Chapter 03 Multiresolution Analysis MRA V 0 V

Chapter 03 Multiresolution Analysis (MRA) V 0 V 1 V 2 1

Multiresolution Gjennomsnitt V 0 V 1 V 2 W 1 W 2 V 3 V 4 Differens W 0 W 3 2

Analysis /Synthesis Example J=5 Antall samplinger: 2 J = 32 3

Analysis Synthesis J=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0 4

Scaling function Example 1 1 1 2 1 3 n n+1 5

Scaling function that span V 0 Scaling function V 0 L 2(R) 6

Scaling Function that span V 0 Example 1 1 2 3 4 5 5 1 4 5 7

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

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

Scaling functions (normalized) Scaling function V 0 V 1 V 2 10

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

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

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

V 1 V 2 Scaling Function that span V 1 and V 2 1 1 14

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

Relation between V 0 and V 1 V 0 V 1 Haar Wavelet - Triangle Wavelet Scaling function 16

Properties of the h-coefficients (1/5) V 0 V 1 17

Properties of the h-coefficients (2/5) V 0 V 1 18

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

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

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

Examples of h-coefficients n=4 One degree of freedom D 2 D 4 22

Examples of h-coefficients n=6 23

Examples of h-coefficients D 6 D 8 24

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

Daubechies Vanishing 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 n = 4 : D 4 Compact support, continuous, not diff. n = 6 : D 6 Compact support, continuous, 1 diff. n = 8 : D 8 Compact support, continuous, 2 diff. . M 0 Mi Mi Mi =0 = 0 i=0. . n/2 -1 26

Wavelet functions Scaling function V 0 V 1 V 2 W 0 W 1 Wavelet function 27

Properties of the g-coefficients Scaling function Wavelet function V 0 V 1 V 2 W 0 W 1 28

Decomposition of V 3 = V 0 + W 1 + W 2 29

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

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

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

Fourier (Standard Frequency Domain Basis) f t 33

Two-band Wavelet Basis f t 34

Analysis /Synthesis Example J=5 Antall samplinger: 2 J = 32 35

Analysis Synthesis J=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0 36

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