Biomedical signal processing: Wavelets

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Biomedical signal processing Wavelets

• Yevhen Hlushchuk,
• 11 November 2004

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Usefull wavelets

• analyzing of transient and nonstationary signals
• EP noise reduction denoising
• compression of large amounts of data (other basis
  functions can also be employed)

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Introduction

• Class of basis functons, known as wavelets,
  incorporate two parameters
• one for translation in time
• another for scaling in time
• main point is to accomodate temporal information
  (crucial in evoked responses (EP) analysis)
• Another definition
• A wavelet is an oscillating function whose energy
  is concentrated in time to better represent
  transient and nonstationary signals
  (illustration).

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Continuous wavelet transform (CWT)
Example of continuous wavelet transform (here we
see the scalogram)
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Other ways to look at CWT

• The CWT can be interpreted as a linear filtering
  operation (convolution between the signal x(t)
  and a filter with impulse response ?(-t/s))
• The CWT can be viewed as a type of bandpass
  analysis where the scaling parameter (s) modifies
  the center frequency and the bandwidth of a
  bandpass filter (Fig 4.36)

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Discrete wavelet transform

• CWT is highly redundant since 1-dimensional
  function x(t) is transformed into 2-dimensional
  function. Therefore, it is Ok to discretize them
  to some suitably chosen sample grid. The most
  popular is dyadic sampling
• s2-j, t k2-j
• With this sampling it is still possible to
  reconstruct exactly the signal x(t).

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Multiresolution analysis

• The signal can be viewed as the sum of
• a smooth (coarse) part reflects main
  features of the signal (approximation signal)
• a detailed (fine) part faster fluctuations
  represent the details of the signal.
• The separation of the signal into 2 parts is
  determined by the resolution.

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Scaling function and wavelet function

• The scaling function is introduced for
  efficiently representing the approximation signal
  xj(t) at different resolutions.
• This function has a unique wavelet function
  related to it.
• The wavelet function complements the scaling
  function by accounting for the details of a
  signal (rather than its approximations)

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Classic example of multiresolution analysis
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What should you want from the scaling and wavelet
function?

• Orthonormality and compact support (concentrated
  in time, to give time resolution)
• Smooth, if modeling or analyzing physiological
  responses (e.g., by requiring vanishing moments
  at certain scale) Daubechies, Coiflets.
• Symmetric (hard to get, only Haar or sinc, or
  switching to biorthogonality)

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Scaling and wavelet functions

• Haar wavelet (square wave, limited in time,
  superior time localization)
• Mexican hat (smooth)
• Daubechies, Coiflet and others (Fig4.44)

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• One more example but now with a smooth function
  Coiflet-4, you see, this one models the response
  somewhat better than Haar ?

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Denoising

• Truncation (denoising is done without sacrificing
  much of the fast changes in the signal, compared
  to linear techniques)
• Hard thresholding (zeroing)
• Soft thresholding (zeroing and shrinking the
  others above the threshold)
• Scale-dependent thresholding
• Time windowing
• Scale-dependent time windowing

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• Example Daubechies-4. Noise in finer
  scales!!! (as usually). Good reason for scale-
  dependent thresholding

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When signal denoising is helpful?

• Producing more accurate measurements of latency
  and time
• Thus, of great value for single-trial analysis
• Improves results of the Woody method (latency
  correction)

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Application of scale-dependent thresholding
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Summary

• The strongest point (as I see) in the wavelets
  is flexibility (2-dimenionality) compared to
  other basis functions analysis we studied.
• Wavelet analysis useful in
• analyzing of transient and nonstationary signals
  (single-trial EPs)
• EP noise reduction denoising
• compression of large amounts of data (other basis
  functions can also be employed)

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Happy end
?

• Oooooopshu!

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Non-covered issues (this and following slides )

• Refinement equation
• Scaling and wavelet coefficients

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Calculating scaling and wavelet coefficients

• Analysis filter bank (top-down, fine-coarse)
• Synthesis filter bank (bottom-up, coarse-fine)