Signal Processing Course : Denoising
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Transcript of Signal Processing Course : Denoising
Linear and Non LinearDenoising
Gabriel Peyréwww.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