Introduction To Wavelets

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 Gene A. Tagliarini

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 Gene A. Tagliarini

4. Another Spatial Representation • One level of a two-dimensional wavelet transform • Image resized for comparison • Contrast is enhanced Image 3 Gene A. Tagliarini

5. Sobel Filtering Of Transformed Image • Source (Image 3) is ¼ the size of Images 1 and 2 • Blocks arise from rescaling Image 4 Gene A. Tagliarini

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 Gene A. Tagliarini

7. One-dimensional Signal • A clean signal • Short duration (circa 200 mSec) • Typical of sonar transients Gene A. Tagliarini

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 Gene A. Tagliarini

9. Common Wavelet Applications • Requiring only decomposition • Signal processing • Image filtering • Feature extraction • Requiring decomposition and reconstruction • Data transmission using compression • Matrix multiplication Gene A. Tagliarini

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 Gene A. Tagliarini

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]) Gene A. Tagliarini

12. f(2x) + f(2x-1) f(x) = c([0,1)) = 0 1 0 0.5 1 A Graphical Example of Dilation Gene A. Tagliarini

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) wheref(x) = c([0,1)) Gene A. Tagliarini

14. The Box Function And The Basic Haar Wavelet f(2x) +f(2x-1) f(2x) -f(2x-1) Haar Wavelet Box Function Gene A. Tagliarini

15. A Normalized Dilation Equation Gene A. Tagliarini

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 Gene A. Tagliarini

17. Matrix Representation Of The Haar Wavelet Transform Gene A. Tagliarini

18. Inverting The Effects Of The Haar Wavelet Transform Gene A. Tagliarini

19. The Inverse Matrix Is A Scaled Version Of The Transpose Gene A. Tagliarini

20. A Sample Signal And Its Decomposition (One Level) Gene A. Tagliarini

21. WT Computation: Low-pass Filter Output And Down-sampling c0 c1 c2 c3 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 Gene A. Tagliarini

22. WT Computation: High-pass Filter Output And Down-sampling d0 d1 d2 d3 hp0 d0 d1 d2 d3 hp1 d0 d1 d2 d3 hp2 d0 d1 d2 d3 hp3 … … s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 Gene A. Tagliarini

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 Gene A. Tagliarini

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 Gene A. Tagliarini

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 Gene A. Tagliarini

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 Gene A. Tagliarini

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 Gene A. Tagliarini

28. Resulting Coefficients Proc. SPIE, Vol. 2762, p. 89 Gene A. Tagliarini

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 Gene A. Tagliarini