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Scaling functions

Scaling functions. ‘or connect the dots’. Fix filter no restrictions yet:. FUND. DEFN:. Scaling Function. relates at two levels of resolution. Basic condition:. Examples so far:. Box:. Tent centered at :. Daubechies D4: does there exist ?.

jason-booth
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Scaling functions

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  1. Scaling functions ‘or connect the dots’ Fix filter no restrictions yet: FUND. DEFN: Scaling Function relates at two levels of resolution. Basic condition:

  2. Examples so far: Box: Tent centered at : Daubechies D4: does there exist ?

  3. Fractal example:

  4. Dyadic rationals: determined at dyadic rationals : Convolution on integers? Powers of 2? KNOW ALL , THENKNOWALL Construct on all as limiting fixed point!

  5. Iterative process: with limit Construct sequence functions such that Then What about convergence? Pointwise, in Energy? Pointwise:start with Tent function In energy:start with Box function

  6. Getting started: Tent function centered at origin: Basic idea: set for suitable

  7. Filter conditions: Need in so that Conditions on : Solve using Fourier Transforms as usual.

  8. Fourier Transforms: Set Then So:

  9. Up-sampling again! Recall Crucial results: where in z-transform notation: Use these to compute .

  10. Connect the dots! Daub-4 Depths: 1, 2, 4, 6

  11. Cascade Algorithm: convergence in energy Start with box function: can exploit orthonormality. with as before, but no Vetterli condition yet. So

  12. Orthonormality: Case: k = 0 Can we recognize sequence: ?

  13. Finally Vetterli! Consider first: Crucial identification: Fourier transform:

  14. Finally Vetterli! When we deduce that So, hence , ORTHONORMAL FAMILY foreachk.

  15. In the limit! When in energy, then so Vetterli ensures orthonormal family in .

  16. Finally wavelets: Fix FIR filter Assume convergence in energy and Vetterli. Set define wavelet by compactly supported if compactly supported. By same argument as for : So need to identify .

  17. More results for wavelets: Recall , so By Vetterli yet again: , so Thus orthonormal family.

  18. Still more results: By same argument yet again: But by Fourier transforms yet again: where remember, . Thus: all .

  19. Main Theorem Part 1: FIR filter If then is a continuous function with derivatives.

  20. Main Theorem Part 2: Suppose also satisfies Vetterli condition. Define wavelet: . Then: 1. , orthonormal families, , 2. 3. . complete orthonormal family in

  21. Main Theorem applied to Daub-4: so hence Daub-4 continuous, not quite differentiable

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