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The First Step To The Wavelets

The First Step To The Wavelets. 2003. 7. 10 Park, Mok-min. Contents. What are wavelets? A Historical Review of Wavelets A Technical Overview of Wavelets An Application Overview of Wavelets References. What are Wavelets? [1]. What are wavelets ?.

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The First Step To The Wavelets

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  1. The First Step To The Wavelets 2003. 7. 10 Park, Mok-min

  2. Contents • What are wavelets? • A Historical Review of Wavelets • A Technical Overview of Wavelets • An Application Overview of Wavelets • References

  3. What are Wavelets? [1]

  4. What are wavelets? • Waves : Waves are oscillating functions of time or space or both. • Wavelets : Wavelets are small waves. • A wavelet has oscillating wave-like characteristics. • Its energy is concentrated in time over relatively small intervals. 1/22

  5. A Historical Review of Wavelets[1],[4]

  6. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • Approximate a complex functions as a weighted sum of simple functions. • Fourier used sinusoids representations provide valuable insight to analysis of complicated functions. • Sinusoids have perfect compact support in frequency domain, but not in time domain. • They cannot be used to approximated non-stationary signals. 2/22

  7. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1946 Gabor : Window Fourier Transform (WFT) • The first modifications to the Fourier Transform to allow analysis of non-stationary signals came as short time Fourier Transform(STFT) • Segmenting the signals by using a time-localized window, and performing the analysis for each segment 2/22

  8. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • Many other time-frequency representation(TFR)s have been developed at this period. • Each of which differed from the other ones only by the selection of the windowing function 2/22

  9. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : Wavelets of constant shape • Different window function for analyzing different frequency bands. 2/22

  10. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) • 1985 Meyer : Orthogonal Wavelet Function • J.O.Stromberg had already discovered the very same wavelets about 5 years ago. 2/22

  11. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1909 Haar : An alternative orthonormal basis for signal decomposition • The first and the simplest orthonormal wavelet basis functions. • Poor frequency localization. • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) 2/22

  12. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1909 Haar : An alternative orthonormal basis for signal decomposition • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) • 1988 Daubechies : Orthonormal bases of compactly supported wavelets 2/22

  13. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1909 Haar : An alternative orthonormal basis for signal decomposition • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) • 1988 Daubechies : Orthonormal bases of compactly supported wavelets • 1989 Mallat : Multiresolution Analysis (MRA) of wavelets 2/22

  14. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1909 Haar : An alternative orthonormal basis for signal decomposition • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) • 1988 Daubechies : Orthonormal bases of compactly supported wavelets • 1989 Mallat : Multiresolution Analysis (MRA) of wavelets • 1992 Coifman : Signal processing by wavelet transforms 2/22

  15. A Historical Review of Wavelets • 1807 Fourier : Orthogonal decomposition of periodic signals • 1909 Haar : An alternative orthonormal basis for signal decomposition • 1946 Gabor : Window Fourier Transform (WFT) • Late 1940s ~ early 1970s • 1980 Morlet & Grossman : wavelets of constant shape • 1984 Meyer : Noticed the similarity between Morlet’s and Calderon(’64) • 1988 Daubechies : Orthonormal bases of compactly supported wavelets • 1989 Mallat : Multiresolution Analysis (MRA) of wavelets • 1992 Coifman : Signal processing by wavelet transforms • 1995 Donoho & Johnstone : Wavelet shrinkage and thresholding 2/22

  16. A Technical Overview of Wavelets[3],[4],[6]

  17. A Technical Overview of Wavelets • Basic Idea • Hierarchical Decomposition of a function into a set of Basis Functions and Wavelet Functions. 3/22

  18. A Technical Overview of Wavelets • The Haar wavelet system 4/22

  19. A Technical Overview of Wavelets • The Haar wavelet system 5/22

  20. A Technical Overview of Wavelets • The Haar wavelet system (example1) 6/22

  21. A Technical Overview of Wavelets • The continuous wavelet transform (CWT) CWT is written as : Where * denotes complex conjugation. A Function f(t) is decomposed into a set of basis functions , called the wavelets. s : scale, τ: translation 7/22

  22. A Technical Overview of Wavelets • The continuous wavelet transform (CWT) The inverse wavelet transform is The wavelets are generated from a single basic wavelet Ψ(t), the so-called mother wavelet, by scaling and translation. Wavelets are is for energy normalization across the different scales. 8/22

  23. A Technical Overview of Wavelets • Wavelet systems • The Haar wavelet system • The Daubechies family • Etc. 9/22

  24. A Technical Overview of Wavelets • The Haar wavelet system (example) Signals : 37, 35, 28, 28, 58, 18, 21, 15 10/22

  25. A Technical Overview of Wavelets • The Haar wavelet system (example) Signals : 37, 35, 28, 28, 58, 18, 21, 15 Decomposition (analysis) : 37 35 28 28 58 18 21 15 36 28 38 181 0 20 3 32284 101 0 20 3 3024 10 1 0 20 3 Averaging Differencing (37+35)/2=36, (37-35)/2=1 (28+28)/2=28, (28-28)/2=0 (58+18)/2=38, (58-18)/2=20 (21+18)/2=15 (21-18)/2=3 11/22

  26. A Technical Overview of Wavelets • The Haar wavelet system (example) Signals : 37, 35, 28, 28, 58, 18, 21, 15 Reconstruction (synthesis) : 3024 10 1 0 20 3 32284 101 0 20 3 36 28 38 181 0 20 3 37 35 28 28 58 18 21 15 30 + 2 = 32, 30 – 2 = 28 12/22

  27. A Technical Overview of Wavelets • The Haar wavelet system (example) Signals : 37, 35, 28, 28, 58, 18, 21, 15 Threshold = 2 : 37 35 28 28 58 18 21 15 3024 10 1 0 20 3 13/22

  28. A Technical Overview of Wavelets • The Haar wavelet system (example) Signals : 37, 35, 28, 28, 58, 18, 21, 15 Threshold = 2 : 37 35 28 28 58 18 21 15 30 2 4 10 1 0 20 3 3004 10 0 0 20 3 30 30 4 10 0 0 20 3 34 26 40 20 0 0 20 3 34 34 26 26 60 20 23 17 Truncate! 14/22

  29. A Technical Overview of Wavelets • The Haar wavelet system (example) Threshold = 2 : 37 35 28 28 58 18 21 15 34 34 26 26 60 20 23 17 15/22

  30. An Application Overview of Wavelets[2],[4],[5]

  31. An Application Overview of Wavelets • Data Compression • Wavelets have good energy concentration properties. • Most DWT coefficients usually are very small. • They can be discarded without incorporating a significant error in the reconstruction stage. 16/22

  32. JPEG Compression Original 4302 Bytes 2245 Bytes 1714 Bytes Wavelet Compression 262179 Bytes 4272Bytes 2256 Bytes 1708 Bytes An Application Overview of Wavelets • Data Compression (example) 17/22

  33. An Application Overview of Wavelets • Denoising • Donoho & Johnstone have devised the wavelet shrinkage denoising(WSD). • Noise will show itself at finer scales. • Discarding the coefficients that falls below a certain threshold at these scales will remove the noise. 18/22

  34. An Application Overview of Wavelets • Denoising (example) Standardized test signals with n = 1,024 19/22

  35. An Application Overview of Wavelets • Denoising (example) Noisy test signals with n = 1,024 SNR = 10 20/22

  36. An Application Overview of Wavelets • Denoising (example) Wavelet shrinkage denoising with ‘SUR’, DROLA(10:5), n=2,048, L=5 21/22

  37. An Application Overview of Wavelets • Etc. • Biomedical Engineering • Nondestructive Evaluation • Numerical Solution of PDEs • Study of Distant Universes • Wavelet Networks • Fractals • Turbulence Analysis • Financial Analysis 22/22

  38. References • [1] Asok Ray, ‘Introduction to Wavelets’ • [2] Donoho, The What, How, and Why of Wavelet Shrinkage Denoising • [3] Ole Nielsen, ‘Introduction to wavelet analysis’, 1998 • [4] Robi Polikar, ‘The Story of Wavelets’ • [5] Wavelets – Image Compression • [6] Daniel Keim, Wavelets and their Applications in Database

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