Wavelets ?

Raghu MachirajuContributions:

Robert Moorhead, James Fowler, David Thompson, Mississippi State University

Ioannis Kakadaris, U of Houston

April 24, 2023 2

State-Of-Affairs

Simulations, scanners

Presentation

Analysis

Representation

Retrospective

Concurrent

April 24, 2023 3

Why Wavelets?•We are generating and measuring larger datasets every year

•We can not store all the data we create (too much, too fast)

•We can not look at all the data (too busy, too hard)

•We need to develop techniques to store the data in better formats

April 24, 2023 4

Data Analysis

• Frequency spectrum correctly shows a spike at 10 Hz

• Spike not narrow - significant component at between 5 and 15 Hz.

•Leakage - discrete data acquisition does not stop at exactly the same phase in the sine wave as it started.

April 24, 2023 5

QuickFix

April 24, 2023 6

Windowing &Filtering

April 24, 2023 7

Image Example

• 8x8 Blocked Window (Cosine) Transform•Each DCT basis waveform represents a fixed frequency in two orthogonal directions

•frequency spacing in each direction is an integer multiple of a base frequency

April 24, 2023 8

Windowing & Filtering

Windows –

fixed in space and frequencies

Cannot resolve all features at all instants

April 24, 2023 9

I (x |) G(, x) I(x) G(,)I(x ) d

= 1 = 16 = 24 = 32

input

Linear Scale Space

April 24, 2023 10

Successive Smoothing

April 24, 2023 11

•Keep 1 of 4 values from 2x2 blocks

•This naive approach and introduces aliasing

•Sub-samples are bad representatives of area

•Little spatial correlation

Sub-sampled Images

April 24, 2023 12

Image Pyramid

April 24, 2023 13

•Average over a 2x2 block

•This is a rather straight forward approach

•This reduces aliasing and is a better representation

•However, this produces 11% expansion in the data

Image Pyramid – MIP MAP

April 24, 2023 14

Image Pyramid – Another Twist

10 2 8 4 6 2 4 0

6 6 4 2

6 34.5

4 2 2 4

4 2 2 4 0 1

4 2 2 4 0 1 1.5

Average DifferencesAverage Sum

Pyramids

This is the Haar Wavelet Transform

April 24, 2023 15

time/space

Frequency

t

sin(nw0t)Impulse

t

Spectrum

Time Frequency Diagram

April 24, 2023 16

Ideally !Create new signal G such that ||F-G|| =

April 24, 2023 17

Wavelet Analysis

D3

D2

D1

A1

A1 D1 D2 D3

x

D1

A2D3

D2

April 24, 2023 18

•We need to develop techniques to analyze data better through noise discrimination

•Wavelets can be used to detect features and to compare features

•Wavelets can provide compressed representations

•Wavelet Theory provides a unified framework for data processing

Why Wavelets? Because …

April 24, 2023 19

Scale-Coherent Structures

Coherent structure - frequencies at all scales

Examples - edges, peaks, ridges

Locate extent and assign saliency

April 24, 2023 20

Wavelets – Analysis

April 24, 2023 21

Wavelets – DeNoising

April 24, 2023 22

Wavelets – Compression

Original 50:1

April 24, 2023 23

Wavelets – Compression

Original 50:1

April 24, 2023 24

1/1 1/4 1/16

Wavelets – Compression

April 24, 2023 25

Yet Another Example

50%

7%

April 24, 2023 26

Final Example

100% 50% 2%

1%

April 24, 2023 27

Information Rate Curve

0.0 0.2 0.4 0.6 0.8 1.0normalized rate0.0

0.2

0.4

0.6

0.8

1.0

norm

alize

d in

form

atio

n(E

)

densityu momentumv momentumw momentumenergy

•Energy Compaction – Few coefficients can efficiently represent functions

•The Curve should be as vertical as possible near 0 rate

April 24, 2023 28

H

G

2

2

H

G

2

2

H

G

2

2

x0

x1

D1

x2

D2

x3

D3

Mallat’s Pyramid Algorithm

Filter Bank Implementation

G: High Pass Filter H: Low Pass Filter

April 24, 2023 29

10 2 8 4 6 2 4 0

3

H2

G2

G2

x3

d2

d3

x1

x0

H2

H = Summation FilterG = Difference Filter

(The Other Half) Synthesis Filter Bank

G2

H2

d1

x2

+ -

0 1

4 2 2 4

1.54.5

6

6

6 4 2

Synthesis Bank

April 24, 2023 30

x

A2

A1

A3

Successive Approximations

April 24, 2023 31

x

D2

D1

D3

Successive Details

April 24, 2023 32

x

D2

D1

A2

Wavelet Representation

April 24, 2023 33

Coefficients

D1D2D3A3

April 24, 2023 34

Lossey Compression

April 24, 2023 35

Lossey Compression

April 24, 2023 36

A Frame

Another Frame

Image Example

April 24, 2023 37

Average

Difference

Image Example

April 24, 2023 38

Rows

ColumnsApprox. at res. j

Low Row, High Column

High Row, Low Column

High Row, High Column

Approx.at res. j+1

Boxes have same meaning!

Wavelet Transform

April 24, 2023 39

LH1

HL1

0

/4

/2

/2 /4

HH1

LL1 LH1

HL1

HH2HL2

LL2 LH2

0

/4

/2

/2 /4

HH1

Frequency Support

April 24, 2023 40

LvLh LvHh

HvLhHvHh

Image Example

April 24, 2023 41

LvLh LvHh

HvLhHvHh

Image Example

April 24, 2023 42

f(x)=

+

+

+

Desired Goal!

A2

D2

D1

0 1

0 1

x 1 0 x 1

0 otherwise

=

x

1– 0 x 12---

1 12--- x 1

0 otherwise

=

How Does One Do This ?

April 24, 2023 43

Dilations•Rescaling Operation t --> 2t

•Down Sampling, n --> 2n

•Halve function support

•Double frequency content

•Octave division of spectrum- Gives rise to different scales and resolutions

•Mother wavelet! - basic function gives rise to differing versions

j x 1

2

j2---

---- x2

j--- =

April 24, 2023 44

box(t)

box (2t)

Finer Resolution, j+1

Coarser Resolution, j

0

sin 2t

sin t

fn = sin an

fn = sin a2n

Rescaling DownSampling

0 1/2

0 1

t

Dilations

April 24, 2023 45

Successive Approximations

f(x)

A2

A1

A3

April 24, 2023 46

box (x-1)

box (2x-1)

box (x)

box (2x-1)

Finer Resolution, j+1

Coarser Resolution, j

x

freq

Translations•Covers space-frequency diagram

•Versions are

jk x 1

2

j2---

---- x k–

2j---------- =

April 24, 2023 47

D1D2D3A3

spectrum

V3

V2 =

+W3

Wavelet Decomposition

•Induced functional Space - Wj.

•Related to Vjs

•Space Wj+1 is orthogonal to Vj+1

•Also

•J-level wavelet decomposition -

Vj Vj 1+ Wj 1+=

Wj 1+ Vj

Vj Vj 2+ Wj 2+ Wj 1+ =

V0 VJ WJ WJ 1– WJ 2– W1+ + + + +=

April 24, 2023 48

Successive Differences

April 24, 2023 49

•Wavelet expansion (Tiling- j: scale, k: translates), Synthesis

•Orthogonal transformation, Coarsest level of resolution - J

•Smoothing function - , Detail function -

•Analysis:

•Commonly used wavelets are Haar, Daubechies and Coiflets

f x aJkJk x k djkjk x

k

j 1=

J+=

aJk f x( )Jk x( ) td–= d jk f x( )jk x( ) td–

=

Wavelet Expansion

April 24, 2023 50

Scaling Functions•Compact support

•Bandlimited - cut-off frequency

•Cannot achieve both

•DC value (or the average) is defined

•Translates of are orthogonal

x xd 1=

x x k– xd k =

spectrum

c

April 24, 2023 51

h(0) = 1/2, h(1) = 1/2

0 1/4 1/2 3/4 1 t

Dilation Equation

coarset

t

t

Scaling Functions

•Nested smooth spaces

•Dilation Equation - Haar

•Generally –

•Frequency Domain

L V V V

t 2h 0 2t 2h 1 2t 1– +=

t 2hk 2t k– =

12

------- H2----

0 1

=

) ) )

April 24, 2023 52

g(0) = 1/2, g(1) = -1/2

0 1/2 1

0 1/2

Wavelet Functions•Wavelet Equation - Haar System: G Filter

•Generally t 2g 0 2t 2g– 1 2t 1– =

t 2gk 2t k– =

April 24, 2023 53

G2

H2

G

H 2

General Case G

2

22

Perfect Reconstruction

•Synthesis and Analysis Filter Banks

•Synthesis Filters - Transpose of Analysis filters•For compact scaling function

h n 2=

April 24, 2023 54

Orthogonal Filter Banks

•Alternating Flip

•Not symmetric - h is even length!

•Example

•Orthogonality conditions

g k 1– kh N k– =

H h0 h1 h2 h3 = HT h3 h2 h1 h0

=

G h3 h2– h1 h0– = GT h0– h1 h2– h3 =

h2 n k = h n g n 2k– 0=

H 2H + 2

+ 2= H2---

1=

April 24, 2023 55

h 0 h 1 + 2=

h2 0 h2 1 + 1=

h 0 12

-------= h 1 12

-------=

Examples

Haar

h 0 h 1 h 2 h 3 + + + 2=

h2 0 h2 1 h2 2 h 3 + + + 1=h 0 h 2 h 1 h 3 + 0=

h 0 1 3+4 2

----------------= h 1 3 3+4 2

----------------=

h 2 3 3–4 2

----------------= h 0 1 3–4 2

----------------=

Daubechies(2)

April 24, 2023 56

Approximation: Vanishing Moments Property

•Function is smooth - Taylor Series expansion

•Wavelets with m vanishing moments

•Function with m derivatives can be accurately represented!

f x fp

0 xp

p!------

p 0==

W f x ; fp

0 tp

p!-----

p m 1+ ==

April 24, 2023 57

Design of Compact Orthogonal Wavelets

•Compute scaling function

•Use Refinement Equation

•N vanishing moments property - H() has a zero of order N at =

•P(y) is pth order polynomial (Daubechies 1992)

•Maxflat filter

12

------- H2----

0 1

=

) ) )

H 1 e i–+2

--------------------- p

Q =

Q P2---- 2sin

=

) )

)

April 24, 2023 58

Example

N=4

April 24, 2023 59

N=16

Example

April 24, 2023 60

Noise•Uncorrelated Gaussian noise is correlated

•Region of correlation is small at coarse scale

•Smooth versions - no noise

•Orthogonal transform - uncorrelated

April 24, 2023 61

Noise Across Scales

April 24, 2023 62

Denoising

•Statistical thresholding methods [Donohoe]

•Assuming Gaussian Noise

•Universal Threshold

•Smoothness guaranteed

•Hard

•Soft

•Works for additive noise since wavelet transform is linear

2 n log=

yhard t x t x t =

ysoft t x t x t –sgn x t =

W a b f ;+ W a b f ; W a b ; +=

April 24, 2023 63

April 24, 2023 64

D1

D3

D2

A3

Discontinuity

April 24, 2023 65

Multi-scale Edges •Mallat and Hwang

•Location - maximas (edges) of wavelet coefficients at all scales

•Maxima chains for each edge

•Ranking - compute Lipschitz coefficient at all points

•Representation - store maximas

•Reconstruction- approximate but works in practice

April 24, 2023 66

Bi-Orthogonal Filter Banks•Analysis/synthesis different

•Aliasing - overlap in spectras

•Alias cancellation

•Distortion Free (phase shift l)

•Alternating Flip condition valid

•Can be odd length, symmetric

H1 H0

+ G1 G0

+ + 0=

H1 H0

G1 G0

+ 2e jl–=

G12

H12

G0 2

H0 2

April 24, 2023 67

Bi-Orthogonal Wavelets•Governing equations

•Spline Wavelets - Many choices of either H0 or H1

•Choose H 0 as spline and solve equations to generate H1

g n 1– nh1 N n– =

g1 n 1– nh N n– =

h n h1 n 2k+ n k =

2

22

21,-2,11,2,-6,2,1

1,2,1-1,2,6,2,-1

April 24, 2023 68

Bi-orthogonal: Lifting Scheme•Lazy wavelet transform: split data in 2 parts

•Keep even part; predict (linear/cubic) odd part

•Lifting - update j+1 with j+1: Maintain properties (moments, avg.)

•Synthesis is just flip of analysis

-

+

split predictupdate

j,k

j+1,k

j+1,k

April 24, 2023 69

Summary Wavelets have good representation property They improve on image pyramid schemes Orthogonal and biorthogonal filter bank

implementations are efficient Wavelets can filter signals They can efficiently denoise signals The presence of singularities can be detected from

the magnitude of wavelet coefficients and their behavior across scales