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Daubechies Wavelets

Daubechies Wavelets. A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001. Introduction. A family of wavelet transforms discovered by Ingrid Daubechies Concepts similar to Haar (trend and fluctuation ) Differs in how scaling functions and wavelets are defined longer supports.

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Daubechies Wavelets

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  1. Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001

  2. Introduction • A family of wavelet transforms discovered by Ingrid Daubechies • Concepts similar to Haar (trend and fluctuation) • Differs in how scaling functions and wavelets are defined • longer supports Wavelets are building blocks that can quickly decorrelate data.

  3. The elements in the synthesis and analysis matrices are Haar Wavelets Revisited

  4. Haar Revisited Synthesis Filter P3 Synthesis Filter Q3

  5. In Other Words

  6. Orthonormal; also lead to energy conservation Averaging Orthogonality Differencing How we got the numbers

  7. How we got the numbers (cont)

  8. Daubechies Wavelets • How they look like: • Translated copy • dilation Scaling functions Wavelets

  9. Obtained from natural basis (n-1) level Scaling functions wrap around at end due to periodicity Each (n-1) level function Support: 4 Translation: 2 Trend: average of 4 values Daub4 Scaling Functions (n-1 level)

  10. Obtained from n-1 level scaling functions Each (n-2) scaling function Support: 10 Translation: 4 Trend: average of 10 values This extends to lower levels Daub4 Scaling Function (n-2 level)

  11. Similar “wrap-around” Obtained from natural basis Support/translation: Same as scaling functions Extends to lower-levels Daub4 Wavelets

  12. Numbers of Scaling Function and Wavelets (Daub4)

  13. Property of Daub4 • If a signal f is (approximately) linear over the support of a Daub4 wavelet, then the corresponding fluctuation value is (approximately) zero. • True for functions that have a continuous 2nd derivative

  14. Property of Daub4 (cont)

  15. MRA

  16. Example (Daub4)

  17. More on Scaling Functions (Daub4, N=8) Synthesis Filter P3

  18. Scaling Function (Daub4, N=16) Synthesis Filter P3

  19. Synthesis Filter P1 Synthesis Filter P2 Scaling Functions (Daub4)

  20. Synthesis Filter Q1 Synthesis Filter Q2 More on Wavelets (Daub4) Synthesis Filter Q3

  21. Summary

  22. Analysis and Synthesis • There is another set of matrices that are related to the computation of analysis/decomposition coefficient • In the Daubechies case, they are also the transpose of each other • Later we’ll show that this is a property unique to orthogonal wavelets

  23. f Analysis and Synthesis

  24. MRA (Daub4)

  25. Energy Compaction (Haar vs. Daub4)

  26. Orthonormal; also lead to energy conservation Orthogonality Averaging Differencing Constant Linear How we got the numbers 4 unknowns; 4 eqns

  27. Supplemental

  28. Define Therefore (Exercise: verify) Conservation of Energy

  29. Energy Conservation • By definition:

  30. By construction Haar is also orthogonal Not all wavelets are orthogonal! Semiorthogonal, Biorthogonal Orthogonal Wavelets

  31. Other Wavelets (Daub6)

  32. Daub6 (cont) • Constraints • If a signal f is (approximately) quadratic over the support of a Daub6 wavelet, then the corresponding fluctuation value is (approximately) zero.

  33. DaubJ • Constraints • If a signal f is (approximately) equal to a polynomial of degree less than J/2 over the support of a DaubJ wavelet, then the corresponding fluctuation value is (approximately) zero.

  34. Comparison (Daub20)

  35. Supplemental on Daubechies Wavelets

  36. Coiflets • Designed for maintaining a close match between the trend value and the original signal • Named after the inventor: R. R. Coifman

  37. Ex: Coif6

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