Different types of wavelets & their properties Compact support Symmetry

1. The Story of WaveletsTheory and Engineering Applications • Different types of wavelets & their properties • Compact support • Symmetry • Number of vanishing moments • Smoothness and regularity • Denoising Using Wavelets

2. 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 support • Symmetry • Number of vanishing moments • Smoothness / regularity

3. 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.

4. DB-4

5. DB-40

6. 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

7. 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!!!

8. 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).

9. 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 =0.28

10. WSD • Three step procedure • Decompose signal using DWT; choose wavelet and number of decomposition levels • Shrink coefficients by thresholding (hard /soft). Do we pick a single threshold or pick different thresholds at different levels? • Reconstruct the signal from thresholded DWT coefficients

11. 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, 2is 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 Length of the DWT coefficients at level k Threshold for subband k Noise std. deviation for subband k [XD, CXD, LXD]=wden(X, TPTR, SORH, SCAL, N, ‘wname’);

12. Denoising Implementation in Matlab First, analyze the signal with appropriate wavelets Hit Denoise

13. Denoising Using Matlab Choose thresholding method Choose noise type Choose thresholds Hit Denoise

14. Denosing Using Matlab