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Signal Denoising with Wavelets

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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

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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.

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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.

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

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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:

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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:

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

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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

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Theorem (Donoho and Johnstone) The risk of a hard/soft threshold estimator will satisfy when

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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:

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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,

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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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Application: Signal Compression Jyun-Ming Chen Spring 2001.

Application: Signal Compression Jyun-Ming Chen Spring 2001.

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