Wavelet Transform:
Wavelet Transform A very brief lookWavelets vs. Fourier Transform:
2 Wavelets vs. Fourier Transform In Fourier transform (FT) we represent a signal in terms of sinusoids FT provides a signal which is localized only in the frequency domain It does not give any information of the signal in the time domainWavelets vs. Fourier Transform:
3 Wavelets vs. Fourier Transform Basis functions of the wavelet transform (WT) are small waves located in different times They are obtained using scaling and translation of a scaling function and wavelet function Therefore, the WT is localized in both time and frequencyWavelets vs. Fourier Transform:
4 Wavelets vs. Fourier Transform In addition, the WT provides a multiresolution system Multiresolution is useful in several applications For instance, image communications and image data base are such applicationsWavelets vs. Fourier Transform:
5 Wavelets vs. Fourier Transform If a signal has a discontinuity, FT produces many coefficients with large magnitude ( significant coefficients) But WT generates a few significant coefficients around the discontinuity Nonlinear approximation is a method to benchmark the approximation power of a transformWavelets vs. Fourier Transform:
6 Wavelets vs. Fourier Transform In nonlinear approximation we keep only a few significant coefficients of a signal and set the rest to zero Then we reconstruct the signal using the significant coefficients WT produces a few significant coefficients for the signals with discontinuities Thus, we obtain better results for WT nonlinear approximation when compared with the FTWavelets vs. Fourier Transform:
7 Wavelets vs. Fourier Transform Most natural signals are smooth with a few discontinuities (are piece-wise smooth ) Speech and natural images are such signals Hence, WT has better capability for representing these signal when compared with the FT Good nonlinear approximation results in efficiency in several applications such as compression and denoisingSeries Expansion of Discrete-Time Signals:
8 Series Expansion of Discrete-Time Signals Suppose that is a square-summable sequence, that is Orthonormal expansion of is of the form Where is the transform of The basis functions satisfy the orthonormality constraintSlide 9:
9 Haar expansion is a two-point avarage and difference operation The basis functions are given as It follows that Haar BasisSlide 10:
10 The transform is The reconstruction is obtained from Haar BasisTwo-Channel Filter Banks:
11 Two-Channel Filter Banks Filter bank is the building block of discrete-time wavelet transform For 1-D signals, two-channel filter bank is depicted belowTwo-Channel Filter Banks:
12 Two-Channel Filter Banks For perfect reconstruction filter banks we have In order to achieve perfect reconstruction the filters should satisfy Thus if one filter is lowpass, the other one will be highpassTwo-Channel Filter Banks:
13 Two-Channel Filter BanksTwo-Channel Filter Banks:
14 Two-Channel Filter Banks To have orthogonal wavelets, the filter bank should be orthogonal The orthogonal condition for 1-D two-channel filter banks is Given one of the filters of the orthogonal filter bank, we can obtain the rest of the filtersHaar Filter Bank:
15 Haar Filter Bank The simplest orthogonal filter bank is Haar The lowpass filter is And the highpass filterHaar Filter Bank:
16 Haar Filter Bank The lowpass output is And the highpass output isHaar Filter Bank:
17 Haar Filter Bank Since and , the filter bank implements Haar expansion Note that the analysis filters are time-reversed versions of the basis functions since convolution is an inner product followed by time-reversalDiscrete Wavelet Transform:
18 Discrete Wavelet Transform We can construct discrete WT via iterated (octave-band) filter banks The analysis section is illustrated below Level 1 Level 2 Level JDiscrete Wavelet Transform:
19 Discrete Wavelet Transform And the synthesis section is illustrated here If is an orthogonal filter and , then we have an orthogonal wavelet transformMultiresolution:
20 Multiresolution We say that is the space of all square-summable sequences if Then a multiresolution analysis consists of a sequence of embedded closed spaces It is obvious thatMultiresolution:
21 Multiresolution The orthogonal component of in will be denoted by : If we split and repeat on , , …., , we have2-D Separable WT:
22 2-D Separable WT For images we use separable WT First we apply a 1-D filter bank to the rows of the image Then we apply same transform to the columns of each channel of the result Therefore, we obtain 3 highpass channels corresponding to vertical, horizontal, and diagonal, and one approximation image We can iterate the above procedure on the lowpass channel2-D Analysis Filter Bank:
23 2-D Analysis Filter Bank diagonal vertical horizontal approximation2-D Synthesis Filter Bank:
24 2-D Synthesis Filter Bank diagonal vertical horizontal approximation2-D WT Example:
25 2-D WT Example Boats image WT in 3 levelsWT-Application in Denoising:
26 WT-Application in Denoising Boats image Noisy image (additive Gaussian noise)WT-Application in Denoising:
27 WT-Application in Denoising Boats image Denoised image using hard thresholdingReference:
28 Reference Martin Vetterli and Jelena Kovacevic, Wavelets and Subband Coding . Prentice Hall, 1995.