Gabriel Peyré
January 01, 2012
1.7k

# Signal Processing Course: Wavelet Processing

January 01, 2012

## Transcript

2. ### Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

•Filter Constraints •2-D Multiresolutions
3. ### Inﬁnite continuous domains: Periodic continuous domains: Inﬁnite discrete domains: Periodic

discrete domains: f0 (t), t R f0 (t), t ⇥ [0, 1] R/Z The Four Settings ˆ f[m] = N 1 n=0 f[n]e 2i N mn ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n Note: for Fourier, bounded periodic. .. . .. . .. . .. . f[n], n Z f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ
4. ### Sampling idealization: Poisson formula: f0 f ˆ f0 ˆ f

sampling periodization cont. FT discr. FT Commutative diagram: f[n] ˆ f0 ( ) ˆ f( ) Sampling and Periodization f[n] = f0 (n/N) ˆ f(⇥) = k ˆ f0 (N(⇥ + 2k )) (a) (c) (b) 1 0 (a) (c) (b) 1 0 (a) (b) (a) f0 (t)

7. ### Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

•Filter Constraints •2-D Multiresolutions

11. ### Haar Multiresolutions 0 0.2 0.4 0.6 0.8 1 0 0.2

0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

16. ### Multiresolutions: Detail Spaces j,n \ j j0, 0 n <

2 j ⇥ ⇥j0,n \ 0 n < 2 j0 ⇥
17. ### Haar Wavelets −0.2 −0.1 0 0.1 0.2 0 0.2 0.4

0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
18. ### Haar Wavelets −0.2 −0.1 0 0.1 0.2 0 0.2 0.4

0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2
19. ### Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

•Filter Constraints •2-D Multiresolutions

26. ### −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

0 0.2 0.4 0.6 0.8 1
27. ### −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2
28. ### −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
29. ### −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.5 0 0.5
30. ### −1.5 −1 −0.5 0 0.5 1 1.5 Discrete Wavelet Coefficients

0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.5 0 0.5

0.5 1
33. ### Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

0.5 1 0 0.5 1 1.5
34. ### Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

0.5 1 0 0.5 1 1.5 0 0.5 1 1.5 2
35. ### Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0

0.5 1 0 0.5 1 1.5 0 0.5 1 1.5 2

46. ### Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

•Filter Constraints •2-D Multiresolutions

49. ### Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
50. ### Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
51. ### Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
52. ### Approximation Filter Constraints {⌅(· n)}n orthogonal ⇥⇤ ⌅ n, ⌅

⇧ ¯ ⌅(n) = [n] ⇥⇤ k | ˆ ⌅(⇤ + 2k⇥)|2 = 1
53. ### { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
54. ### { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
55. ### { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
56. ### { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
57. ### { (· n)}n orthogonal ⇥ k | ˆ ⇥(⇤ +

2k )|2 = 1 n, ⇥ ⇤ ⇥(n) = [n] Detail Filter Constraint
58. ### Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1
59. ### Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1
60. ### Vanishing Moment Constraint −0.2 −0.1 0 0.1 0.2 −0.2 −0.1

0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1

63. ### Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform

•Filter Constraints •2-D Multiresolutions