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Chapter 03 Multiresolution Analysis (MRA)

Chapter 03 Multiresolution Analysis (MRA). V 0  V 1  V 2. Multiresolution. Gjennomsnitt. V 0. V 1. V 2. V 3. V 4. Differens. W 0. W 1. W 2. W 3. J=5 Antall samplinger: 2 J = 32. Analysis /Synthesis Example. Analysis Synthesis J=5 Sampling: 2 5 = 32. j=5. j=4. j=3.

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Chapter 03 Multiresolution Analysis (MRA)

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  1. Chapter 03Multiresolution Analysis (MRA) V0  V1  V2

  2. Multiresolution Gjennomsnitt V0 V1 V2 V3 V4 Differens W0 W1 W2 W3

  3. J=5 Antall samplinger: 2J = 32 Analysis /SynthesisExample

  4. AnalysisSynthesisJ=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0

  5. Scaling functionExample 1 1 1 2 1 3 n n+1

  6. Scaling function that span V0 Scaling function V0  L2(R)

  7. Scaling Function that span V0Example 1 1 2 3 4 5 5 1 1 2 3 4 5

  8. Scaling Function (unnormalized) that span Vj 1 1 Dilation Translation 1 1 1 1 1 1 1 1 1 1 1 1

  9. Scaling Function (normalized) that span Vj 1 1 Dilation Translation 2 1/2 2 1/2 2 2 2 2

  10. Scaling functions (normalized) Scaling function V0  V1  V2

  11. Normalization of scaling functions Scaling function Inner product Norm Scaling functions (Orthonormal)

  12. Haar Scaling Functions (unnormalized) that span Vj k 0 1 2 3 j 0 1 1 For hver j: Basisfunksjoner: (2jt-k) k = 0,…2 j-1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1

  13. Haar Scaling Functions (normalized) that span Vj k 0 1 2 3 j 0 1 1 For hver j: Basisfunksjoner: j,k(t) k = 0,…2 j-1 2 1/2 2 1/2 1 1 1 2 2 2 2 2 1 1 1 1 2 3/2 2 3/2 2 3/2 2 3/2 3 1 1 1 1

  14. V1  V2 Scaling Function that span V1 and V2 1 1

  15. Haar Scaling Functions that span Vj j = 0,1,2,3 j 0 1 2 3 k 0 1 2 3 4 5 6 7

  16. V0  V1 Relation between V0 and V1Haar Wavelet - Triangle Wavelet Scaling function

  17. Properties of the h-coefficients (1/5) V0  V1

  18. Properties of the h-coefficients (2/5) V0  V1

  19. Properties of the h-coefficients (3/5)

  20. Properties of the h-coefficients (5/5)

  21. Examples of h-coefficients n=2 D2 Haar scaling function n=3 n odd --> one coefficient = 0 n=4 D4 Daubechies four-tap solution One degree of freedom

  22. Examples of h-coefficients n=4 One degree of freedom D2 D4

  23. Examples of h-coefficients n=6

  24. Examples of h-coefficients D6 D8

  25. DaubechiesVanishing moments The continuous wavelet transform (CWT) Taylor series at t=0 until order n (b=0 for simplicity) Moments of the Wavelet

  26. DaubechiesVanishing moments Wavelet until Daubechies: - Haar Compact support, but discontinuous - Shannon Smooth, but extend the whole real line - Linear spline Continuous, but infinite support Daubechies: Hierarchy of Wavelets: n = 2 : Haar Compact support, discontinuous M0 = 0 n = 4 : D4 Compact support, continuous, not diff. Mi = 0 i=0..n/2-1 n = 6 : D6 Compact support, continuous, 1 diff. Mi = 0 i=0..n/2-1 n = 8 : D8 Compact support, continuous, 2 diff. Mi = 0 i=0..n/2-1 ...

  27. Wavelet functions Scaling function V0  V1  V2 W0 W1 Wavelet function

  28. Properties of the g-coefficients Scaling function V0  V1  V2 W0 Wavelet function W1

  29. Decomposition of V3 = V0 + W0 + W1 + W2

  30. Analysis - From Fine Scale to Coarse Scale j=5 j=4

  31. Analysis - From Fine Scale to Coarse Scale Synthesis - From Coarse Scale to Fine Scale Analysis Synthesis

  32. Dirac Delta Function (Standard Time Domain Basis) f t

  33. Fourier (Standard Frequency Domain Basis) f t

  34. Two-band Wavelet Basis f t

  35. J=5 Antall samplinger: 2J = 32 Analysis /SynthesisExample

  36. AnalysisSynthesisJ=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0

  37. END

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