Signal Denoising with Wavelets

1
Signal Denoising with Wavelets
2
Wavelet Threholding

• Assume an additive model for a noisy signal,
  yfn
• K is the covariance of the noise
• Different options for noise
• i.i.d
• White
• Most common model Additive white Gaussian noise

3
Wavelet transform of noisy signals

• Wavelet transform of a noisy signal yields small
  coefficients that are dominated by noise, large
  coefficients carry more signal information.
• White noise is spread out equally over all
  coefficients.
• Wavelets have a decorrelation property.
• Wavelet transform of white noise is white.

4
Wavelet transform of noisy signal

• B(XfW)
• ByBfBn
• Covariance of noise
• If B is orthogonal and W is white, i.i.d, SK.

5
Hard Thresholding vs. Soft Thresholding

• Hard Thresholding Let BXu
• Soft Thresholding

6
How to Choose the Threshold

• Diagonal Estimation with Oracles A diagonal
  operator estimates each fB from XB.
• Find am that minimizes the risk of the
  estimator

7
Threshold Selection

• Since am depends on fB, this is not realizable
  in practice.
• Simplify
• Linear Projection am is either 1 or 0
• Non-linear Projection Not practical
• The risk of this projector

8
Hard Thresholding

• Threshold the observed coefficients, not the
  underlying
• Nonlinear projector
• The risk is greater than equal to the risk of an
  oracle projector

9
Soft Thresholding

• Attenuation of the estimator, reduces the added
  noise
• Choose T appropriately such that the risk of
  thresholding is close to the risk of an oracle
  projector

10
Theorem (Donoho and Johnstone)

• The risk of a hard/soft threshold estimator will
  satisfy
  
• when

11
Thresholding Refinements

• SURE Thresholds (Stein Unbiased Risk Estimator)
  Estimate the risk of a soft thresholding
  estimator, rt(f,T) from noisy data X
• Estimate
• The risk is

12
SURE

• It can be shown that for soft thresholding, the
  risk estimator is unbiased.
• To find the threshold that minimizes the SURE
  estimator, the N data coefficients are sorted in
  decreasing amplitude.
• To minimize the risk, choose T the smallest
  possible,

13
Extensions

• Estimate noise variance from data using the
  median of the finest scale wavelet coefficients.
• Translation Invariant Averaging estimators for
  translated versions of the signal.
• Adaptive (Multiscale) Thresholding Different
  thresholds for different scales
• At low scale T should be smaller.