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Haar Wavelets

Haar Wavelets. A first look Ref: Walker (ch1) Jyun-Ming Chen, Spring 2001. Introduction. Simplest; hand calculation suffice A prototype for studying more sophisticated wavelets Related to Haar transform, a mathematical operation.

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Haar Wavelets

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  1. Haar Wavelets A first look Ref: Walker (ch1) Jyun-Ming Chen, Spring 2001

  2. Introduction • Simplest; hand calculation suffice • A prototype for studying more sophisticated wavelets • Related to Haar transform, a mathematical operation

  3. Assume discrete signal (analog function occurring at discrete instants) Assume equally spaced samples (number of samples 2n) Decompose the signal into two sub-signals of half its length Running average (trend) Running difference (fluctuation) Haar Transform

  4. Running difference Denoted by: Meaning of superscript explained later Running average Multiplication by is needed to ensure energy conservation (see later) Haar transform, 1-level

  5. Example

  6. Inverse Transform

  7. Small Fluctuation Feature • Magnitudes of the fluctuation subsignal (d) are often significantly smaller than those of the original signal • Logical: samples are from continuous analog signal with very short time increment • Has application to signal compression

  8. Energy Concerns • Energy of signals • The 1-level Haar transform conserves energy

  9. Proof of Energy Conservation

  10. f Haar Transform, multi-level

  11. Compaction of Energy • Compare with 1-level • Can be seen more clearly by cumulative energy profile

  12. Definition Cumulative Energy Profile

  13. Algebraic Operations • Addition & subtraction • Constant multiple • Scalar product

  14. 1-level Haar wavelets “wavelet”: plus/minus wavy nature Translated copy of mother wavelet support of wavelet =2 The interval where function is nonzero Haar Wavelets Property 1. If a signal f is (approximately) constant over the support of a Haar wavelet, then the fluctuation value is (approximately) zero.

  15. 1-level scaling functions Graph: translated copy of father scaling function Support = 2 Haar Scaling Functions

  16. 2-level Haar scaling functions support = 4 2-level Haar wavelets support = 4 Haar Wavelets (cont)

  17. Natural basis: Therefore: Multiresolution Analysis (MRA)

  18. Note: the coefficient vectors MRA

  19. If do it all the way through, representing the average of all data MRA

  20. Example

  21. Example (cont) Decomposition coefficients obtained by inner product with basis function

  22. Haar MRA

  23. They are in fact related Pj is called the synthesis filter (more later) More on Scaling Functions (Haar)

  24. Synthesis Filter P3 Ex: Haar Scaling Functions

  25. Synthesis Filter P1 Synthesis Filter P2 Ex: Haar Scaling Functions

  26. They are in fact related Qj is called the synthesis filter (more later) More on Wavelets (Haar)

  27. Synthesis Filter Q3 Ex: Haar Wavelets

  28. Ex: Haar Wavelets Synthesis Filter Q1 Synthesis Filter Q2

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

  30. Analysis/Decomposition (Haar) A2 A3 B2 Analysis Filter Aj Analysis Filter Bj B3 A1 B1

  31. On the other hand, synthesis filters have to do with reconstructing the signal from MRA results Synthesis Filters

  32. Q1 P1 Q2 P2 Q3 P3 Synthesis/Reconstruction (Haar) Synthesis Filter Pj Synthesis Filter Qj

  33. Conclusion/Exercise

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