1. Signal Denoising with Wavelets

2. Wavelet Threholding Assume an additive model for a noisy signal, y=f+n 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(X=f+W) By=Bf+Bn Covariance of noise: If B is orthogonal and W is white, i.i.d, S=K.

5. Hard Thresholding vs. Soft Thresholding Hard Thresholding: Let BX=u Soft Thresholding:

6. How to Choose the Threshold Diagonal Estimation with Oracles: A diagonal operator estimates each f B from X B. Find a[m] that minimizes the risk of the estimator:

7. Threshold Selection Since a[m] depends on fB, this is not realizable in practice. Simplify: –Linear Projection: a[m] 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, r t (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.