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Signal Denoising with Wavelets

Signal Denoising with Wavelets. 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. Wavelet transform of noisy signals.

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Signal Denoising with Wavelets

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  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 fB from XB. • 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, 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.

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