Processing for compression - The MPEG - page 91

performed in parallel. In this way small objects are not missed because they will generate high-frequency
components in the transform.
Although the matching process is simplified by adopting phase correlation, the Fourier transforms themselves
require complex calculations. The high performance of phase correlation would remain academic if it were too
complex to put into practice. However, if realistic values are used for the motion speeds which can be handled, the
computation required by block matching actually exceeds that required for phase correlation.
The elimination of amplitude information from the phase-correlation process ensures that motion estimation
continues to work in the case of fades, objects moving into shade or flashguns firing.
The details of the Fourier transform have been described in section 3.5 .
A one-dimensional example of phase correlation will be given here by way of introduction. A line of luminance,
which in the digital domain consists of a series of samples, is a function of brightness with respect to distance
across the screen. The Fourier transform converts this function into a spectrum of spatial frequencies (units of
cycles per picture width) and phases.
All television signals must be handled in linear-phase systems. A linear-phase system is one in which the delay
experienced is the same for all frequencies. If video signals pass through a device which does not exhibit linear
phase, the various frequency components of edges become displaced across the screen. Figure 3.56 shows what
phase linearity means. If the left-hand end of the frequency axis (DC) is considered to be firmly anchored, but the
right-hand end can be rotated to represent a change of position across the screen, it will be seen that as the axis
twists evenly the result is phase-shift proportional to frequency. A system having this characteristic is said to
display linear phase.
Figure 3.56: The definition of phase linearity is that phase shift is proportional to frequency. In phase-linear
systems the waveform is preserved, and simply moves in time or space.
In the spatial domain, a phase shift corresponds to a physical movement. Figure 3.57 shows that if between fields a
waveform moves along the line, the lowest frequency in the Fourier transform will suffer a given phase shift, twice
that frequency will suffer twice that phase shift and so on. Thus it is potentially possible to measure movement
between two successive fields if the phase differences between the Fourier spectra are analysed. This is the basis
of phase correlation.
Figure 3.57: In a phase-linear system, shifting the video waveform across the screen causes phase shifts in each
component proportional to frequency.
Figure 3.58 shows how a one-dimensional phase correlator works. The Fourier transforms of two lines from
successive fields are computed and expressed in polar (amplitude and phase) notation (see section 3.5 ). The
phases of one transform are all subtracted from the phases of the same frequencies in the other transform. Any
frequency component having significant amplitude is then normalized, or boosted to full amplitude.