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Linear and Non Linear Denoising Gabriel Peyré www.numerical-tours.com

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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

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Noise in Images

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Denoising Problem

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Denoising Problem

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Additive Noise Model

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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

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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

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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

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Data-dependent Noise

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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

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Linear Denoising Estimator

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Fourier and Denoising

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Optimal Filter Choice

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Oracle Estimation of Optimal Filter

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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

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Diagonal Thresholding

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Wavelet Diagonal Hard Thresholding

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Sparse Signal Estimation

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Optimal Threshold Selection

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W unit variance white noise. Non-linear Approximation and Estimation

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Hard vs. Soft Thresholding

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Hard vs. Soft Thresholding

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Optimal Threshold

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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

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Translation Invariant Denoising

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Translation Invariant Wavelets

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Translation Invariant Haar (1D)

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Translation Invariant Transform (2D)

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Translation Invariant Thresholding

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Optimal Invariant Threshold

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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

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Between Hard and Soft Thresholding

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Stein Quadratic-Soft Thresholder

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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

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Block Thresholding

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Optimal Block Choice

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Comparison

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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

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Poisson Noise

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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

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Multiplicative Noise

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Multiplicative Noise Stabilization 0 0.5 1 1.5 2 2.5 −1.5 −1 −0.5 0 0.5 1 1.5

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Conclusion