Information Technology Reference
In-Depth Information
Figure 3.50:
Unlike discrete Fourier transforms, wavelet basis functions are scaled so that they contain the same
number of cycles irrespective of frequency. As a result their frequency discrimination ability is a constant proportion
of the centre frequency.
Figure 3.51:
(a) In transforms greater certainty in the time domain leads to less certainty in the frequency domain
and vice versa. Some transform coders split the spectrum as in (b) and use different window lengths in the two
bands. In the recently developed wavelet transform the window length is inversely proportional to the frequency,
giving the advantageous time/frequency characteristic shown in (c).
The dual of this temporal behaviour is that the frequency discrimination of the wavelet transform is a constant
fraction of the signal frequency. In a filter bank such a characteristic would be described as 'constant
Q
'.
Figure
3.52
shows the division of the frequency domain by a wavelet transform is logarithmic whereas in the Fourier
transform the division is uniform. The logarithmic coverage is effectively dividing the frequency domain into
octaves.
Figure 3.52:
Wavelet transforms divide the frequency domain into octaves instead of the equal bands of the
Fourier transform.
When using wavelets with two-dimensional pixel arrays, vertical and horizontal filtering is required.
Figure 3.49
(
b)
shows that after a single stage, decimation by two in two axes means that the low-pass output contains one quarter
of the original data, whereas the difference data require the other three quarters.
