Processing for compression - The MPEG - page 85

[ 16 ] De With, P.H.N., Data Compression Techniques for Digital Video Recording , PhD Thesis, Technical University of
Delft (1992)
3.14 The wavelet transform
The wavelet transform was not discovered by any one individual, but has evolved via a number of similar ideas and
was only given a strong mathematical foundation relatively recently. [ 17 ][ 18 ] Much of the early work was performed in
France, where the term ondelette was coined. Wavelet is an anglicized equivalent. The Fourier transform is based
on periodic signals and endless basis functions and requires windowing to a short- term transform in practical use.
The wavelet transform is fundamentally windowed and there is no assumption of periodic signals. The basis
functions employed are not endless sine waves, but are finite on the time axis. Wavelet transforms do not use a
fixed window, but instead the window period is inversely proportional to the frequency being analysed.
As a result a useful combination of time and frequency resolutions is obtained. High frequencies corresponding to
transients in audio or edges in video are transformed with short basis functions and therefore are accurately
located. Low frequencies are transformed with long basis functions which have good frequency resolution. [ 19 ][ 20 ]
Figure 3.46 shows that in the pure time (or space in imaging) domain, there are only samples or pixels. These are
taken at vanishingly small instants and so cannot contain any frequency information. At the other extreme is the
Fourier domain. Here there are only coefficients representing the amplitude of endless sinusoids which cannot
carry any time/space information.
Figure 3.46: When in the pure time domain, nothing is known about frequency and vice versa. These pure
domains are academic and require infinitely large and small quantities to implement. Wavelets operate in the real
world where time and frequency information is simultaneously present.
The Heisenberg uncertainty principle simply observes that at any position on the continuum between extremes a
certain combination of time and frequency accuracy can be obtained. Clearly the human senses operate with a
combination of both. Wavelets are transforms which operate in the time/frequency domain as shown in the figure.
There are pixels at one end of the scale and coefficients at the other. Wavelet parameters fall somewhere in
between the pure pixel and the pure coefficient. In addition to describing the amount of energy at a certain
frequency, they may also describe where that energy existed. There seems to be no elegant contraction of
coefficient and pixel to produce an English term for a wavelet parameter, but the Franglais terms pixelette and
ondel have a certain charm.
The fundamental component of a wavelet transform is the filter decimator shown in Figure 3.47 ( a). This is a
complementary high- and low-pass filter which divides the input bandwidth in half. The high-pass filter impulse
response is a wavelet. If a block of samples or pixels of a certain size is input, each output is decimated by a factor
of two so that the same number of samples is present on the output. There is a strong parallel with the quadrature
mirror filter here. The low-pass output is a reduced resolution signal and the high-pass output represents the detail
which must be added to make the original signal. The high-pass process is a form of differentiation. Figure 3.48
shows that these two sample streams can be recombined. Each one is upsampled by a factor of two by an
appropriate form of interpolator and then the two sample streams are added.
Figure 3.47: (a) The fundamental component of a wavelet transform is a filter decimator. (b) These stages can be
cascaded.