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

1 / 1

Review: Wavelets in a Nutshell

Three Wavelet Examples

–1.0

0.0

1.0

–4 0t

4

0.0

1.0

–4 0 4t

ψ

–1.0

–0.5

0.0

0.5

1.0

–6 –4 –2 0 2 4 6

t

Wavelets = packets

Nonstationary signals

Basis functions

All oscillate

Varied functional forms

Vary scale & center

Finite ∆ τ ∆ ω

∆τ ∆ω ≥ 2π

2 / 1

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

3 / 1

How to Discretize DWT?

Auto Scalings, Translations = ♥Wavelets

Discrete scaling s, discrete time translation τ :

s = 2j , τ =k2j, 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)

4 / 1

Time & Frequency Sampling

Sample y(t) in Time & Frequency Ranges

Time

Freq

uen

cy

High ω ↑ rangeHigh ω for detailsFew low ω for shapeEach t , ∆ scales

Uncertainty Prin:∆ω∆t ≥ 2πDon’t be wasteful!⇒ H ×W = Const

5 / 1

Multi Resolution Analysis (MRA)

Digital Wavelet Transform ≡ Filter

L

L

H H

LL

LHH

2

2

2

2

DataInput 2

2

2

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

g(t) =∫ +∞−∞

dτ h(t − τ) y(τ) (Filter)

Y (s, τ) =∫ +∞−∞

dt Ψ∗(

t − τs

)y(t) '

∑wiψiy(ti ) (Transform)

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

6 / 1

MRA via Filter Tree (Pyramid Algorithm)

Filtering with Decimation

L

L

H H

LL

LHH

2

2

2

2

DataInput 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

7 / 1

Example from Appendix

High → Medium → Low Resolution

8 / 1

Summary

Pyramid DWT algorithm compresses data, separates hi resSmooth info in low-ω (large s) componentsDetailed info in high-ω (small s) componentsHigh-res reproduction: more info on details than shapeDifferent resolution components = independent⇒ Lower data storageRapid reproduction/inversion (JPEG2)

9 / 1

Pyramid Algorithm Graphically (see text)

InputN 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

CoefficientsCoefficients

Coefficients

Coefficients

L

L

L

L

H

H

H

H

L & H via matrix mult (TFs)Decimated H output saved

Downsample: ↓ #, ∆ scaleEnds with 2 H, L points

10 / 1

N = 8 Example Matrices

y1y2y3y4y5y6y7y8

filter−→

s(1)1d (1)1s(1)2d (1)2s(1)3d (1)3s(1)4d (1)4

order−→

s(1)1s(1)2s(1)3s(1)4d (1)1d (1)2d (1)3d (1)4

filter−→

s(2)1d (2)1s(2)2d (2)2d (1)1d (1)2d (1)3d (1)4

order−→

s(2)1s(2)2d (2)1d (2)2d (1)1d (1)2d (1)3d (1)4

11 / 1

Pyramid Algorithm Matrices

Pyramid Algorithm Successive Operations1 Mult N-D vector of Y by c matrix2 (See text for ci derivation)

Y0Y1Y2Y3

=

c0 c1 c2 c3c3 −c2 c1 −c0c2 c3 c0 c1c1 −c0 c3 −c2

y0y1y2y3

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

12 / 1

Inversion Y → y

Using transpose (inverse) of transfer matrix at each stagey0y1y2y3

=

c0 c3 c2 c1c1 −c2 c3 −c0c2 c1 c0 c3c3 −c0 c1 −c2

Y0Y1Y2Y3

.

13 / 1

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

14 / 1

Daubechies Daub4 Wavelet (Derivation in Text)

–0.1

–0.06

–0.02

0.02

0.06

0.1

0 400 800 1200

c0 =1 +√

34√

2, c1 =

3 +√

34√

2

c2 =3−√

34√

2, c3 =

1−√

34√

2

15 / 1

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

16 / 1

ReviewProblem: Optimize Wavelet Transform and InverseDiscrete