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Wavelet Processing Gabriel Peyré www.numerical-tours.com

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Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform •Filter Constraints •2-D Multiresolutions

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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

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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)

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Sampling and Periodization (a) (c) (d) (b) 1 0

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Sampling and Periodization: Aliasing (b) (c) (d) (a) 0 1

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Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform •Filter Constraints •2-D Multiresolutions

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Multiresolutions: Approximation Spaces

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Multiresolutions: Approximation Spaces

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Multiresolutions: Approximation Spaces

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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

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Multiresolutions: Detail Spaces

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Multiresolutions: Detail Spaces

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Multiresolutions: Detail Spaces

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Multiresolutions: Detail Spaces

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Multiresolutions: Detail Spaces j,n \ j j0, 0 n < 2 j ⇥ ⇥j0,n \ 0 n < 2 j0 ⇥

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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

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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

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Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform •Filter Constraints •2-D Multiresolutions

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Computing the Wavelet Coefficients

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Computing the Wavelet Coefficients

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Computing the Wavelet Coefficients

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Computing the Wavelet Coefficients

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Computing the Wavelet Coefficients

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Discrete Wavelet Coefficients 0 0.2 0.4 0.6 0.8 1

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

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−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

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−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

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−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

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−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

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Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1

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Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0 0.5 1

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Fast Wavelet Transform 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 1.5

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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

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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

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Haar Refinement

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Haar Refinement

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Haar Transform

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Haar Transform

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Haar Transform

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Haar Transform

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Haar Transform

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Inverting the Transform

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Inverting the Transform

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Inverting the Transform

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Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform •Filter Constraints •2-D Multiresolutions

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Approximation Filter Constraints

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Approximation Filter Constraints

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

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

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

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

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

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

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

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

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

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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

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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

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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

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Daubechies Family

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Daubechies Family

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Overview •Review : Fourier transforms •1-D Multiresolutions •1-D Wavelet Transform •Filter Constraints •2-D Multiresolutions

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Anisotropic Wavelet Transform

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Anisotropic Wavelet Transform

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Anisotropic Wavelet Transform

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Anisotropic Wavelet Transform

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Anisotropic Wavelet Transform

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2D Multi-resolutions

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2D Multi-resolutions

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2D Multi-resolutions

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2D Wavelet Basis

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Discrete 2D Wavelets Coefficients

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Discrete 2D Wavelets Coefficients

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Discrete 2D Wavelets Coefficients

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Discrete 2D Wavelets Coefficients

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Discrete 2D Wavelets Coefficients

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Examples of Decompositions

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Separable vs. Isotropic

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Fast 2D Wavelet Transform

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Fast 2D Wavelet Transform

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Fast 2D Wavelet Transform

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Fast 2D Wavelet Transform

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Inverse 2D Wavelet Transform

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Inverse 2D Wavelet Transform

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Conclusion

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Conclusion

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Conclusion