Signal Representations DS-GA 1013 / MATH-GA 2824 Mathematical Tools - PowerPoint PPT Presentation

Haar wavelet basis The Haar father wavelet ϕ is a constant vector, such that 1 √ ϕ [ j ] := , 1 ≤ j ≤ N N The mother wavelet µ satisfies  − 1 2 , j = 1 , √    1 µ [ j ] := 2 , j = 2 , √   0 , j > 2  Other options: Meyer, Daubechies, coiflets, symmlets, etc.

Haar wavelets Scale Basis functions 2 0 2 1 2 2

Multiresolution decomposition V K W 2 W 1 W 0 P V k x is an approximation of x at scale 2 k

Vertical line (column 135) 1.0 0.8 0.6 0.4 0.2 0 100 200 300 400 500

Scale 2 9 Approximation Coefficients 1.0 Data Approximation 16 0.8 14 12 0.6 10 8 6 0.4 4 2 0.2 0 0.04 0.02 0.00 0.02 0.04 0 100 200 300 400 500

Scale 2 8 Approximation Coefficients 1.0 Data Approximation 4.0 0.8 3.5 3.0 2.5 0.6 2.0 1.5 0.4 1.0 0.5 0.2 0.0 0.04 0.02 0.00 0.02 0.04 0 100 200 300 400 500

Scale 2 7 Approximation Coefficients 1.0 Data Approximation 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.0 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500

Scale 2 6 Approximation Coefficients 1.0 Data Approximation 1.4 0.8 1.2 1.0 0.6 0.8 0.6 0.4 0.4 0.2 0.0 0.2 0 1 2 3 0 100 200 300 400 500

Scale 2 5 Approximation Coefficients 1.0 Data Approximation 0.8 0.6 0.5 0.6 0.4 0.3 0.2 0.4 0.1 0.0 0.2 0 2 4 6 0 100 200 300 400 500

Scale 2 4 Approximation Coefficients 1.0 Data Approximation 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.0 0.2 0.2 0 5 10 15 0 100 200 300 400 500

Scale 2 3 Approximation Coefficients 1.0 Data Approximation 0.4 0.8 0.3 0.6 0.2 0.1 0.4 0.0 0.2 0.1 0 10 20 30 0 100 200 300 400 500

Scale 2 2 Approximation Coefficients 1.0 Data Approximation 0.8 0.25 0.20 0.6 0.15 0.10 0.05 0.4 0.00 0.05 0.2 0 20 40 60 0 100 200 300 400 500

Scale 2 1 Approximation Coefficients 1.0 Data Approximation 0.20 0.8 0.15 0.10 0.6 0.05 0.00 0.4 0.05 0.10 0.2 0 25 50 75 100 125 0 100 200 300 400 500

Scale 2 0 Approximation Coefficients 1.0 Data Approximation 0.8 0.2 0.1 0.6 0.0 0.4 0.1 0.2 0.2 0 50 100 150 200 250 0 100 200 300 400 500

Haar wavelets in the frequency domain Width: 200 0.16 Width: 100 0.14 Width: 50 0.12 0.10 0.08 0.06 0.04 0.02 0.00 200 150 100 50 0 50 100 150 200 Frequency

Time-frequency support of basis vectors STFT Wavelets

2D Wavelets Extension to 2D by using outer products of 1D basis vectors To build a 2D basis vector at scale ( m 1 , m 2 ) and shift ( s 1 , s 2 ) we set � T � v 2D [ s 1 , s 2 , m 1 , m 2 ] := v 1D v 1D , [ s 1 , m 1 ] [ s 2 , m 2 ] where v 1D can refer to 1D father or mother wavelets Nonseparable designs: steerable pyramid, curvelets, bandlets...

2D Haar wavelet basis vectors

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2D Haar wavelet decomposition Approximation Coefficients 350 340 330 320 310 300

2D Haar wavelet decomposition Approximation Coefficients 100 100 80 80 60 60 40 40 20 20 0 0 100 80 60 40 20 0

2D Haar wavelet decomposition Approximation Coefficients 30 30 25 25 20 20 15 15 10 10 5 5 0 0 5 5 30 25 20 15 10 5 0 5

2D Haar wavelet decomposition Approximation Coefficients 15 15 10 10 5 5 0 0 5 5 15 10 5 0 5

2D Haar wavelet decomposition Approximation Coefficients 6 6 4 4 2 2 0 0 2 2 4 4 6 6 6 4 2 0 2 4 6

2D Haar wavelet decomposition Approximation Coefficients 4 4 2 2 0 0 2 2 4 4 6 6 4 2 0 2 4 6

2D Haar wavelet decomposition Approximation Coefficients 3 3 2 2 1 1 0 0 1 1 2 2 3 2 1 0 1 2

2D Haar wavelet decomposition Approximation Coefficients 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

2D Haar wavelet decomposition Approximation Coefficients 1.0 1.0 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0

2D Haar wavelet decomposition Approximation Coefficients 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.6 0.4 0.2 0.0 0.2 0.4 0.6

Windowing Short-time Fourier transform Multiresolution analysis Denoising via thresholding

Denoising Aim: Estimate signal x from data of the form y = x + z

Motivation STFT coefficients of audio and wavelet coefficients of images are sparse Coefficients of noise are dense Idea: Get rid of small entries in Ay = Ax + A � z

Why are noise coefficients dense? If ˜ z is Gaussian with mean µ and covariance matrix Σ , then for any A , z is Gaussian with mean A µ and covariance matrix A Σ A ∗ A ˜ If A is orthogonal, iid zero mean noise is mapped to iid zero mean noise

Example Data Signal

Thresholding Hard-thresholding operator � v [ j ] if | v [ j ] | > η H η ( v ) [ j ] := 0 otherwise

Denoising via hard thresholding Estimate Signal

Denoising via hard thresholding Given data y and a sparsifying linear transform A 1. Compute coefficients Ay 2. Apply the hard-thresholding operator H η : C n → C n to Ay 3. Invert the transform x est := L H η ( Ay ) , where L is a left inverse of A

Speech signal 7500 5000 2500 0 2500 5000 7500 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 Time (s)

STFT coefficients 4000 10 1 3000 Frequency (Hz) 10 2 2000 10 3 1000 10 4 0 10 5 0 2 4 6 Time (s)

Noisy signal 10000 7500 5000 2500 0 2500 5000 7500 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 Time (s)

STFT coefficients 4000 10 1 3000 Frequency (Hz) 10 2 2000 10 3 1000 10 4 0 0 2 4 6 Time (s)

Thresholded STFT coefficients 4000 10 1 3000 Frequency (Hz) 10 2 2000 10 3 1000 10 4 0 10 5 0 2 4 6 Time (s)

Denoised signal 7500 5000 2500 0 2500 5000 7500 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 Time (s)

Denoised signal 3000 2000 1000 0 1000 Signal 2000 STFT thresholding 3000 Noisy data 4.3600 4.3625 4.3650 4.3675 4.3700 4.3725 4.3750 4.3775 Time (s)

Denoised signal (Wiener filtering) 3000 2000 1000 0 1000 Signal 2000 Wiener denoising 3000 Noisy data 4.3600 4.3625 4.3650 4.3675 4.3700 4.3725 4.3750 4.3775 Time (s)

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