lecture10: Discrete-Wavelet filters
We now take wavelets onto the domain of discrete-time
signals (all of the work in lecture 9 concerns wavelets on
continuous functions). We call the wavelet transform on a
discrete-time signal a Discrete Wavelet Filter. When we consider
discrete-time signals, there are significant computational
advantages to the wavelet transform, in particular we will discuss
the pyramidal filter-bank approach to computing wavelets.
Topics covered
- Discrete wavelet filters (DWF)
- Continuous to discrete
- Wavelets as convolution
- How to get wavelet filter: sampling
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- Sampling the wavelet
- Possible range of scales
- Example: Haar wavelets
- Discrete wavelet filters (DWF)
- Pyramidal decomposition algorithm
- Example
- Representation
- Initialization
- Deriving the filters directly
- Example: Haar
- Finite signals
- Haar
- Filter properties
- Vanishing moments
- Support
- Regularity
- Daubechies wavelets
- Daubechies wavelets
- Daubechies wavelet filters
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- Daubechies wavelets
- Shannon wavelets
- Other wavelets
- Symmlets
- Battle-Lemari\'{e}
- Applications
- Tonebursts
- De-noising
- Dyadic grid
- Test signal: Blocks
- Test signal: Blocks with noise
- Haar Wavelet transform
- Histogram of details at scale 1
- Thresholded details
- Thresholded transform
- Reconstructed signal
- De-noising
- Edge Detection
- Code