Development of a wavelet-based algorithm to detect and determine ...

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 49
Figure 6.13: The dyadic lattice in the time-scale plane
rithms that concern the convolution theorem. For example, it is considerably faster to
do the calculation in the frequency domain with the help of the Fourier transform: one
can calculate the wavelet coefficients for a given scale at all sections simultaneously
(step 2 and 3 at the same time) and efficiently.
6.2.3 Discretization of Continuous Wavelet Transform
As mentioned above, to carry out the continuous wavelet transform is a fair amount
of work and it generates large amount of data. Therefore, the continuous wavelet
transform is primarily employed to derive properties and the discrete form is necessary
for reducing redundancy and for most computer implementations.
In general, the discrete forms of the continuous wavelet transforms are generated by
the sampling from the corresponding continuous wavelet transform. For example, the
scale is sampled by a = a j
0, and the translation by b = nb0a j
0, where a0 and b0 are the
discrete scale and the translation step sizes respectively and j ∈ Z. If a0 = 2 and b0 = 1,
then the scales and positions will be based on the power of two, i.e. a = 2 j and b = n2 j .
The wavelet coefficients then correspond to the dyadic lattice points in the time-scale
plane, as shown in Figure 6.13. This lattice can be indexed by two integers: j and n,
where j corresponds to the discrete scale levels and n to the discrete translation steps.