Wavelets ? - PowerPoint PPT Presentation

About This Presentation
Title:

Wavelets ?

Description:

Simulations, scanners. Presentation. Analysis. Representation. Retrospective. Concurrent ... We are generating and measuring larger datasets every year ... – PowerPoint PPT presentation

Number of Views:86
Avg rating:3.0/5.0
Slides: 70
Provided by: Torsten77
Category:

less

Transcript and Presenter's Notes

Title: Wavelets ?


1
Wavelets ?
  • Raghu Machiraju
  • Contributions Robert Moorhead, James Fowler,
    David Thompson, Mississippi State University
  • Ioannis Kakadaris, U of Houston

2
State-Of-Affairs
  • Presentation
  • Concurrent
  • Retrospective
  • Analysis
  • Representation

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

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.

5
QuickFix
6
Windowing Filtering
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

8
Windowing Filtering
Windows fixed in space and frequencies
Cannot resolve all features at all instants
9
Linear Scale Space
input
s 1
s 16
s 24
s 32
10
Successive Smoothing
11
Sub-sampled Images
  • Keep 1 of 4 values from 2x2 blocks
  • This naive approach and introduces aliasing
  • Sub-samples are bad representatives of area
  • Little spatial correlation

12
Image Pyramid
13
Image Pyramid MIP MAP
  • 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

14
Image Pyramid Another Twist
15
Time Frequency Diagram
16
Ideally !
Create new signal G such that F-G e
17
Wavelet Analysis
  • A1 D1 D2 D3
  • D3
  • D2
  • D1
  • A1

18
Why Wavelets? Because
  • 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

19
Scale-Coherent Structures
  • Coherent structure - frequencies at all scales
  • Examples - edges, peaks, ridges
  • Locate extent and assign saliency

20
Wavelets Analysis
21
Wavelets DeNoising
22
Wavelets Compression
Original
501
23
Wavelets Compression
Original
501
24
Wavelets Compression
25
Yet Another Example
50
7
26
Final Example
50
100
1
27
Information Rate Curve
  • Energy Compaction Few coefficients can
    efficiently represent functions
  • The Curve should be as vertical as possible near
    0 rate

28
Filter Bank Implementation
  • G High Pass Filter
  • H Low Pass Filter

29
Synthesis Bank
30
Successive Approximations
31
Successive Details
32
Wavelet Representation
33
Coefficients
34
Lossey Compression
35
Lossey Compression
36
Image Example
A Frame
Another Frame
37
Image Example
Average
Difference
38
Wavelet Transform
39
Frequency Support
40
Image Example
LvLh
LvHh
HvHh
HvLh
41
Image Example
LvLh
LvHh
HvHh
HvLh
42
How Does One Do This ?
43
  • Dilations
  • Rescaling Operation t --gt 2t
  • Down Sampling, n --gt 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

44
Dilations
45
Successive Approximations
46
  • Translations
  • Covers space-frequency diagram
  • Versions are

47
  • Wavelet Decomposition
  • Induced functional Space - Wj.
  • Related to Vjs
  • Space Wj1 is orthogonal to Vj1
  • Also
  • J-level wavelet decomposition -

48
Successive Differences
49
Wavelet Expansion
  • Wavelet expansion (Tiling- j scale, k
    translates), Synthesis
  • Orthogonal transformation, Coarsest level of
    resolution - J
  • Smoothing function - f, Detail function - y
  • Analysis
  • Commonly used wavelets are Haar, Daubechies and
    Coiflets

50
  • Scaling Functions
  • Compact support
  • Bandlimited - cut-off frequency
  • Cannot achieve both
  • DC value (or the average) is defined
  • Translates of f are orthogonal

51
  • Scaling Functions
  • Nested smooth spaces
  • Dilation Equation - Haar
  • Generally
  • Frequency Domain

52
  • Wavelet Functions
  • Wavelet Equation - Haar System G Filter
  • Generally

53
Perfect Reconstruction
  • Synthesis and Analysis Filter Banks
  • Synthesis Filters - Transpose of Analysis filters
  • For compact scaling function

54
  • Orthogonal Filter Banks
  • Alternating Flip
  • Not symmetric - h is even length!
  • Example
  • Orthogonality conditions

55
Examples
Haar
Daubechies(2)
56
  • Approximation Vanishing Moments Property
  • Function is smooth - Taylor Series expansion
  • Wavelets with m vanishing moments
  • Function with m derivatives can be accurately
    represented!

57
  • Design of Compact Orthogonal Wavelets
  • Compute scaling function
  • Use Refinement Equation
  • N vanishing moments property - H(w) has a zero of
    order N at wp
  • P(y) is pth order polynomial (Daubechies 1992)
  • Maxflat filter

58
Example
N4
59
Example
N16
60
  • Noise
  • Uncorrelated Gaussian noise is correlated
  • Region of correlation is small at coarse scale
  • Smooth versions - no noise
  • Orthogonal transform - uncorrelated

61
Noise Across Scales
62
  • Denoising
  • Statistical thresholding methods Donohoe
  • Assuming Gaussian Noise
  • Universal Threshold
  • Smoothness guaranteed
  • Hard
  • Soft
  • Works for additive noise since wavelet transform
    is linear

63
(No Transcript)
64
Discontinuity
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

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

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

68
  • Bi-orthogonal Lifting Scheme
  • Lazy wavelet transform split data in 2 parts
  • Keep even part predict (linear/cubic) odd part
  • Lifting - update lj1 with gj1 Maintain
    properties (moments, avg.)
  • Synthesis is just flip of analysis

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
Write a Comment
User Comments (0)
About PowerShow.com