1. ECE 501 Introduction to BME ECE 501 Dr. Hang

2. Part V Biomedical Signal Processing Introduction to Wavelet Transform ECE 501 Dr. Hang

3. ECE 501 Dr. Hang Fourier Analysis Introduction

4. ECE 501 Dr. Hang Fourier Analysis Introduction A serious drawback: time information is lost Cannot handle transitory characteristics

5. ECE 501 Dr. Hang Short-Time Fourier Analysis Introduction A compromise between the time- and frequency-based views of a signal: analyze a small section of the signal at a time A drawback: The window is the same for all frequencies

6. ECE 501 Dr. Hang Wavelet Analysis Introduction A windowing technique with variable-sized regions: long time interval for low-frequency information, shorter regions for high-frequency information Time-scale region

7. ECE 501 Dr. Hang What is Wavelet Analysis Introduction A wavelet is a waveform of effectively limited duration that has an average value of zero Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (mother) wavelet.

8. ECE 501 Dr. Hang Fourier Analysis The sum over all time of the signal multiplied by a complex exponential Continuous Wavelet Transform

9. ECE 501 Dr. Hang CWT The sum over all time of the signal multiplied by scaled, shifted version of the wavelet function Continuous Wavelet Transform

10. ECE 501 Dr. Hang Scaling Scaling a wavelet: stretching or compressing it a: scaling factor Continuous Wavelet Transform

11. ECE 501 Dr. Hang Scaling Low scale High frequency High scale Low frequency Continuous Wavelet Transform

12. ECE 501 Dr. Hang Shifting Continuous Wavelet Transform

13. ECE 501 Dr. Hang Five Steps to a CWT 1.Take a wavelet and compare it to a section at the start of the original signal 2. Calculate the wavelet coefficient C Continuous Wavelet Transform

14. ECE 501 Dr. Hang Five Steps to a CWT 3.Shift the wavelet to the right and repeat steps 1 and 2 until the whole signal is covered. Continuous Wavelet Transform

15. ECE 501 Dr. Hang Five Steps to a CWT 4.Scale the wavelet and repeat steps 1 through 3 Continuous Wavelet Transform

16. ECE 501 Dr. Hang Five Steps to a CWT 5.Repeat steps 1 through 4 for all scales Continuous Wavelet Transform

17. ECE 501 Dr. Hang Plot CWT coefficients Continuous Wavelet Transform

18. ECE 501 Dr. Hang Plot CWT coefficients Continuous Wavelet Transform

19. ECE 501 Dr. Hang Dyadic scales and positions: Mallat algorithm: fast algorithm via filtering Accurate analysis: compression, denoising Discrete Wavelet Transform

20. ECE 501 Dr. Hang One-Stage filtering: Approximations and Details Discrete Wavelet Transform Not Efficient!

21. ECE 501 Dr. Hang One-Stage filtering: Approximations and Details Discrete Wavelet Transform Efficient!

22. ECE 501 Dr. Hang One-Stage filtering: Approximations and Details Discrete Wavelet Transform

23. ECE 501 Dr. Hang One-Stage filtering: Approximations and Details Discrete Wavelet Transform

24. ECE 501 Dr. Hang Multiple-Level Decomposition Discrete Wavelet Transform

25. ECE 501 Dr. Hang Multiple-Level Decomposition Discrete Wavelet Transform

26. ECE 501 Dr. Hang Wavelet Reconstruction Discrete Wavelet Transform Up Sampling

27. ECE 501 Dr. Hang Wavelet Reconstruction Discrete Wavelet Transform

28. ECE 501 Dr. Hang Wavelet Reconstruction Discrete Wavelet Transform

29. ECE 501 Dr. Hang Wavelet Families Daubechies family

30. ECE 501 Dr. Hang Wavelet Families Symlets

31. ECE 501 Dr. Hang Denoising 1.Decompose 2.Threshold detail coefficients 3.Reconstruct

32. ECE 501 Dr. Hang Denoising Two thresholding method: (1) Soft (2) Hard