Review: Wavelets in a Nutshell Three Wavelet Examples Varied - - PowerPoint PPT Presentation

SLIDE 1

Discrete Wavelet Transforms⊙

Industrial-Strength, Technology-Enabling Computing (look, listen, read) Rubin H Landau

Sally Haerer, Producer-Director

Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation

Course: Computational Physics II

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SLIDE 2

Review: Wavelets in a Nutshell

Three Wavelet Examples

–1.0 0.0 1.0 –4

t

4 0.0 1.0 –4 4

t ψ

–1.0 –0.5 0.0 0.5 1.0 –6 –4 –2 2 4 6 t

Wavelets = packets Nonstationary signals Basis functions All oscillate Varied functional forms Vary scale & center Finite ∆ τ ∆ ω ∆τ ∆ω ≥ 2π

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SLIDE 3

Problem: Determine ≤ N Indep Wavelet TFs Yi,j

The Discrete Wavelet Transform (DWT) Y(s, τ) = +∞

−∞

dt ψ∗

s,τ(t) y(t)

(Wavelet Transform) Given: N signal measurements: y(tm) ≡ ym, m = 1, . . . , N Compute no more DWTs than needed Hint: Lossless: consistent with uncertainty principle Hint: Lossy: consistent with required resolution

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SLIDE 4

How to Discretize DWT?

Auto Scalings, Translations = ♥ Wavelets Discrete scaling s, discrete time translation τ:

s = 2j, τ = k 2j , k, j = 0, 1, . . . (Dyadic Grid) (1) ψj,k(t) = 1 √ 2j Ψ t − k2j 2j

• (Wavelets T = 1)

(2) Yj,k = +∞

−∞

dt ψj,k(t) y(t) (3) ≃

• m

ψj,k(tm)y(tm)h (DWT) (4)

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SLIDE 5

Time & Frequency Sampling

Sample y(t) in Time & Frequency Ranges

Time Frequency

High ω ↑ range High ω for details Few low ω for shape Each t, ∆ scales Uncertainty Prin: ∆ω ∆t ≥ 2π Don’t be wasteful! ⇒ H × W = Const

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SLIDE 6

Multi Resolution Analysis (MRA)

Digital Wavelet Transform ≡ Filter

L L H H LL LH H

2 2 2 2

D a t a I n p u t

2 2 2

Filter: ∆ relative ω strengths ≡ analyze ∆ scale: MRA Sample → Filter → Sample · · ·

g(t) = +∞

−∞

dτ h(t − τ) y(τ) (Filter) Y(s, τ) = +∞

−∞

dt Ψ∗ t − τ s

• y(t) ≃
• wiψiy(ti)

(Transform)

wi = integration weight + wavelet values = “filter coeff”

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SLIDE 7

MRA via Filter Tree (Pyramid Algorithm)

Filtering with Decimation

L L H H LL L H H

2 2 2 2

Data Input

2 2 2

H: highpass filters L: lowpass filters Ea filter: lowers scale ↓2: rm 1/2 signal Factor-of-2 “decimation” "Subsampling" Keeps area constant Need little large-s info

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SLIDE 8

Example from Appendix

High → Medium → Low Resolution

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SLIDE 9

Summary

Pyramid DWT algorithm compresses data, separates hi res Smooth info in low-ω (large s) components Detailed info in high-ω (small s) components High-res reproduction: more info on details than shape Different resolution components = independent ⇒ Lower data storage Rapid reproduction/inversion (JPEG2)

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SLIDE 10

Pyramid Algorithm Graphically (see text)

Input

N Samples

N/2 N/4 N/8 2 N/2 N/4 N/8 2 c c c c d d d d

(1) (2) (3) (n) (1) (2) (3) (n)

Coefficients Coefficients Coefficients Coefficients Coefficients Coefficients Coefficients Coefficients

L L L L H H H H

L & H via matrix mult (TFs) Decimated H output saved Downsample: ↓ #, ∆ scale Ends with 2 H, L points

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SLIDE 11

N = 8 Example Matrices

               y1 y2 y3 y4 y5 y6 y7 y8                filter − →                   s(1)

1

d(1)

1

s(1)

2

d(1)

2

s(1)

3

d(1)

3

s(1)

4

d(1)

4

                 

• rder

− →                   s(1)

1

s(1)

2

s(1)

3

s(1)

4

d(1)

1

d(1)

2

d(1)

3

d(1)

4

                  filter − →                   s(2)

1

d(2)

1

s(2)

2

d(2)

2

d(1)

1

d(1)

2

d(1)

3

d(1)

4

                 

• rder

− →                   s(2)

1

s(2)

2

d(2)

1

d(2)

2

d(1)

1

d(1)

2

d(1)

3

d(1)

4

                 

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SLIDE 12

Pyramid Algorithm Matrices

Pyramid Algorithm Successive Operations

1

Mult N-D vector of Y by c matrix

2

(See text for ci derivation)

     Y0 Y1 Y2 Y3      =      c0 c1 c2 c3 c3 −c2 c1 −c0 c2 c3 c0 c1 c1 −c0 c3 −c2           y0 y1 y2 y3     

3

Mult (N/2)-D smooth vector by c matrix

4

Reorder: new 2 smooth on top, new detailed, older detailed

5

Repeat until only 2 smooth remain

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SLIDE 13

Inversion Y → y

Using transpose (inverse) of transfer matrix at each stage      y0 y1 y2 y3      =      c0 c3 c2 c1 c1 −c2 c3 −c0 c2 c1 c0 c3 c3 −c0 c1 −c2           Y0 Y1 Y2 Y3      .

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SLIDE 14

Chirp Example Graphical

1024 sin(60t2) 1024 thru H & L Downsample → 512 L, 512 H Save details Each step ↓ 2× Connected dots End: 2 ↓ detail

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SLIDE 15

Daubechies Daub4 Wavelet (Derivation in Text)

–0.1 –0.06 –0.02 0.02 0.06 0.1 400 800 1200

c0 = 1 + √ 3 4 √ 2 , c1 = 3 + √ 3 4 √ 2 c2 = 3 − √ 3 4 √ 2 , c3 = 1 − √ 3 4 √ 2

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SLIDE 16

Summary: Wavelet Transforms

Continuous → Discrete → Pyramid Algorithm Y(s, τ) = +∞

−∞

dt ψ∗

s,τ(t) y(t)

Discrete: measurements,

• →

i

Transform → digital filter → coefficients Multiple scales → series H & L filters Compression: N independent components Further compression: Variable resolution

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