0. Optimal Adaptive Wavelet Bases
Mahdi Amiri
Advisors
Dr. H. R. Rabiee
Dr. M. T. Manzuri
February 2002
Sharif University of Technology

1. Optimal Adaptive Wavelet Bases
Presentation Outline
Motivation and Goals
Road to the Optimal Adaptive Wavelet Bases
Why Wavelet-domain detection?
Wavelets for Object Detection Literature
The Proposed System
Conclusions and Future Works
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Optimal Adaptive Wavelet Bases

2. Optimal Adaptive Wavelet Bases
Motivation and Goals
Our Focus:
Image Understanding (IU) Applications
Specifically:
Object Detection
Object Recovery
Image Enhancement
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Optimal Adaptive Wavelet Bases

3. Optimal Adaptive Wavelet Bases
Motivation and Goals
Key Applications
Quality Control
Fabric Defect Detection
[LamB97]
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Optimal Adaptive Wavelet Bases

4. Optimal Adaptive Wavelet Bases
Motivation and Goals
Key Applications
Medical imaging
Tumor detection, Microcalcification detection
[WangK98]
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Optimal Adaptive Wavelet Bases

5. Optimal Adaptive Wavelet Bases
Motivation and Goals
Key Applications
Military Applications
Automatic or Aided Target Detection
[KimL98]
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Optimal Adaptive Wavelet Bases

6. Optimal Adaptive Wavelet Bases
Motivation and Goals
Key Applications
Military Applications
An ultimate example
[MesRK99]
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Optimal Adaptive Wavelet Bases

7. Optimal Adaptive Wavelet Bases
Motivation and Goals
Key Applications
Security Applications
Face detection, Face recognition
Autonomous Navigation
Real-time object detection for autonomous robots
Human-Computer Interaction
Recognizing the emotion of users
Data Compression
Efficient communication and compact coding
Cartography
Geographical survey, classifying landscape
Pattern Analysis
Web searches of images
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Optimal Adaptive Wavelet Bases

8. Optimal Adaptive Wavelet Bases
Road to the OAWB
Road to the
Optimal Adaptive Wavelet Bases
(OAWB)
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Optimal Adaptive Wavelet Bases

9. Optimal Adaptive Wavelet Bases
Road to the OAWB
Fourier, 1807
Haar, 1910
Math World
Orthogonal
Bi-orthogonal
Structural sets
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Optimal Adaptive Wavelet Bases

10. Optimal Adaptive Wavelet Bases
Road to the OAWB
What kind of Could be useful?
Impulse Function (Haar): Best time resolution
Sinusoids (Fourier): Best frequency resolution
We want both of the best resolutions
Heisenberg, 1930
Uncertainty Principle
There is a lower bound for (An intuitive prove in [Mac91])
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Optimal Adaptive Wavelet Bases

11. Optimal Adaptive Wavelet Bases
Road to the OAWB
Gabor, 1945
Short Time Fourier Transform (STFT)
Disadvantage: Fixed window size
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Optimal Adaptive Wavelet Bases

12. Optimal Adaptive Wavelet Bases
Road to the OAWB
How to Construct A Wavelet?
There was already some methods
But not interesting for engineers
Daubechies, 1988
Compactly Supported Wavelets
Maximum number of vanishing moments for their support width
Based on Filter Banks
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Optimal Adaptive Wavelet Bases

13. Optimal Adaptive Wavelet Bases
Road to the OAWB
Wavelet Construction
Start with a pair of filters
Iterate the filter bank
The equivalent filter leads us to the “wavelet”
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Optimal Adaptive Wavelet Bases

14. Theory of Multiresolution Signal Decomposition
Road to the OAWB
A Major Breakthrough
Mallat, 1989:
Theory of Multiresolution Signal Decomposition
Fast Algorithm for the Computation of Wavelet Transform Coefficients using Filter Banks
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Optimal Adaptive Wavelet Bases

15. Road to the OAWB Multiresolution Signal Representation
Coarse version (Approximation)
more useful than the Detail
Browsing image databases on the web
Signal transmission for communication
Denoising
Wavelet Tree Decomposition
We may lose what is in the Detail
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Optimal Adaptive Wavelet Bases

16. Road to the OAWB Full Tree Decomposition
Which decomposition path could be the best choice?
The answer leads us to the Best Basis
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Optimal Adaptive Wavelet Bases

17. Optimal Adaptive Wavelet Bases
Road to the OAWB
Best Basis Selection Criterions
Cut if:
Entropy: Shannon, Norm, Log Entry, Threshold, SURE
Coifman, Meyer, Wickerhauser (Patent 1995)
Rate-Distortion
Subband Coding (Vetterli, 1995)
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Optimal Adaptive Wavelet Bases

18. Detection in Wavelet Domain
So far we have reached to
Optimal Adaptive Wavelet Bases
Next Discussion
Why Wavelet Domain Detection?
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Optimal Adaptive Wavelet Bases

19. Find a test statistic and classify
Detection in Wavelet Domain
Signal Detection
Find a test statistic and classify
Matched Filter, Correlation Detector
Each view of the object may require a unique template
Mathematical Morphology
A priori knowledge of the resolution level of the object
Local Filtering Techniques
Fine tuning of several parameters related to local image statistics, Result in large number of false positives
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Optimal Adaptive Wavelet Bases

20. Optimal Adaptive Wavelet Bases
Detection in Wavelet Domain
Discrete Wavelet Transform (DWT)
Rapid processing
Adaptation to changing local image statistics
Representation of abrupt changes
Precise position information
Adapts to high background noise
Adapts to uncertainty about object properties
Relative independence to object-to-sensor distance
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Optimal Adaptive Wavelet Bases

21. Detection in Wavelet Domain
Wavelets
“frequency compact” energy
into a small set of low frequency coefficients
“spatially compact” energy
into a small set of high frequency coefficients
Compression, Denoising, “Feature Detection”
Here, Compression is not among our desired applications
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Optimal Adaptive Wavelet Bases

22. Wavelets for Object Detection Literature
General Approach
Forward Wavelet Transform
Nonlinearly transform or
adaptively weight
the wavelet coefficients
Inverse Wavelet Transform
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Optimal Adaptive Wavelet Bases

23. Optimal Adaptive Wavelet Bases
Wavelets for Object Detection Literature
Early works
Isolating singularities caused by edges in noisy data
Denoising algorithm:
Remove all maxima whose amplitude increase on average when the scale decreases [MalH92]
Detecting faint optical targets (trucks) in multispectral satellite images [YuRK92]
Uses feature vectors
Parameters: Spectral, Orientation, Scale
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Optimal Adaptive Wavelet Bases

24. Optimal Adaptive Wavelet Bases
Wavelets for Object Detection Literature
Early works
Wavelet and Gabor transform coefficients
Detection in infrared images [CasSY92]
Wavelet transform = Bank of matched filters
[SzuSc92]
Ideal spatial adaptation via wavelet shrinkage
A major breakthrough [DonJ94]
Undecimated wavelet transform
Wavelet thresholding
Donoho
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Optimal Adaptive Wavelet Bases

25. Optimal Adaptive Wavelet Bases
Wavelets for Object Detection Literature
State of the Art (Two Step)
Classification of transient sonar signals [JinXB96]
Adaptive Wavelet  Feature Vectors (FV)
Classification: Neural net
Microcalcification detection [StrH97]
Biorthogonal spline wavelet  FV
Linear FV Classification
Adaptive texture representation for ATR [MesRK99]
Wavelets and PCA  FV
Model based classification
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Optimal Adaptive Wavelet Bases

26. Optimal Adaptive Wavelet Bases
Wavelets for Object Detection Literature
Adaptive Thresholding
Spatially adaptive thresholding in wavelet domain with context modeling
Results are significantly superior to uniform thresholding
[ChaYV00]
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Optimal Adaptive Wavelet Bases

27. The Proposed System Contribution Adaptive thresholding
Among all of the works two groups are interesting
Adaptive thresholding
[ZhaD01], [MadTR01]
Adaptive wavelets
[ZhiPY01], [LiBH01]
We will take advantage from both of these adaptivities
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28. The Proposed System Block Diagram Undecimated Wavelet Packet Tree
Input Image
Undecimated Wavelet Packet Tree
Adaptive Best Basis
Feature Selection
Detection
Classifier
Threshold
Nonlinear Mapping of Details
Recovery
Inverse Wavelet Transform
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Optimal Adaptive Wavelet Bases

29. The Proposed System Biorthogonal Decomposition
For Undecimated Wavelet Packet Tree
Analysis
Synthesis
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Optimal Adaptive Wavelet Bases

30. Optimal Adaptive Wavelet Bases
The Proposed System
Feature Vectors
Feature Vector for each pixel:
Components: A group of the following coefficients after Entropy based best basis selection
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Optimal Adaptive Wavelet Bases

31. The Proposed System Classifier (Method 1) Hottelling Observer
Fischer’s discriminant:
Mean of “signal present” class FVs:
Mean of “signal absent” class FVs:
Provides maximum separation between
distributions in an N class problem [DudH73]
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Optimal Adaptive Wavelet Bases

32. Widely used in pattern classification applications
The Proposed System
Classifier (Method 2)
Mahalanobis Distance
Model FVs: Gaussian distribution
Mean: Covariance matrix:
Widely used in pattern classification applications
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Optimal Adaptive Wavelet Bases

33. Optimal Adaptive Wavelet Bases
The Proposed System
Threshold for detection
Apply a weighting factor for the identified pixels
Compute inverse wavelet transform for recovery
Compare Performance
ROC (Receiver Operating Characteristic)
X axis: False Positive Fraction
Y axis: 1 – True Positive Fraction
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Optimal Adaptive Wavelet Bases

34. Optimal Adaptive Wavelet Bases
The Proposed System
Disadvantages
Logarithmic distance of decomposition scales
Use “Voices”
May need some pre/post processing
Pre: Contrast Enhancement
Post: Decrease false positive rate in conjunction with non-wavelet based methods (e.g. Spatial edge detection)
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Optimal Adaptive Wavelet Bases

35. Optimal Adaptive Wavelet Bases
Conclusions and Future Works
Simulating Adaptive Best Basis Selection Methods
Simulating Classifiers and Adaptive Thresholding
Hardware Implementation
Just for prove of concept, based on TMS320Cx
FIND OUT MORE AT... 1. 2.
(References may be found there in the accompanied report)
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Optimal Adaptive Wavelet Bases

36. Optimal Adaptive Wavelet Bases
Mahdi Amiri
Advisors
Dr. H. R. Rabiee
Dr. M. T. Manzuri
“You can't predict the future, but you can invent it.” Dennis Gabor