Editing Wavelet (section)

=== Wavelet denoising ===

[[File:Wavelet denoising.svg|thumb|Signal denoising by wavelet transform thresholding]]

Suppose we measure a noisy signal <math>x = s + v </math>, where <math>s</math> represents the signal and <math>v</math> represents the noise. Assume <math>s</math> has a sparse representation in a certain wavelet basis, and <math>v \ \sim\ \mathcal{N}(0,\,\sigma^2I)</math>

Let the wavelet transform of <math>x</math> be <math>y = W^T x = W^T s + W^T v = p + z</math>, where <math>p = W^T s</math> is the wavelet transform of the signal component and <math>z = W^T v</math> is the wavelet transform of the noise component.

Most elements in <math>p</math> are 0 or close to 0, and <math>z \ \sim\ \ \mathcal{N}(0,\,\sigma^2I)</math>

Since <math>W</math> is orthogonal, the estimation problem amounts to recovery of a signal in iid [[Gaussian noise]]. As <math>p</math> is sparse, one method is to apply a Gaussian mixture model for <math>p</math>.

Assume a prior <math>p \ \sim\ a\mathcal{N}(0,\,\sigma_1^2) +(1- a)\mathcal{N}(0,\,\sigma_2^2)</math>, where <math>\sigma_1^2</math> is the variance of "significant" coefficients and <math>\sigma_2^2</math> is the variance of "insignificant" coefficients.

Then <math>\tilde p = E(p/y) = \tau(y) y</math>, <math>\tau(y)</math> is called the shrinkage factor, which depends on the prior variances <math>\sigma_1^2</math> and <math>\sigma_2^2</math>. By setting coefficients that fall below a shrinkage threshold to zero, once the inverse transform is applied, an expectedly small amount of signal is lost due to the sparsity assumption. The larger coefficients are expected to primarily represent signal due to sparsity, and statistically very little of the signal, albeit the majority of the noise, is expected to be represented in such lower magnitude coefficients... therefore the zeroing-out operation is expected to remove most of the noise and not much signal. Typically, the above-threshold coefficients are not modified during this process. Some algorithms for wavelet-based denoising may attenuate larger coefficients as well, based on a statistical estimate of the amount of noise expected to be removed by such an attenuation.

At last, apply the inverse wavelet transform to obtain <math> \tilde s = W \tilde p</math>