Wavelets: Introduction, Applications, and Developments

1. Chapter 01Introduction to Wavelets

2. Wavelets = New mathematical method Wavelets is a relative new mathematical method with many interesting applications.

3. Mathematical operation - New information Transformed Function Function We want a suitable representation of a function - Mathematical operation of a function - Draw new information from a function

4. Wavelets = Small Waves Wavelets = Small Waves

5. Wavelets = Building blocks At the present day it is almost impossible to give a precise definition of wavelets. The research field is growing so fast and novel contributions are made at such a rate that even if one manages to give a definition today, it might be obsolute tomorrow. One, very vague, way of thinking about wavelets could be: Wavelets are building blocks that can quickly decorrelate data. • Wavelets are building blocks for general functions. • Wavelets have space-frequency localization. • Wavelets have fast transform algorithms.

6. Frequency / Transient signals / Discontinuity Adopting a whole new mindset or perspective in prosessing data Data • Wavelets are mathematical functions that can cut up data into different frequency components, and then study each component with a resolution matched to its scale. • Wavelets have advantages over traditional Fourier methods in analyzing physical situation where the signal is transient or contains discontinuities and sharp spikes.

7. Wavelets - Different scales

8. Interesting applicationsThe subject of Wavelets is expanding at a tremendous rate • Wavelet transform has been perhaps the most exciting development in the last decade to bring together researchers in several different fields: Seismic Geology Signal processing (frequency study, compression, …) Image processing (image compression, video compression, ...) Denoising data Communications Computer science Mathematics Electrical Engineering Quantum Physics Magnetic resonance Musical tones Diagnostic of cancer Economics …

9. History • Before 1930: The main branch of mathematics leading to wavelets began with Joseph Fourier (1807) with his theories of frequency analysis. • 1930: Several groups working independently researced the representation of functions using scale-varying basis functions. Physicists Paul Levy was studying small complicated details in Brownian motion using Haar basis function. Paley and Stein discovered a scale-varying function that conserve the energy of the function. This function was used by David Marr in numerical image processing in early 1980. • 1980- : S. Mallat discovered som relationships between quadrature mirror filters, pyramid algorithms, and orthonormal wavelet bases. Y. Meyer constructed the first non-trivial wavelets. Meyer wavelets are continuously differentiable, but do not have compact support. I. Daubechies constructed orthonormal wavelet basis funcions that has become the comberstone of wavelet applications today. • 1995: A new philosophy in biorthogonal Wavelet construction: The Lifting Scheme.

10. Properties • New technology - Rediscovered by I. Daubechies in 1987. • Signal analysis - Weighted sum of basis functions. • Infinitely many possible sets of wavelets. • Wavelet-coefficients contain information about the signal. • Basis functions containing information about both the time and frequency. (Heisenberg inequality: Resolution in time and frequency cannot both be made arbitrarily small.)

11. Software • Wavelab at Standford University: Matlab library. • Wavelet Workbench from Research Systems, Inc. • Liftpack from Gabriel Fernandez, Senthil Periaswamy, and Wim Sweldens: C-routins. • Mathematica Lifting Notebook by Paul Abbott. • …..

12. Analysis - Synthesis Analysis Synthesis

13. ComponentsBread Brød = 1 kg Hvetemel + 1/2 kg Grovt mel + 1 1/2 ts Salt + 50 g Gjær + 100 g Margarin + 1 1/2 l Vann/Melk Koeffisienter Basisfunksjoner Analysis Synthesis

14. ComponentsBlood Blod = 0.45 % Blodlegemer = + 0.55 % Blodplasma BlodPlasma = 7 % Proteiner + 0.9 % Salter + 0.1 % Glukose + … Koeffisienter Basisfunksjoner

15. Components - Components / Positions Interested in components, but not in the positions. Interested in components, and in the positions.

16. Frequency - Frequency / TimeMusic Analysis Synthesis Tools for analysis / synthesis: - Fourier transformation (frequence) - Wavelet transformation (frequence / time) - …

17. Components / PostitionsFourier / Wavelets Components = Freqyency Fourier Components = Freqyency Wavelets Positions = Place or Time

18. Potential of Wavelet Analysis Engineers, physicists, astronomers, geologists, medical researchers, and others have begun exploring the extraordinary array of potential applications of wavelet analysis, ranging from signal and image processing to data analysis. Wavelet analysis, in contrast to Fourier analysis, uses approximating functions that are localized in both time and frequency space.

19. Seismic trace

20. Original Fingerprints Reconstructed from 26:1 compression Without wavelet technology, digitizing the FBI's constantly growing database of over 200 million fingerprint records (originally stored as inked impressions on paper cards) would have required an unmanageable 2,000 terabytes (1 Tb = 1000 Mb) of storage and filled over a billion 3.5-inch high-density floppy disks. Faced with this digital storage dilemma, the FBI researched a variety of image compression techniques before finally settling on one robust enough to preserve vital fine-scale fingerprint image details--a breakthrough wavelet-based image coding algorithm developed in cooperation with Los Alamos National Laboratory researchers answered the call.

21. FingerprintOriginal - JPEG - Wavelet Original JPEG Wavelet

22. FingerprintOriginal

23. FingerprintJPEG

24. FingerprintWavelet

25. Compression 1.7 Mb 4 kb 4 kb Original JPEG Wavelet

26. Wavelet transformation • From a signal processing standpoint, one may view an image as a signal that has • high-frequency (high-spatial detail) and • low-frequency (smooth) components. The algorithm filters the signal and then iterates the process.

27. Originalt Compression 1:50 JPEG Wavelet

28. Wavelets and Telemedicine • Massachusetts General Hospital: No clinically significant image degradation was identified in radiologi images up to 30:1. • Wavelet-based compression technology is superior toall other compression technologies (keep details, high compression ratio).

29. Denoising Noisy Data

30. Sea Surface Temperature

31. CommunicationCompression

32. WavesConstruction of boats

33. Medical imageUltrasound / ECG Ultrasound ECG

34. Medical imageThresholding - Segmentation

35. Medical imageUltrasound - Operation in the brain

36. DNR L i ne æ r a k s e l e r a t o r B i l d e b h a n d l i n g

37. Stråleterapi - Pasientposisjon Referansebilde Kontrollbilde

38. Bildebehandling - Histogram

39. Bildebehandling - Gråtoneskalaer

40. Bildebehandling - Convolution

41. Bildebehandling - Fourier transformasjon I

42. Bildebehandling - Fourier transformasjon II

43. Bilderepresentasjon Pixel Bilderepresentasjon vha pixel-verdier i intervallet [0,255]

44. Fourier-transformation of a square wave f(x) square wave (T=2) N=1 N=2 N=10

45. Frequence Sinuswave with frequence f1 = 1 f1 < f2 Sinuswave with frequence f2 = 2

46. Signals and FT FT FT FT

47. Stationary / Non-stationary signals Stationary FT Non stationary FT The stationary and the non-stationary signal both have the same FT. FT is not suitable to take care of non-stationary signals to give information about time.

48. WaveletsLocalization both in frequency and time WT is suitable to take care of non-stationary signals to give information about time.

49. Dissimilarities of Fourier and Wavelet Transforms

50. End