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Signal Processing Course: Denoising
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Gabriel Peyré
January 01, 2012
Research
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Signal Processing Course: Denoising
Gabriel Peyré
January 01, 2012
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Transcript
Linear and Non Linear Denoising Gabriel Peyré www.numerical-tours.com
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Noise in Images
Denoising Problem
Denoising Problem
Additive Noise Model
Noise Distributions −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3
−0.2 −0.1 0 0.1 0.2 0.3
Noise Distributions −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3
−0.2 −0.1 0 0.1 0.2 0.3
Noise Distributions −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3
−0.2 −0.1 0 0.1 0.2 0.3
Data-dependent Noise
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Linear Denoising Estimator
Fourier and Denoising
Optimal Filter Choice
Oracle Estimation of Optimal Filter
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Diagonal Thresholding
Wavelet Diagonal Hard Thresholding
Sparse Signal Estimation
Optimal Threshold Selection
W unit variance white noise. Non-linear Approximation and Estimation
Hard vs. Soft Thresholding
Hard vs. Soft Thresholding
Optimal Threshold
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Translation Invariant Denoising
Translation Invariant Wavelets
Translation Invariant Haar (1D)
Translation Invariant Transform (2D)
Translation Invariant Thresholding
Optimal Invariant Threshold
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Between Hard and Soft Thresholding
Stein Quadratic-Soft Thresholder
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Block Thresholding
Optimal Block Choice
Comparison
Overview • Noise in Signals and Images • Linear Denoising
by Blurring • Non-linear Wavelet Denoising • Translation Invariant Thresholding • Other Diagonal Thresholders • Non-diagonal Block Thresholding • Data-dependent Noise
Poisson Noise
Poisson Noise Variance Stabilization 1 2 3 4 5 6
7 8 9 10 0.75 0.8 0.85 0.9 0.95 1 1.05
Multiplicative Noise
Multiplicative Noise Stabilization 0 0.5 1 1.5 2 2.5 −1.5
−1 −0.5 0 0.5 1 1.5
Conclusion