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Wavelets ? Raghu Machiraju Contributions: Robert Moorhead, James Fowler, David Thompson, Mississippi State University Ioannis Kakadaris, U of Houston
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Wavelets ?

Mar 13, 2016

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Wavelets ?. Raghu Machiraju Contributions: Robert Moorhead, James Fowler, David Thompson, Mississippi State University Ioannis Kakadaris, U of Houston. Simulations, scanners. State-Of-Affairs. Concurrent. Presentation. Retrospective. Analysis. Representation. Why Wavelets? - PowerPoint PPT Presentation
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Page 1: Wavelets ?

Wavelets ?

Raghu MachirajuContributions:

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

Ioannis Kakadaris, U of Houston

Page 2: Wavelets ?

April 24, 2023 2

State-Of-Affairs

Simulations, scanners

Presentation

Analysis

Representation

Retrospective

Concurrent

Page 3: Wavelets ?

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

Page 4: Wavelets ?

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.

Page 5: Wavelets ?

April 24, 2023 5

QuickFix

Page 6: Wavelets ?

April 24, 2023 6

Windowing &Filtering

Page 7: Wavelets ?

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

Page 8: Wavelets ?

April 24, 2023 8

Windowing & Filtering

Windows –

fixed in space and frequencies

Cannot resolve all features at all instants

Page 9: Wavelets ?

April 24, 2023 9

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

= 1 = 16 = 24 = 32

input

Linear Scale Space

Page 10: Wavelets ?

April 24, 2023 10

Successive Smoothing

Page 11: Wavelets ?

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

Page 12: Wavelets ?

April 24, 2023 12

Image Pyramid

Page 13: Wavelets ?

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

Page 14: Wavelets ?

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

Page 15: Wavelets ?

April 24, 2023 15

time/space

Frequency

t

sin(nw0t)Impulse

t

Spectrum

Time Frequency Diagram

Page 16: Wavelets ?

April 24, 2023 16

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

Page 17: Wavelets ?

April 24, 2023 17

Wavelet Analysis

D3

D2

D1

A1

A1 D1 D2 D3

x

D1

A2D3

D2

Page 18: Wavelets ?

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 …

Page 19: Wavelets ?

April 24, 2023 19

Scale-Coherent Structures

Coherent structure - frequencies at all scales

Examples - edges, peaks, ridges

Locate extent and assign saliency

Page 20: Wavelets ?

April 24, 2023 20

Wavelets – Analysis

Page 21: Wavelets ?

April 24, 2023 21

Wavelets – DeNoising

Page 22: Wavelets ?

April 24, 2023 22

Wavelets – Compression

Original 50:1

Page 23: Wavelets ?

April 24, 2023 23

Wavelets – Compression

Original 50:1

Page 24: Wavelets ?

April 24, 2023 24

1/1 1/4 1/16

Wavelets – Compression

Page 25: Wavelets ?

April 24, 2023 25

Yet Another Example

50%

7%

Page 26: Wavelets ?

April 24, 2023 26

Final Example

100% 50% 2%

1%

Page 27: Wavelets ?

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

Page 28: Wavelets ?

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

Page 29: Wavelets ?

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

Page 30: Wavelets ?

April 24, 2023 30

x

A2

A1

A3

Successive Approximations

Page 31: Wavelets ?

April 24, 2023 31

x

D2

D1

D3

Successive Details

Page 32: Wavelets ?

April 24, 2023 32

x

D2

D1

A2

Wavelet Representation

Page 33: Wavelets ?

April 24, 2023 33

Coefficients

D1D2D3A3

Page 34: Wavelets ?

April 24, 2023 34

Lossey Compression

Page 35: Wavelets ?

April 24, 2023 35

Lossey Compression

Page 36: Wavelets ?

April 24, 2023 36

A Frame

Another Frame

Image Example

Page 37: Wavelets ?

April 24, 2023 37

Average

Difference

Image Example

Page 38: Wavelets ?

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

Page 39: Wavelets ?

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

Page 40: Wavelets ?

April 24, 2023 40

LvLh LvHh

HvLhHvHh

Image Example

Page 41: Wavelets ?

April 24, 2023 41

LvLh LvHh

HvLhHvHh

Image Example

Page 42: Wavelets ?

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 ?

Page 43: Wavelets ?

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--- =

Page 44: Wavelets ?

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

Page 45: Wavelets ?

April 24, 2023 45

Successive Approximations

f(x)

A2

A1

A3

Page 46: Wavelets ?

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---------- =

Page 47: Wavelets ?

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+ + + + +=

Page 48: Wavelets ?

April 24, 2023 48

Successive Differences

Page 49: Wavelets ?

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

Page 50: Wavelets ?

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

Page 51: Wavelets ?

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

=

) ) )

Page 52: Wavelets ?

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– =

Page 53: Wavelets ?

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=

Page 54: Wavelets ?

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=

Page 55: Wavelets ?

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)

Page 56: Wavelets ?

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+ ==

Page 57: Wavelets ?

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

=

) )

)

Page 58: Wavelets ?

April 24, 2023 58

Example

N=4

Page 59: Wavelets ?

April 24, 2023 59

N=16

Example

Page 60: Wavelets ?

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

Page 61: Wavelets ?

April 24, 2023 61

Noise Across Scales

Page 62: Wavelets ?

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 ; +=

Page 63: Wavelets ?

April 24, 2023 63

Page 64: Wavelets ?

April 24, 2023 64

D1

D3

D2

A3

Discontinuity

Page 65: Wavelets ?

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

Page 66: Wavelets ?

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

Page 67: Wavelets ?

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

Page 68: Wavelets ?

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

Page 69: Wavelets ?

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