ECE 501 Introduction to BME

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

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Fourier Analysis

Introduction

• A serious drawback: time information is lost

• Cannot handle transitory characteristics

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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

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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

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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.

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Fourier Analysis The sum over all time of the signal multiplied by a complex exponential

Continuous Wavelet Transform

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CWT The sum over all time of the signal multiplied by scaled , shifted version of the wavelet function

Continuous Wavelet Transform

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Scaling• Scaling a wavelet: stretching or compressing it• a: scaling factor

Continuous Wavelet Transform

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Scaling• Low scale High frequency • High scale Low frequency

Continuous Wavelet Transform

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Shifting

Continuous Wavelet Transform

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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

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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

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Five Steps to a CWT

4. Scale the wavelet and repeat steps 1 through 3

Continuous Wavelet Transform

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Five Steps to a CWT

5. Repeat steps 1 through 4 for all scales

Continuous Wavelet Transform

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Plot CWT coefficients

Continuous Wavelet Transform

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Plot CWT coefficients

Continuous Wavelet Transform

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• Dyadic scales and positions:

• Mallat algorithm: fast algorithm via filtering

• Accurate analysis: compression, denoising

Discrete Wavelet Transform

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One-Stage filtering: Approximations and Details

Discrete Wavelet Transform

Not Efficient!

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One-Stage filtering: Approximations and Details

Discrete Wavelet Transform

Efficient!

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One-Stage filtering: Approximations and Details

Discrete Wavelet Transform

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One-Stage filtering: Approximations and Details

Discrete Wavelet Transform

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Multiple-Level Decomposition

Discrete Wavelet Transform

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Multiple-Level Decomposition

Discrete Wavelet Transform

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Wavelet Reconstruction

Discrete Wavelet Transform

Up Sampling

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Wavelet Reconstruction

Discrete Wavelet Transform

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Wavelet Reconstruction

Discrete Wavelet Transform

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Wavelet Families

Daubechies family

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Wavelet Families

Symlets

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Denoising

1. Decompose

2. Threshold detail coefficients

3. Reconstruct

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Denoising

Two thresholding method: (1) Soft (2) Hard