Wavelets: Theory and Applications

1. Wavelets: Theory and Applications Maria Elena Velasquez Idaho State University 3 May 2006 Committee: Dr. Ken Bosworth Dr. Don Cresswell Dr. Charles R. Peterson Dr. Habib Sadid Dr. Rob Van Kirk

2. Fourier Analysis: Joseph Fourier (1768-1830) France • 1780-Entered the Royal Military Academy • French Revolution-joined the revolutionary committee of Auxerre • After the revolution- Taught in Paris • Accompanied Napoleon to Egypt • 1802-Returned to France and become the prefect of Isare

3. Fourier Analysis • 1802-Returned to France and became the Prefect of Isere • 1807-Began writing “la Theorie Analytique de la Chaleur” • 1817-Was elected to the Academy of Science • 1822-Published “la Theorie Analytique de la Chaleur”

4. Fourier’s Work Mathematical statement: Any periodic function can be represented as a Sum of sines and cosines.

5. Formalized Work- Fourier Analysis

6. Theorem

7. Problems with Fourier Transform • Time to compute coefficients • Nonlinear problems • Fourier transform hides information about time • Local characteristics of the signal become global characteristic of the transform • Fourier transform is very vulnerable to errors

8. Some Solutions To The Problems • FFT Fast Fourier Transform • WFT Window Fourier Transform • Wavelet Theory

9. What are Wavelets? Wavelets are functions that satisfy certain requirements Admissibility condition: • The function should integrate to zero (“wavy”) • The function has to be well localized (diminutive connotation of wavelet)

10. Daubechies wavelet family

11. Other Wavelets

12. Stretches and translations Horizontal or vertical stretch Translation

13. Continuous Wavelet Transform

14. Inverse Wavelet Transform

15. Discrete Wavelet Transform

16. MultiResolution Analysis in MRA

17. How do we do it? Define the first space as follows: Define the rest of the space as follows: Where:

18. W W V 0 0 0 W W V 1 1 1 V 2 Wavelets enter the picture

20. Wavelets enter the picture

21. Steps for the Discrete Wavelet Transform • Take wavelet and compare it to the beginning section of the original signal • Calculate the coefficient, which represents how closely correlated the wavelet is with this section of the signal • Shift wavelet to the right and repeat steps 1 and 2 until you have covered the entire signal • Scale (Stretch) the wavelet and repeat steps 1 through 3 • Repeat steps 1 through 4 for all scales • At the end we will have coefficients produced at different scales by different sections of the signal.

22. Discrete Wavelet transform Translation: Localization on time, translations of mother wavelet Scale: Localization on frequency (1/freq), dilations of mom wavelet Scaling: dilate ==large scale ==low frequency ==global view compress ==small scale ==high frequency == detailed view The scaling factor in the wavelets is in the denominator.

23. V2 WH2 V1 WH1 WH1 L SI WV2 WD2 L WV1 WD1 WV1 WD1 H f H Discrete Wavelet and Filter Banks

24. Similarities Between FT and WT • FFT and DWT are both linear operations that generate log(n) segments of various lengths, usually filling and transforming it into a different data vector of length 2^n • The inverse transform matrix of both is the transpose of the original matrix. • FFT is a rotation in discrete function space to a new domain spanned by the basis functions sines and cosines. DWT domain is spanned by more complicated functions: scaled-shifted wavelets. • In both cases the basis functions are localized in frequency.

25. Dissimilarities Between FT and WT The most interesting dissimilarity between the Fourier Transform and the Wavelet Transform is that individual wavelet functions are localized in space. Fourier sine and cosine functions are not. Therefore, The Wavelet transform is localized in both time and frequency.

26. Applications: • Data compression (fingerprinting) • Signal processing • Transformational algebras • Analysis of spectra • PDE and Integral equations • Numerical solvers for dynamical systems • Imaging • Human vision • Speech recognition • Solutions for stochastic models applied to Biology • Seismic Theory • Geosciences • Non-destructive assay techniques

27. Detection of Single Isotopes in Composite Gamma-ray Spectrum • Existing software applications for spectra analysis • can be described as follows : • Preprocess (calibration) • Peak finding and fitting • Peak analysis (calculation of areas under peaks) • Peak verification (calculation of activities)

28. Wavelet signatures for Gamma-raySpectra Phase I: construction and storage of the signatures: Construction of the wavelet approximation Construction of the non-parametric local smooth Detection of significant peaks Phase II: Construction of the signature for the composite Ranking detection of the single isotopes Verification of detection

29. Wavelet Decomposition: db1 level 4

30. Non-Parametric Smoothing

31. Detection of the Significant Peaks

32. Signature Example:

33. Detection of single isotopes:

34. Automated Photographic Identification of Amphibians • Method Description: • Image Pre-processing • Signature Construction • Signature Comparison

35. Image Pre-processing: “Best’’ bounding ellipse (user can adjust manually). Recommended white or gray background. Automatic edge detection for irrelevant white-zones. Orientation to major/minor axes.

36. Warping the “frog” Impose a polar-elliptical coordinate system:

37. Signature and Mask Construction The stardarized image was transformed using db4 at Level 3. Coefficients with magnitude below a threshold (0.1) were ignored and a mask created (only significant coefficients are compared). The resulting coefficients were encoded using a binary code.

38. Signature Construction Semisoft threshold (0.1):

39. Signature Construction

40. Signature Comparison Fractional Hamming distance is a measure of dissimilarity between two binary vectors

41. Proof of Concept: Gamma-ray

42. ADVVS Output File

43. Proof of Concept: “frog” identification Identifrog: