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Haar Wavelets

Haar Wavelets. Contents. Introduction The Haar Transform Conservation and Compaction of Energy Haar Wavelets Multiresolution Analysis Signal Compression Removing Noise. Introduction. A haar wavelet : 가장 간단한 type 의 wavelet The haar transform Discrete form of haar wavelets 와 관련

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Haar Wavelets

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  1. Haar Wavelets

  2. Contents • Introduction • The Haar Transform • Conservation and Compaction of Energy • Haar Wavelets • Multiresolution Analysis • Signal Compression • Removing Noise

  3. Introduction • A haar wavelet : 가장 간단한 type의 wavelet • The haartransform • Discrete form of haarwavelets와 관련 • 모든 wavelettransform의 prototype • 손 계산 가능

  4. The Haar Transform (1) • Analyze될 signals : discrete signals • N: f의 길이로서 positive even integer • f의값 : N개의 실수 • Equally spaced sample values or simply sample values • 아날로그 signal g를등 간격인시간 t = t1, t2, … , tN 에서 sampling

  5. The Haar Transform (2) • Haar transform • Discrete signal을 길이가 반인 두개의 subsignal로 분해 • Running average (trend) • First trend : a1 = (a1, a2, … , aN/2 ) • a1= (f1+f2)/2 * => m = 1,2,3,…,N/2 • Running difference (fluctuation) • First fluctuation : d1 = (d1, d2, … , dN/2 ) • d1=(f1-f2)/2 * => m = 1,2,3,…,N/2

  6. The Haar Transform (3) • The Haar transform : 여러 stages or levels로 수행 • 첫번째 level 의 mapping H1 : • Inverse of H1 : (a1| d1) → f • Small fluctuations feature : fluctuation subsignal 값의 크기는 originalsignal 값의 크기보다 상당히 작다. • 의 평균값 : 7 • 의 평균값 : • 6.6배 차이

  7. The Haar Transform (4) • Small fluctuations feature • 로부터 1024개 값 추출 • g가매우 작은 time 간격을 가진다면,

  8. Conservation and Compaction of Energy (1) • Haartransform의 중요한 두가지 properties • Energy conservation of signals • Energy of f : • : • 1-level haar transform : • Compaction of the energy of signals • Trend : • 440/446 ⇒ 98.7% • Fluctuation : • Trend subsignal a1 의 energy가 transformedsignal (a1| d1)의 energy의많은 부분을 차지함

  9. Conservation and Compaction of Energy (2) • Multiplelevels transform • Conservation of energy • : • Compaction or localization of the energy of f • a2 : • Energy : 90% of f • Length : 1/4 • a3 : • Energy : 87.89% of f • Length : 1/8

  10. Conservation and Compaction of Energy (3) • Cumulative energy profile :

  11. Conservation and Compaction of Energy (4) • 수학적 증명으로 보는 energyconservation

  12. Haar Wavelets (1) • 1-level haar wavelets • 성질 • 각각 energy가 1 • 평균값 0을 가진 두 값 사이에서 빠른 fluctuation으로구성 • 첫번째 haarwavelet 의 짝수 timetranslation

  13. Haar Wavelets (2) • Scalarproduct • 첫번째 fluctuationsubsignal d1 using haar wavelets : • 두번째 fluctuationsubsignal d2 using haar wavelets : • 1-levelhaar wavelets 을 가지고첫번째 fluctuation 표현 가능

  14. Haar Wavelets(3) • 1-level haar scaling signals • 1-levelhaar scaling signals을 가지고첫번째 trend 표현가능 : • 성질 • 각각 energy가 1 • 두개의 연속적인 time index로 구성된 support를 가짐 • 첫번째 haarscalingsignal 의 짝수 timetranslation

  15. Haar Wavelets(4) • 2-level haar scaling signals • 2-level trend • 2-level haar wavelets • 2-level fluctuation

  16. Multiresolution Analysis (1) • 두 signal 에 대해

  17. Multiresolution Analysis (2) • Basic idea of MRA • Signal f : a lower resolution signal(5,5,11,11,7,7,5,5)과 fluctuationsignal(-1,1,-1,1,1,-1,0,0)의 합으로 표현 에서 a1 ,a2,…,aN/2 과 d1 ,d2,…, dN/2 분리

  18. Multiresolution Analysis (3) • 2-level ofa MRA of a signal f • k-level ofa MRA of a signal f

  19. Multiresolution Analysis (4) • 10-levels of MRA

  20. Signal Compression (1) • Audio signal • Method of wavelet transform compression • Signal에 wavelet transform • Thresholding • Transform된 값의크기를 큰 값부터 정렬 • Threshold보다 작은 값은 0 • Transmit • Transformed data + significancemap (0 or 1) • Inverse wavelet transform

  21. Signal Compression (2) • Original signal을 복원하려면 energy의 99.99%이상이 포함되도록 threshold를선택해야 함 • 1024 : 51 ≒ 20 : 1 압축 4096 : 410 ≒ 10 : 1 (99.99% 이상이되려면 2.3 : 1 이상이 되야 함, 즉, 1782개 이상)

  22. Removing Noise (1) • Contaminated signal = original signal + noise • f = s + n • Randomnoise만 고려 • Noisesignal : highly oscillatory, 평균값위아래로 빠르게 변함 • Transform에 의해 originalsignal은 적은 개수의 높은 에너지로 집약되고 노이즈는 낮은 에너지를 가지게 됨 • Threshold method of wavelet denoising • s의 energy 측정 : 대부분의 에너지가 형성되는 thresholdTs > 0 찾음 • Noisesignal의 transform값을 모두 포함하는 Ts보다작은 thresholdTn 값 • Tn 값보다작은 값은 0 • Root Mean Square Error (RMS Error)

  23. Removing Noise (2) • RMS : 0.057 -> 0.011 0.057 -> 0.035

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