Wavelet Decomposition Basics
Characteristic of wavelet expansions
- A wavelet expansion is formed by a two-dimensional expansion of a signal. It should be noticed that the dimension of the signal itself is not determinant in the wavelet representation.
- A wavelet expansion provides a dual time-frequency localization of the input signal. This implies that most of the energy of the signal will be captured by a few coefficients.
- The computational complexity of the discrete wavelet transform is at most O(nlog(n)) i.e. as bad as for the discrete Fourier transform (DFT) when calculated using the Fast Fourier Transform (FFT). For some particular types of wavelets, the complexity can be as low as O(n).
- The basis functions in a wavelet expansion are generated from the mother wavelet by scaling and translation operations. The indexing in two dimensions is achieved using this expression:ψj,k(t)=2j/2ψ(2jt−k)j,k∈Z(4)
- Most wavelets basis functions satisfy multiresolution conditions. This property guarantees that if a set of signals can be represented by basis functions generated from a translation ψ(t−k) of the mother wavelet, then a larger set of functions, including the original, can be represented by a new set of basis functions ψ(2t−k). This feature is used in the algorithm of the fast wavelet transform, FWT the equivalent of the FFT algorithm for wavelets decomposition.
- The lower resolution coefficients can be calculated from the higher resolution coefficients using a filter bank algorithm. This property contributes to the efficiency of the calculation of the DWT.
Denoising with Wavelets
Classical approach to wavelet thresholding
The minimax threshold
The universal threshold
The translation invariant method
The SureShrink Threshold
Classical Methods: Block Thresholding
Overlapping Block Thresholding Estimator
Bayesian Approach to wavelet shrinkage and thresholding
Shrinkage estimates based on deterministic/stochastic decompositions
Description of the Scheme
- Root Mean Squared Error: The mean squared error defined as (Equation Equation 37)is computed for each realization and averaged over the 100 samples. Then, its square root is taken.
1 N N ∑ i=1
- Maximum Deviation: The average over the 100 samples of max|f,(xi),−,fdn,(xi)|
- VisuShrink-Hard: Universal threshold with hard thresholding rule
- VisuShrink-Soft: Universal threshold with soft thresholding rule
- SureShrink: SureShrink threshold
- Translation-Invariant-Hard: Translation invariant threshold with hard thresholding rule
- Translation-Invariant-Soft: Translation invariant threshold with soft thresholding rule
- Minimax-Hard: Minimax threshold with hard thresholding rule
- Minimax-Soft: Minimax threshold with soft thresholding rule
- NeighBlock: Overlapping block thresholding (with L0=[logn/2], λ=4.50524)
- Linear Penalization: Term-by-term thresholding using linear shrinking
- Deterministic/Stochastic: Bayesian thresholding method for shrinkage estimates