32
ECE 501 Introduction to BME ECE 501 Dr. Hang
ECE 501 Introduction to BME
ECE 501 Dr. Hang
Part V Biomedical Signal Processing Introduction to Wavelet Transform
ECE 501 Dr. Hang
ECE 501 Dr. Hang
Fourier Analysis
Introduction
ECE 501 Dr. Hang
Fourier Analysis
Introduction
• A serious drawback: time information is lost
• Cannot handle transitory characteristics
0 200 400 600 800 1000 1200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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
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
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.
ECE 501 Dr. Hang
Fourier Analysis The sum over all time of the signal multiplied by a complex exponential
Continuous Wavelet Transform
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
ECE 501 Dr. Hang
Scaling• Scaling a wavelet: stretching or compressing it• a: scaling factor
Continuous Wavelet Transform
ECE 501 Dr. Hang
Scaling• Low scale High frequency • High scale Low frequency
Continuous Wavelet Transform
ECE 501 Dr. Hang
Shifting
Continuous Wavelet Transform
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
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
ECE 501 Dr. Hang
Five Steps to a CWT
4. Scale the wavelet and repeat steps 1 through 3
Continuous Wavelet Transform
ECE 501 Dr. Hang
Five Steps to a CWT
5. Repeat steps 1 through 4 for all scales
Continuous Wavelet Transform
ECE 501 Dr. Hang
Plot CWT coefficients
Continuous Wavelet Transform
ECE 501 Dr. Hang
Plot CWT coefficients
Continuous Wavelet Transform
ECE 501 Dr. Hang
• Dyadic scales and positions:
• Mallat algorithm: fast algorithm via filtering
• Accurate analysis: compression, denoising
Discrete Wavelet Transform
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
Not Efficient!
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
Efficient!
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
ECE 501 Dr. Hang
Multiple-Level Decomposition
Discrete Wavelet Transform
ECE 501 Dr. Hang
Multiple-Level Decomposition
Discrete Wavelet Transform
ECE 501 Dr. Hang
Wavelet Reconstruction
Discrete Wavelet Transform
Up Sampling
ECE 501 Dr. Hang
Wavelet Reconstruction
Discrete Wavelet Transform
ECE 501 Dr. Hang
Wavelet Reconstruction
Discrete Wavelet Transform
ECE 501 Dr. Hang
Wavelet Families
Daubechies family
ECE 501 Dr. Hang
Wavelet Families
Symlets
ECE 501 Dr. Hang
Denoising
1. Decompose
2. Threshold detail coefficients
3. Reconstruct
ECE 501 Dr. Hang
Denoising
Two thresholding method: (1) Soft (2) Hard
