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