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

2. State-Of-Affairs Concurrent Presentation Retrospective Analysis
Simulations, scanners
Concurrent
Presentation
Retrospective
Analysis
Representation
November 12, 2018

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
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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.
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5. QuickFix
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6. Windowing &Filtering
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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
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8. Cannot resolve all features at all instants
Windowing & Filtering
Windows –
fixed in space and frequencies
Cannot resolve all features at all instants
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9. Linear Scale Space
input
s = 1
s = 16
s = 24
s = 32
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10. Successive Smoothing
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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
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12. Image Pyramid
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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
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14. Image Pyramid – Another Twist
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15. Time Frequency Diagram
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16. Create new signal G such that ||F-G|| = e
Ideally !
Create new signal G such that ||F-G|| = e
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17. Wavelet Analysis
A1 D1 D2 D3 D3 D2 D1 A1
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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
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19. Scale-Coherent Structures
Coherent structure - frequencies at all scales
Examples - edges, peaks, ridges
Locate extent and assign saliency
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20. Wavelets – Analysis
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21. Wavelets – DeNoising
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22. Wavelets – Compression
Original
50:1
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23. Wavelets – Compression
Original
50:1
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24. Wavelets – Compression
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25. Yet Another Example
50% 7%
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26. Final Example 2%
50%
100% 1%
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27. Information Rate Curve
0.0
0.2
0.4
0.6
0.8
1.0
normalized rate n o r m a l i z e d f t ( E )
density
u momentum
v momentum
w momentum
energy
Energy Compaction – Few coefficients can efficiently represent functions
The Curve should be as vertical as possible near 0 rate
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28. Filter Bank Implementation
G: High Pass Filter
H: Low Pass Filter
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29. Synthesis Bank
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30. Successive Approximations
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31. Successive Details
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32. Wavelet Representation
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33. Coefficients
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34. Lossey Compression
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35. Lossey Compression
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36. Image Example
A Frame
Another Frame
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37. Image Example
Average
Difference
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38. Wavelet Transform
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39. Frequency Support
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40. Image Example
LvLh
LvHh
HvHh
HvLh
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41. Image Example
LvLh
LvHh
HvHh
HvLh
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42. How Does One Do This ?
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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
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44. Dilations
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45. Successive Approximations
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46. Translations Covers space-frequency diagram Versions are
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47. Wavelet Decomposition
Induced functional Space - Wj.
Related to Vjs
Space Wj+1 is orthogonal to Vj+1
Also
J-level wavelet decomposition -
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48. Successive Differences
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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
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50. Scaling Functions Compact support Bandlimited - cut-off frequency
Cannot achieve both
DC value (or the average) is defined
Translates of f are orthogonal
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51. Scaling Functions Nested smooth spaces Dilation Equation - Haar
Generally –
Frequency Domain
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52. Wavelet Functions Wavelet Equation - Haar System: G Filter Generally
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53. Perfect Reconstruction
Synthesis and Analysis Filter Banks
Synthesis Filters - Transpose of Analysis filters
For compact scaling function
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54. Orthogonal Filter Banks
Alternating Flip
Not symmetric - h is even length!
Example
Orthogonality conditions
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55. Examples
Haar
Daubechies(2)
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56. Approximation: Vanishing Moments Property
Function is smooth - Taylor Series expansion
Wavelets with m vanishing moments
Function with m derivatives can be accurately represented!
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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 w=p
P(y) is pth order polynomial (Daubechies 1992)
Maxflat filter
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58. Example
N=4
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59. Example
N=16
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60. Noise Uncorrelated Gaussian noise is correlated
Region of correlation is small at coarse scale
Smooth versions - no noise
Orthogonal transform - uncorrelated
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61. Noise Across Scales
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62. Denoising Statistical thresholding methods [Donohoe]
Assuming Gaussian Noise
Universal Threshold
Smoothness guaranteed
Hard
Soft
Works for additive noise since wavelet transform is linear
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63. November 12, 2018

64. Discontinuity
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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
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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
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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
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68. Bi-orthogonal: Lifting Scheme
Lazy wavelet transform: split data in 2 parts
Keep even part; predict (linear/cubic) odd part
Lifting - update lj+1 with gj+1: Maintain properties (moments, avg.)
Synthesis is just flip of analysis
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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
November 12, 2018