Usage of Wavelets in Computational Science

1. Usage of Waveletsin Computational Science

2. REUSummer 2005Florida State University • Dr. Gordon Erlebacher (Florida State University) • Shahrzad Farshi (Wellesley College)

3. What Are Wavelets? • Wavelets are functions that satisfy certain mathematical requirements, and are used to represent data or other functions. • They work similarly to Fourier transforms. Fourier transforms use sine's and cosines to represent other functions. • Wavelet functions are localized in space and are used to represent a signal or data function.

4. Studying Haar Transforms • The Haar wavelet transform, is a transform that makes use of the Haar function.

5. At the scale of 2-j centered at x=k2-j

6. Thresholding • Thresholding is a method often used to de-noise (smooth) data. • The idea of thresholding is to set all the wavelet coefficients that are less than a certain magnitude, or within a certain bound, to zero. • These remaining coefficients are used in an inverse wavelet transformation to reconstruct the smooth version of the original data.

7. I wrote a program that calculated the 1D Haar Transform, and then did a thresholding on a specific function. • The output of the program was plotted. • Below is an example of the plot, and it’s de-noised Haar transform using different values of threshold. • You can see that the Haar Transform of the function is much smoother than the actual function.

8. Note that the thresholding with the Haar wavelets produces C0 curves. The equation of the curve is: • See the differences in the functions and their corresponding Haar transforms.

9. Figure 1 has a threshold fraction of 0.98 and random coefficient of   0.1 Figure 1

10. Figure 2 has a threshold fraction of 0.995 and random coefficient of   0.05 Figure 2

11. Figure 3 has a threshold fraction of 0.98 and random coefficient of   0.05 Figure 3

12. Fire Models and Stimulations • High Resolution Numerical Models of wildfires are critical to the environment. • Uncontrolled fire can damage life and property, whereas controlled fire is critical to the diversity of the natural habitats. • Today, Scientists are interested in modeling the behavior of wildfires and their smoke impacts. • The smoke plumes usually affect a larger population than the fire itself does.

13. Dr. Cunningham of the department of Meteorology and the Geophysical Fluid Dynamics Institute did a research on understanding wildfire behavior and smoke transport. • He analyzed density, velocity and temperature and how they interacted and evolved overtime.

14. Dr. Erlebacher and I were interested in turning Dr. Cunningham’s research into visual models, using the visualization program Amira. • We thought that by visualizing the physical process and analyzing the data at different scales, we can better understand what happens during the fire.

15. Note the difference in the pictures as the boundaries of thresholding changes. • Here, the wavelets are C2. • Here the curves are sorted by magnitude, not scale.

16. Figure 4 Figure 4 shows the visualization of the fire plume when the thresholding boundary is from 1 to 10, and the temperature is 1.6.

17. Figure 5 shows the visualization of the fire plume when the thresholding boundary is from 1 to 4, and the temperature is 1.2. Figure 5 .

18. Figure 6 Figure 6 shows the visualization of the fire plume when the thresholding boundary is from 0.2 to 5, and the temperature is 1.17.

19. Figure 7 Figure 7 shows the visualization of the fire plume when the thresholding boundary is from 0.2 to 0.9, and the temperature is 1.17.

20. Experiences • At the beginning of the program, I didn’t even know how to program with C++. • I had to teach myself to work with Linux, C++, gnuplot, and computer problems that used to be alien to me. • I had to compile and fix the programs that crashed.

21. I feel that I have gained experience and have a better understanding of what real research is like. • Now, I appreciate researchers’ work much more than I used to, knowing what they go through. • If I had more time, I would like to implement sorting the wavelet coefficients based on scale rather than magnitude.

22. References • http://www.amara.com/IEEEwave/IEEEwavelet.html • http://mathworld.wolfram.com/HaarTransform.html • Cunningham, Phil. “Advances in the Understanding of Wildfire Behaviour and Smoke Transport Using High-Resolution Numerical Models.” (2005)