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Wavelet-Based Denoising Using Hidden Markov Models

Wavelet-Based Denoising Using Hidden Markov Models. M. Jaber Borran and Robert D. Nowak Rice University. Some properties of DWT. Primary Locality  Match more signals Multiresolution Compression  Sparse DWT’s Secondary Clustering  Dependency within scale

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Wavelet-Based Denoising Using Hidden Markov Models

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  1. Wavelet-Based Denoising Using Hidden Markov Models M. Jaber Borran and Robert D. Nowak Rice University

  2. Some properties of DWT • Primary • Locality  Match more signals • Multiresolution • Compression Sparse DWT’s • Secondary • Clustering Dependency within scale • Persistence  Dependency across scale

  3. pS(1) fW|S(w|1) pS(2) fW|S(w|2) fW (w) S W Probabilistic Model for an Individual Wavelet Coefficient • Compression many small coefficients few large coefficients

  4. t f Probabilistic Model for a Wavelet Transform Ignoring the dependencies  Independent Mixture (IM) Model Clustering  Hidden Markov Chain Model Persistence  Hidden Markov Tree Model

  5. 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

  6. Signal w1 w1 Wavelet t t T T t 0 t T w2 w2 T/2 T/2 t 0 t T/2 w2 w2 T T/2 T t t T/2 T/2 T Dependency between Signs of Wavelet Coefficients

  7. pS(2) fW|S(w|2) pS(4) fW|S(w|4) pS(1) fW|S(w|1) pS(3) fW|S(w|3) fW (w) S W New Probabilistic Model for Individual Wavelet Coefficients • Use one-sided functions as conditional probability densities

  8. Proposed Mixture PDF • Use exponential distributions as components of the mixture distribution m even m odd

  9. 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, m even m odd

  10. 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).

  11. Model Training Using Expectation Maximization Algorithm • and then, • Define the set of complete data, x = (y,s)

  12. 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.

  13. Denoising • MAP estimate:

  14. Denoising (continued) • Conditional Mean estimate:

  15. Conclusion • We observed a high correlation between the signs of the wavelet coefficients in adjacent scales. • We used one-sided distributions as mixture components for individual wavelet coefficients. • We used hidden Markov tree model to capture the dependencies. • The proposed method achieves better MSE in denoising and the denoised signals are much smoother.

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