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The first summary is given in four tables of the basic relationships and equations, primarily developed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , for the scaling function $\phi \left(t\right)$ , scaling coefficients $h\left(n\right)$ , and their Fourier transforms $\text{\Phi}\left(\omega \right)$ and $H\left(\omega \right)$ for the multiplier $M=2$ or two-band multiresolution system. The various assumptions and conditions are omitted in order to see the “big picture"and to see the effects of increasing constraints.
Case | Condition | $\phi \left(t\right)$ | $\text{\Phi}\left(\omega \right)$ | Signal Space |
1 | Multiresolution | $\phi \left(t\right)=\sum h\left(n\right)\sqrt{2}\phi (2t-n)$ | $\text{\Phi}\left(\omega \right)=\prod \frac{1}{\sqrt{2}}H\left(\frac{\omega}{{2}^{k}}\right)$ | distribution |
2 | Partition of 1 | $\sum \phi (t-n)=1$ | $\text{\Phi}\left(2\pi k\right)=\delta \left(k\right)$ | distribution |
3 | Orthogonal | $\int \phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}\phi (t-k)\phantom{\rule{0.166667em}{0ex}}dt=\delta \left(k\right)$ | ${\sum \left|\text{\Phi}(\omega +2\pi k)\right|}^{2}=1$ | ${L}^{2}$ |
5 | SF Smoothness | $\frac{{d}^{\left(\ell \right)}\phi}{dt}<\infty $ | poly $\in {\mathcal{V}}_{j}$ | |
6 | SF Moments | $\int {t}^{k}\phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=0$ | Coiflets |
Case | Condition | $h\left(n\right)$ | $H\left(\omega \right)$ | Eigenval.{ T } |
1 | Existence | $\sum h\left(n\right)=\sqrt{2}$ | $H\left(0\right)=\sqrt{2}$ | |
2 | Fundamental | $\sum h\left(2n\right)=\sum h(2n+1)$ | $H\left(\pi \right)=0$ | $\mathrm{EV}=1$ |
3 | QMF | $\sum h\left(n\right)\phantom{\rule{0.166667em}{0ex}}h(n-2k)=\delta \left(k\right)$ | ${\left|H\left(\omega \right)\right|}^{2}+{\left|H(\omega +\pi )\right|}^{2}=2$ | $\mathrm{EV}\le 1$ |
4 | Orthogonal | $\sum h\left(n\right)\phantom{\rule{0.166667em}{0ex}}h(n-2k)=\delta \left(k\right)$ | ${\left|H\left(\omega \right)\right|}^{2}+{\left|H(\omega +\pi )\right|}^{2}=2$ | one $\mathrm{EV}=1$ |
${L}^{2}$ Basis | and $H\left(\omega \right)\ne 0,\left|\omega \right|\le \pi /3$ | others $<1$ | ||
6 | Coiflets | $\sum {n}^{k}h\left(n\right)=0$ |
Case | Condition | $\text{\psi}\left(t\right)$ | $\text{\Psi}\left(\omega \right)$ | Signal Space |
1 | MRA | $\text{\psi}\left(t\right)=\sum {h}_{1}\left(n\right)\sqrt{2}\phi (2t-n)$ | $\text{\Psi}\left(\omega \right)=\prod \frac{1}{\sqrt{2}}{H}_{1}\left(\frac{\omega}{{2}^{k}}\right)$ | distribution |
3 | Orthogonal | $\int \varphi \left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi (t-k)\phantom{\rule{0.166667em}{0ex}}dt=0$ | ${L}^{2}$ | |
3 | Orthogonal | $\int \psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi (t-k)\phantom{\rule{0.166667em}{0ex}}dt=\delta \left(k\right)$ | ${L}^{2}$ | |
5 | W Moments | $\int {t}^{k}\phantom{\rule{0.166667em}{0ex}}\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=0$ | poly $not\in {\mathcal{W}}_{j}$ |
Case | Condition | ${h}_{1}\left(n\right)$ | ${H}_{1}\left(\omega \right)$ | Eigenval.{ T } |
2 | Fundamental | $\sum {h}_{1}\left(n\right)=0$ | ${H}_{1}\left(0\right)=0$ | |
3 | Orthogonal | ${h}_{1}\left(n\right)={(-1)}^{n}h(1-n)$ | $|{H}_{1}\left(\omega \right)|=|H(\omega +\pi )|$ | |
3 | Orthogonal | $\sum {h}_{1}\left(n\right){h}_{1}(2m-n)=\delta \left(m\right)$ | $|{H}_{1}{\left(\omega \right)|}^{2}+{\left|H\left(\omega \right)\right|}^{2}=2$ | |
5 | Smoothness | $\sum {n}^{k}{h}_{1}\left(n\right)=0$ | $H\left(\omega \right)={(\omega -\pi )}^{k}\phantom{\rule{0.166667em}{0ex}}\tilde{H}\left(\omega \right)$ | $1,\frac{1}{2},\frac{1}{4},\cdots $ |
The different “cases" represent somewhat similar conditions for the stated relationships. For example, in Case 1, Table 1, themultiresolution conditions are stated in the time and frequency domains while in Table 2 the corresponding necessary conditionson $h\left(n\right)$ are given for a scaling function in ${L}^{1}$ . However, the conditions are not sufficient unless general distributions areallowed. In Case 1, Table 3, the definition of a wavelet is given to span the appropriate multiresolution signal space butnothing seems appropriate for Case 1 in Table 4. Clearly the organization of these tables are somewhat subjective.
If we “tighten" the restrictions by adding one more linear condition, we get Case 2 which has consequences in Tables 1, 2, and 4 but does not guarantee anything better that adistribution. Case 3 involves orthogonality, both across scales and translations, so there are two rows for Case 3 in the tablesinvolving wavelets. Case 4 adds to the orthogonality a condition on the frequency response $H\left(\omega \right)$ or on the eigenvalues of the transition matrix to guarantee an ${L}^{2}$ basis rather than a tight frame guaranteed for Case 3. Cases 5 and 6concern zero moments and scaling function smoothness and symmetry.
In some cases, columns 3 and 4 are equivalent and others, they are not. In some categories, a higher numbered case assumes alower numbered case and in others, they do not. These tables try to give a structure without the details. It is useful torefer to them while reading the earlier chapters and to refer to the earlier chapters to see the assumptions and conditionsbehind these tables.
Here we try to present a structured list of the various classes of wavelet systems in terms of modification and generalizations of thebasic $M=2$ system. There are some classes not included here because the whole subject is still an active research area, producing newresults daily. However, this list plus the table of contents, index, and references will help guide the reader through the maze. Therelevant section or chapter is given in parenthesis for each topic.
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