 1/2 L 2, l 2 : Integrals / sums converge Image processing: is L 2 a good metric for perception? Alternatives: f L 2 or f L 1. 3

4 SVD Singular Value Decomposition Images are matrices of pixel values. Truncations: A = UΣV T Ã = U ΣV T Classical approach Optimal in average sense. Data specific: Image as matrix Good for discrimination 4

5 Fourier transform 5

6 Figure 1: Fourier, before and after [include Fourier before and after here] Fourier transform on R ˆf(ξ) = f(t) e 2πitξ dt = f, cos 2πtξ + i sin 2πtξ Fourier inversion formula f(t) = ˆf(ξ)e 2πitξ dξ In what sense does this representation converge? Unitary: f, g = ˆf, ĝ. 6

7 ˆf 2 = f 2 Fourier series: periodic functions f(t +1)=f(t) ˆf[n] = 1 0 f(t) e 2πint dt f(t) = n= ˆf[n] e 2πint 7

8 Properties on R Translation and modulation Dilation: forλ>0: F(f( α))(ξ) =e 2πiαξ F(f( ))(ξ); F(e 2πiα f( ))(ξ) =F(f( ))(ξ α) F(f(λ ))(ξ) = 1 λ F(f( )) ( ξ λ Periodization: f p (t) = n= f(t + n) ) f p [n] = ˆf(n) Plancherel; Parseval: f, g = ˆf, ĝ 8

9 Discrete signals Digital signals come from sampling Discrete Fourier transform matrix: F jk = 1 N ω jk, ω = e 2πi/N, j,k =0,...,N 1 9

10 Figure 2: John Tukey. Developed FFT (O(N logn)) with J.W. Cooley 10

11 Wavelets: A little history 11

12 Figure 3: Alfred Haar. First wavelet basis in

13 Philip Franklin (1928): Periodic continuous wavelets... Yves Meyer (early 80s): Can one discretize to get an ONB? J.O. Strömberg: Franklin wavelets function spaces 13

14 Why weren t wavelets developed sooner? Data explosion: more recent FFT: good enough? Why did wavelets become so popular so quickly? Engineering: Esteban et al: subband coding for speech processing; parallel implementations Seismic imaging: Morletetal(CWT) Mathematics: nice for analyzing function spaces, PDEs Approximation theory: subdivision, splines, etc. Other areas 14

15 Applications/directions FBI fingerprint standard JPEG 0 PDE Different directions: Wavelet packets: Coifman-Meyer. Algorithms: Wickerhauser. Local trigonometric bases: Coifman-Meyer; Malvar. Time-frequency tilings Brushlets: Coifman et al. chirplets Curvelets...: Donoho, Càndes et. al. 15

16 The holy grail: what is the right way of representing a signal? Grand challenge obtain accurate models of naturally occurring sources of data, obtain optimal representations of such models and rapidly compute such optimal representations. Why STILL wavelets? 16

17 Wavelets to mathematicians: Easy properties Scaling equation: ϕ(x) =2 k h k ϕ(2x k) 17

18 Linear spline H(x) = = ½ H(2x) + + H(2x - 1) + ½ H(2x - 2) 18

19 ϕ(x) = lim T n f 0 Tf(x) = 2 k h k f(2x k) subdivision scheme 19

20 D4 scaling, level 3 D4 wavelet Figure 5: Level 3 20

21 D4 scaling, level 4 D4 wavelet Figure 6: Level 4 21

22 D4 scaling, level 6 D4 wavelet Figure 7: Level 6 22

23 D4 scaling, level 8 D4 wavelet Figure 8: Level 8 23

24 D4 scaling, level 10 D4 wavelet Figure 9: Level 10 24

25 V ϕ = { k c kϕ(x k) : k Z c2 k < } V ϕ j = {f(2 j x): f(x) V ϕ } Scaling implies V ϕ j V ϕ j+1. 25

26 MultiResolution Analysis properties: V ϕ j V ϕ j+1 V j = lim j V j = {0} V j dense in L 2 (R). Abstract Hilbert space theory: spaces W j : V 1 = V 0 W 0 V 2 = V 1 W 1 = V 0 W 0 W V N = V 0 W 0 W N L 2 (R) = j= W j 26

27 What one wants Approximation: Polynomials up to some degree nearly in V ϕ j Regularity: ϕ has some derivatives ϕ(x) = lim T n f 0 Tf(x) = 2 k h k f(2x k) converges in a suitable norm Orthogonality: ϕ( ), ϕ( k) = δ 0k 27

28 Construction of ϕ from {h k } 1 2 ϕ( x ) 2 = k h k ϕ(2x k) ˆϕ(2ξ) = k h k e 2πikξ ˆϕ(ξ) H(ξ) = k = H(ξ)ˆϕ(ξ) h k e 2πikξ Iterate... ˆϕ(2ξ) =ˆϕ(ξ) H(ξ/2 j ) j=1 28

29 Orthogonality: ϕ( ), ϕ( k) = δ 0k ϕ( ), ϕ( k) = ϕ(x)ϕ(x k) dx = = = l= l 1 l= 1 0 ˆϕ(ξ)ˆϕ(ξ) e 2πikξ dξ l+1 0 ˆϕ(ξ) 2 e 2πikξ dξ ˆϕ(ξ + l) 2 e 2πikξ dξ Φ(ξ) 2 e 2πikξ dξ Orthogonality plus Fourier uniqueness: Φ 1. 29

30 Orthogonality and H Break into odd and even and using Φ 1: 1 = l ˆϕ(2ξ + l) 2 = l ( H ξ + l ) ( 2 ˆϕ ξ + l ) = l = l ( ) ( ) ( H ξ + l 2 ˆϕ ξ + l 2 + H ξ + l + 1 ) ( 2 ˆϕ ξ + l + 1 ) ( ) ( ) ( H ξ 2 ˆϕ ξ + l 2 + H ξ + 1 ) ( 2 ˆϕ ξ + l + 1 ) ( = H(ξ) 2 + H ξ + 1 )

31 Condition of orthogonality: H(ξ) 2 + H ( ξ + 1 ) Plus some subtleties! 31

32 Conditions of regularity Depends on eigenvalues of transition matrix Example Daubechies 4-coefficient systems H(z) = 1 1+ν 2 ( ν(ν 1) + (1 ν)z +(1+ν)z 2 + ν(1 + ν)z 3) 32

33 Dnu scaling nu =.001 Dnu wavelet

34 0.12 Dnu scaling nu =.2 Dnu wavelet

35 Dnu scaling nu =.5 Dnu wavelet

36 Dnu scaling nu =.7 Dnu wavelet

37 Dnu scaling nu =.9 Dnu wavelet

38 37-1 Dnu scaling nu =.99 Dnu wavelet

39 Wavelets to engineers Low pass H(ω) = k h k e 2πikω High pass G(ω) =e πiω k ( 1) k h 1 k e 2πikω = e πiω H(ω +1/2) 38

40 Mother wavelet ψ(x) =2 k g k ψ(2x k) Wavelet basis ψ jk (x) =2 j ψ(2 j x k) 39

41 A catalog of wavelets Look in folder orthogonal ; see also biorthogonal, interpolating For multiwavelets, see mwmp, coefs and multiplot Principal examples: (1) Meyer (2) Battle-Lemarie (3) Daubechies (4) Deslaurier s-dubuc (5) spline wavelets Multiwavelets: good symmetry and support properties Custom designed MRA (i) one benefits from the scaling perspective here! (ii) idea of local approximation (iii) symmetry comes from symmetry of the coefficients. 40

42 41

43 0.2 Haar Wavelet 0.2 D4 Wavelet C3 Coiflet 0.2 S8 Symmlet

44 9 Some S8 Symmlets at Various Scales and Locations (7,95) (6,43) (6,32) (6,21) (5,13) (4, 8) (3, 5) (3, 2)

45 GHM scaling #1 GHM wavelet # GHM scaling #2 GHM wavelet #

46 Discrete implementation 45

47 Discrete convolution/decimation filters H, G : l 2 (Z) l 2 (Z) (Ha) k = 2 l h l 2k a l (Ga) k = 2 l ḡ l 2k a l. 46

48 Fast wavelet transform c N H c N 1 H c N 2 H c N 3 c L G G G d N 1 d N 2 d N 3 d L Inverse Fast wavelet transform c L H H c L+1 H c L+2 c L+3 c N 1 H c N G G G G G d L d L+1 d L+2 d N 2 d N 1 47

49 log(resolution) Object Doppler WT[Doppler] position 48 Wavelet Components of Object Doppler (8,10) (8, 9) 40 (7,11) (7,10) (7, 9) (7, 8) (7, 6) 35 (6,12) (6,10) (6, 9) (6, 8) (6, 7) 30 (6, 6) (6, 5) (6, 4) (6, 3) (5,10) 25 (5, 9) (5, 8) (5, 7) (5, 6) (5, 5) 20 (4,13) (4, 8) (4, 7) (4, 6) (4, 5) 15 (4, 4) (4, 3) (4, 2) (3, 7) (3, 6) 10 (3, 5) (3, 4) (3, 3) (3, 2) (3, 1) 5 (3, 7) (3, 6) (3, 5) (3, 4) (3, 3)

50 The crime of wavelets Sample/pixel values treated as {c N } 49

51 Multivariate wavelets: tensor products Simple: tensor products. 2 n 1 wavelets: V 1 V 1 =(V 0 W 0 ) (V 0 W 0 )= (V 0 V 0 ) [(W 0 V 0 ) (V 0 W 0 ) (W 0 W 0 )] 50

52 Comparison of Methods I: Zebra 51

53

54 reconstruction from largest8% Fourier coefficientsreconstruction from largest8 % wavelet coefficients

55 reconstruction from largest4% Fourier coefficientsreconstruction from largest4 % wavelet coefficients

56 reconstruction from largest2% Fourier coefficientsreconstruction from largest2 % wavelet coefficients

57 Comparison of Methods II: Max 56

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59

60 Eigenvector approximations of Max 59

61 reconstruction from largest0.5% eigenvalues Reconstruction Error (rescaled intensity) 60

62 reconstruction from largest2% eigenvalues Reconstruction Error (rescaled intensity) 61

63 reconstruction from largest4% eigenvalues Reconstruction Error (rescaled intensity) 62

64 reconstruction from largest8 % eigenvalues Reconstruction Error (rescaled intensity) 63

65 reconstruction from largest16 % eigenvalues Reconstruction Error (rescaled intensity) 64

66 reconstruction from largest32 % eigenvalues Reconstruction Error (rescaled intensity) 65

67 Fourier Approximations of Max 66

68 reconstruction from largest5e 005 % Fourier coefficientsreconstruction Error (rescaled intensity) 67

69 reconstruction from largest % Fourier coefficientsreconstruction Error (rescaled intensity) 68

70 reconstruction from largest0.005 % Fourier coefficients Reconstruction Error (rescaled intensity) 69

71 reconstruction from largest0.05 % Fourier coefficients Reconstruction Error (rescaled intensity) 70

72 reconstruction from largest0.5 % Fourier coefficients Reconstruction Error (rescaled intensity) 71

73 reconstruction from largest2 % Fourier coefficients Reconstruction Error (rescaled intensity) 72

74 reconstruction from largest4 % Fourier coefficients Reconstruction Error (rescaled intensity) 73

75 reconstruction from largest8 % Fourier coefficients Reconstruction Error (rescaled intensity) 74

76 reconstruction from largest16 % Fourier coefficients Reconstruction Error (rescaled intensity) 75

77 Wavelet approximations of Max 76

78 Wavelet Transform of Max

79 0 Nonzero Pattern in Sparsification of WT[Max] nz =

80 reconstruction from largest5e 005 % waveletsreconstruction Error (rescaled intensity) 79

81 reconstruction from largest % waveletsreconstruction Error (rescaled intensity) 80

82 reconstruction from largest0.005 % wavelets Reconstruction Error (rescaled intensity) 81

83 reconstruction from largest0.05 % wavelets Reconstruction Error (rescaled intensity) 82

84 reconstruction from largest0.5 % wavelets Reconstruction Error (rescaled intensity) 83

85 reconstruction from largest2 % wavelets Reconstruction Error (rescaled intensity) 84

86 reconstruction from largest4 % wavelets Reconstruction Error (rescaled intensity) 85

87 reconstruction from largest8 % wavelets Reconstruction Error (rescaled intensity) 86

88 Max: Fourier versus wavelet 87

89 10 10 Wavelet Compression vs. DCT Compression 10 9 DCT sum(error 2 ) DWT Number of Coefficients Retained 88

90 reconstruction from largest0.5% Fourier coefficientsreconstruction from largest0.5 % wavelet coefficients 89

91 reconstruction from largest1% Fourier coefficients reconstruction from largest1 % wavelet coefficients 90

92 reconstruction from largest2% Fourier coefficients reconstruction from largest2 % wavelet coefficients 91

93 reconstruction from largest4% Fourier coefficients reconstruction from largest4 % wavelet coefficients 92

94 Homework: make the world a better place Can you use wavelets to make better cats and dogs? 93

95 Figure 56: Man s best friends? 94

96 Some basics of matlab Software resources try this first 95

97 Bounded variation and decay Cohen, DeVore, Petrushev and Yu : If f L 1 (R n )thenβ(j, k) =2 j(1 n 2 ) f,ψ jk defines a sequence in l 1, (Z Z n ), that is, for each λ>0, #{Q Q : β(q) >λ} c(n) λ f dx Corollary A deep improvement of the Sobolev embedding theorem. 96

98 Further issues Progressive transmission and reconstruction Entropy and source coding 97