• Different types of wavelets & their properties

• Compact support

• Symmetry

• Number of vanishing moments

• Smoothness and regularity

• Denoising Using Wavelets

The Story of WaveletsTheory and Engineering Applications

Wavelet Selection Criteria

There are a multitude of wavelets with different properties. It is important to choose the one with appropriate properties for a given application.

Most important properties are:Compact supportSymmetryNumber of vanishing momentsSmoothness / regularity

Compactly Supported Wavelets

Compact support: Finite duration wavelet (FIR filter). If a wavelet is compactly supported in time domain, it is not bandlimited in frequency domain. Most compactly supported wavelets are designed to have a rapid fall-off, so that they can be considered as bandlimited

Examples: Daubechies, Symlets, Coiflets, etc. Compact support allows

Reduced computation complexity Better time resolution, but Poorer frequency resolution

Narrowband wavelets, are compactly supported in frequency, but not in time. IIR Filters are constructed from narrowband wavelets.

Examples: Meyer wavelets.

DB-4

DB-40

Symmetry

Symmetric or antisymmetric wavelets give rise to linear – phase filters.

Orthogonal wavelets with compact support cannot be symmetric

Linear phase FIR filters can be constructed from biorthogonal wavelets.

Most orthogonal wavelets also satisfy biorthogonality. However, such wavelets do NOT satisfy symmetry requirements, e.g., Daubechies wavelets

Vanishing Moments

The mth moment of a wavelet is defined as

If the first M moments of a wavelet are zero, then all polynomial type signals of the form

have (near) zero wavelet / detail coefficients. Why is this important? Because if we use a wavelet with enough number of

vanishing moments, M, to analyze a polynomial with a degree less than M, then all detail coefficients will be zero excellent compression ratio.

But most practical signals do not look anything like polynomials…?, you ask…

Recall that any function can be written as a polynomial when expanded into its Taylor series. This is what makes wavelets so successful in compression!!!

dtttm )(

Mm

mmtctx

0

)(

Smoothness /Regularity

We have talked about how to obtain h[] and g[] filter coefficients from scaling and wavelet functions using the two-scale equation.

How about the reverse procedure? Can we start with a set of filter coefficients, iterate using the two-scale equation to obtain a scaling function? Would this function be smooth ? Under certain conditions, the answer is YES!

Regularity / smoothness is roughly the number of times a function can be differentiated at any given point.

Regularity is closely related to the number of vanishing moments. The more the number of vanishing moments, the smoother the wavelet. Why is this important? Smoothness provide numerical stability Better reconstruction properties Necessary for certain applications, such as solution of diff.

Equations For Daubechies wavelets, smoothness ~ N/5 for large N (the number of vanishing

moments).

)2(][2)(~

ntnhtn

Denosing Using Wavelets:Wavelet Shrinkage Denosing

Based on reducing the values of certain coefficients (at each level) that are believed to correspond to noise.

Better then regular filtering, because no significant signal information is lost, even when signal and noise spectra overlap !

Two types of thresholding are used:

Hard Thresholding Soft Thresholding

|)(| ,0

|)(| ),()(

tx

txtxtyhard

|)(| ,0

|)(| ,)(sgn)(

tx

txtxtxtysoft

=0.28

WSD

Three step procedure1. Decompose signal using DWT; choose wavelet

and number of decomposition levels 2. Shrink coefficients by thresholding (hard

/soft). Do we pick a single threshold or pick different thresholds at different levels?

3. Reconstruct the signal from thresholded DWT coefficients

How Do we Choose the Threshold?

There are many models, such as Stein’s unbiased risk estimate, universal threshold estimate, combination of the above two, minimax criterion, etc. Among them, universal threshold estimate is the one used most often.

According to this model, x[n]=s[n]+ e[n], where s[n] is the clean signal, e[n] is the noise, 2 is the noise power, and x[n] is the noisy signal. The noise is considered as Additive White Gaussian Noise (AWGN).

This model estimates the universal threshold (subband dependent, of course) as

)log(2 kkk N

Threshold forsubband k

Noise std. deviationfor subband k

Length of the DWT coefficients at level k

[XD, CXD, LXD]=wden(X, TPTR, SORH, SCAL, N, ‘wname’);

Denoising Implementation in Matlab

First, analyze the signal with appropriate wavelets

Hit Denoise

Denoising Using Matlab

Choose thresholdingmethod

Choose noise type

Choose thresholds

Hit Denoise

Denosing Using Matlab