<< Chapter < Page | Chapter >> Page > |
[link] gives the same information for the length-6, 4, and 2 Daubechies scaling coefficients, wavelet coefficients, scaling coefficient moments,and wavelet coefficient moments. Again notice how many discrete wavelet moments are zero.
[link] shows the continuous moments of the scaling function and wavelet for the Daubechies systems with lengths six and four. The discrete moments are the momentsof the coefficients defined by [link] and [link] with the continuous moments defined by [link] and [link] calculated using [link] and [link] with the programs listed in Appendix C.
Daubechies | |||||
0 | 0.33267055295008 | -0.03522629188571 | 1.414213 | 0 | 0 |
1 | 0.80689150931109 | -0.08544127388203 | 1.155979 | 0 | 1 |
2 | 0.45987750211849 | 0.13501102001025 | 0.944899 | 0 | 2 |
3 | -0.13501102001025 | 0.45987750211849 | -0.224341 | 3.354101 | 3 |
4 | -0.08544127388203 | -0.80689150931109 | -2.627495 | 40.679682 | 4 |
5 | 0.03522629188571 | 0.33267055295008 | 5.305591 | 329.323717 | 5 |
Daubechies | |||||
0 | 0.48296291314453 | 0.12940952255126 | 1.414213 | 0 | 0 |
1 | 0.83651630373781 | 0.22414386804201 | 0.896575 | 0 | 1 |
2 | 0.22414386804201 | -0.83651630373781 | 0.568406 | 1.224744 | 2 |
3 | -0.12940952255126 | 0.48296291314453 | -0.864390 | 6.572012 | 3 |
Daubechies | |||||
0 | 0.70710678118655 | 0.70710678118655 | 1.414213 | 0 | 0 |
1 | 0.70710678118655 | -0.70710678118655 | 0.707107 | 0.707107 | 1 |
0 | 1.4142135 | 0 | 1.0000000 | 0 |
1 | 1.1559780 | 0 | 0.8174012 | 0 |
2 | 0.9448992 | 0 | 0.6681447 | 0 |
3 | -0.2243420 | 3.3541019 | 0.4454669 | 0.2964635 |
4 | -2.6274948 | 40.6796819 | 0.1172263 | 2.2824642 |
5 | 5.3055914 | 329.3237168 | -0.0466511 | 11.4461157 |
0 | 1.4142136 | 0 | 1.0000000 | 0 |
1 | 0.8965755 | 0 | 0.6343975 | 0 |
2 | 0.5684061 | 1.2247449 | 0.4019238 | 0.2165063 |
3 | -0.8643899 | 6.5720121 | 0.1310915 | 0.7867785 |
4 | -6.0593531 | 25.9598790 | -0.3021933 | 2.0143421 |
5 | -23.4373939 | 90.8156100 | -1.0658728 | 4.4442798 |
These tables are very informative about the characteristics of wavelet systems in general as well as particularities of the Daubechies system.We see the of [link] and [link] that is necessary for the existence of a scaling function solution to [link] and the of [link] and [link] that is necessary for the orthogonality of the basis functions. Ortho normality requires [link] which is seen in comparison of the and , and it requires from [link] and [link] . After those conditions are satisfied, there are degrees of freedom left which Daubechies uses to set wavelet moments equal zero. For length-6 we have two zero wavelet moments and for length-4, one. For all longer Daubechies systems we have exactly zero wavelet moments in addition to the one for a total of zero wavelet moments. Note as will be explained in [link] and there exist relationships among some of the values of the even-ordered scaling functionmoments, which will be explained in [link] through [link] .
As stated earlier, these systems have a maximum number of zero moments of the wavelets which results in a high degree ofsmoothness for the scaling and wavelet functions. [link] and [link] show the Daubechies scaling functions and wavelets for . The coefficients were generated by the techniques described in Section: Parameterization of the Scaling Coefficients and Chapter: Regularity, Moments, and Wavelet System Design . The Matlab programs are listed in Appendix C and values of can be found in [link] or generated by the programs. Note the increasing smoothness as is increased. For , the scaling function is not continuous; for , it is continuous but not differentiable; for , it is barely differentiable once; for , it is twice differentiable, and similarly for longer . One can obtain any degree of differentiability for sufficiently long .
Notification Switch
Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?