Different types of wavelets

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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

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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

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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.

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DB-4
5
DB-40
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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

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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!!!

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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).

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Denosing Using WaveletsWavelet 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
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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

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How Do we Choose the Threshold?

• There are many models, such as Steins 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, xnsn? en, where
  sn is the clean signal, en is the noise, ?2
  is the noise power, and xn 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, LXDwden(X, TPTR, SORH, SCAL, N,
wname)
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Denoising Implementation in Matlab
First, analyze the signal with appropriate
wavelets
Hit Denoise
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Denoising Using Matlab
Choose thresholding method
Choose noise type
Choose thresholds
Hit Denoise
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Denosing Using Matlab