Wavelet-Based Denoising Using Hidden Markov Models

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Wavelet-Based Denoising Using Hidden Markov Models

• ELEC 631 Course Project
• Mohammad Jaber Borran

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Some properties of DWT

• Primary
• Locality
• ? Match more
  signals
• Multiresolution
• Compression ? Sparse DWTs
• Secondary
• Clustering ? Dependency within scale
• Persistence ? Dependency across scale

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Probabilistic Model for an Individual Wavelet
Coefficient

• Compression ? many small coefficients
• few large
  coefficients

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Probabilistic Model for a Wavelet Transform
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Parameters of HMT Model

• pmf of the root node
• transition probability
• (parameters of the)
• conditional pdfs
• e.g. if Gaussian Mixture is used

q Model Parameter Vector
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Dependency between Signs of Wavelet Coefficients
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New Probabilistic Model for Individual Wavelet
Coefficients

• Use one-sided functions as conditional
  probability densities

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Proposed Mixture PDF

• Use exponential distributions as components of
  the mixture distribution

If m is even
If m is odd
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PDF of the Noisy Wavelet Coefficients
Wavelet transform is orthonormal, therefore if
the additive noise is white and zero-mean
Gaussian process with variance s2, then we have
Noisy wavelet coefficient,
If m is even
If m is odd
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Training the HMT Model

• y Observed noisy wavelet coefficients
• s Vector of hidden states
• q Model parameter vector
• Maximum likelihood parameter estimation

Intractable, because s is unobserved (hidden).
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Model Training Using Expectation Maximization
Algorithm

• Define the set of complete data, x (y,s)

• and then,

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EM Algorithm (continued)

• State a posteriori probabilities are calculated
  using Upward-Downward algorithm
• Root state a priori pmf and the state transition
  probabilities are calculated using Lagrange
  multipliers for maximizing U.
• Parameters of the conditional pdf may be
  calculated analytically or numerically, to
  maximize the function U.

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Denoising

• MAP estimate

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Denoising (continued)

• Conditional mean estimate

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Conclusion

• Mixture distributions for individual wavelet
  coefficients can effectively model the
  nonGaussian nature of the coefficients.
• Hidden Markov Models can serve as a powerful tool
  for wavelet-based statistical signal processing.
• One-sided exponential distributions for mixture
  components along with hidden Markov Tree model
  can achieve better performance in denoising.

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