1. Introduction To Wavelets
Transforming Problem Representation

2. Why Representation Is Key?
The curse of dimensionality
Dealing with distractors
Signal enhancement goals such as increasing low contrast
Feature identification
Image 1
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3. Sobel Filtering of Raw Image Data
Sobel filtered raw data image
Note speckling
Median filters can reduce speckle (this image is not median filtered)
Image 2
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4. Another Spatial Representation
One level of a two-dimensional wavelet transform
Image resized for comparison
Contrast is enhanced
Image 3
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5. Sobel Filtering Of Transformed Image
Source (Image 3) is ¼ the size of Images 1 and 2
Blocks arise from rescaling
Image 4
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6. Sobel Filtering Of Raw Image Data
Sobel filtered raw data image
Note speckling
Median filters can reduce speckle (this image is not median filtered)
Image 2
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7. One-dimensional Signal
A clean signal
Short duration (circa 200 mSec)
Typical of sonar transients
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8. Noisy One-dimensional Signal
The transient occurs slightly after the center time displayed
Detection-determining that a transient occurred
Classification-determining which transient occurred
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9. Common Wavelet Applications
Requiring only decomposition
Signal processing
Image filtering
Feature extraction
Requiring decomposition and reconstruction
Data transmission using compression
Matrix multiplication
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10. Wavelet Transform Properties
Analogous to Fourier Transforms—but different
Constant ratio of scale versus constant difference of frequencies
Open choice of basis functions versus fixed choice of basis functions (sines and cosines)
Compact support versus non-compact support
Provides a decomposition for square integrable
Localize in time and scale versus time and frequency
Readily computable
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11. The Basic Dilation Equation
Two simple solutions:
c0 = 2, implies f(x) = d(x)
c0=c1=1, implies f(x) = f(2x) + f(2x-1) and
f(x) = c([0,1])
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12. A Graphical Example of Dilation
f(x) = c([0,1)) =
f(2x) + f(2x-1) 1
0.5
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13. The Basic Wavelet Equation
Uses differences and the scaling function
The Haar wavelet (based on the box function) is given by W(x)= f(2x) - f(2x-1) where f(x) = c([0,1))
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14. The Box Function And The Basic Haar Wavelet
f(2x) + f(2x-1)
f(2x) - f(2x-1)
Haar Wavelet
Box Function
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15. A Normalized Dilation Equation
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16. The Goal Of The Transformation Process
Write one function (the signal S) as a linear combination of the scaling and wavelet functions
where the wj,k are the wavelet scalars and Wj,k(x) are translated and dilated wavelets
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17. Matrix Representation Of The Haar Wavelet Transform
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18. Inverting The Effects Of The Haar Wavelet Transform
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19. The Inverse Matrix Is A Scaled Version Of The Transpose
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20. A Sample Signal And Its Decomposition (One Level)
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21. WT Computation: Low-pass Filter Output And Down-sampling
lp0 c0 c1 c2 c3
lp1 c0 c1 c2 c3
lp2 c0 c1 c2 c3
lp3 … … s0 s1 s2 s3 s4 s5 s6 s7 s8 s9
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22. WT Computation: High-pass Filter Output And Down-sampling… … s0 s1 s2 s3 s4 s5 s6 s7 s8 s9
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23. For Two-Dimensional Signals (Using Separable Wavelets)LL LH LP HP
Original HL HH
Process each row, storing LP results on the left and HP results on the right
Process each column of the previous, storing LP results at the top and HP at the bottom
One level of a 2-D transform
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24. A Two-level Two-dimensional Example
Images from the TRIM-2 data set acquired from the Night Vision and Electronic Sensors Directorate, Night Vision Laboratories, Ft. Belvoir
Images are in ARF format using 480x640 pixels, each byte representing one of 256 possible gray-scale levels
Terrain board images simulate scenes viewed in infrared
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25. So, Where Is The Compression?
Suppose the signal is represented with N samples
Low-pass filtering produces n/2 values
High-pass filtering produces n/2 values
Compression arises from quantization and encoding of HP filter output
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26. How Can One Generate Wavelet Filter Coefficients?
Rigorous mathematical analysis
Daubechies coefficients
Exploit parameterizations
TeKolste
Pollen
C0 = [(1 + cos a + sin a)(1 – cos b – sin b) + 2 sin b cos a]/4
C1 = [(1 - cos a + sin a)(1 + cos b – sin b) - 2 sin b cos a]/4
C2 = [1 + cos(a – b) + sin(a – b)]/2
C3 = [1 + cos(a – b) - sin(a – b)]/2
C4 = 1 – c2 – c0
C5 = 1 – c3 – c1
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27. What Might One Ask Using The Pollen Parameterization?
What wavelet provides the best basis for compression?
Can one minimize variability or magnitudes in the high-pass filter output?
What wavelet might emerge using a small set of samples from a piece-wise linear function?
32 samples
Corresponds to image regions having smooth gradients or homogeneous contents
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28. Resulting Coefficients
Proc. SPIE, Vol. 2762, p. 89
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29. Concluding Comments
Wavelets provide an approach to transforming problems from the time domain to a scale domain
The transform can be tailored to meet processing objectives
Optimization techniques can lead to results that correspond to (possibly difficult to obtain) analytical results
There is much more to say about things like:
Orthogonality, bi-orthogonality, and orthonormality
Separability and nonseparability
Super-wavelets
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