DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Figure 17.6 Cubic interpolation
of a signal starts with
equally spaced samples, in
this example 47, 42, 43,
and 46. The underlying
function is estimated to
be a cubic polynomial that
passes through (“interpolates”)
all four samples.
The polynomial is evaluated
between the two
central samples, as shown
by the black segment. Here,
evaluation is at phase offset
�. If the underlying function
isn’t a polynomial,
small errors are produced.
Eq 17.5
Sample value
x − x0
� = ; x0≤ � ≤ x
x1−x 1
0
47
46
45
44
43
42
41
s -1
x-1 x0 � x1 x2 Sample coordinate
outside the interval [x -1 , x 2 ] would be called extrapolation.)
In digital video and audio, we limit x to the range
between x 0 and x 1 , so as to estimate the signal in the
interval between the central two samples. To evaluate
outside this interval, we substitute the input sample
values [s -1 , s 0 , s 1 , s 2 ] appropriately – for example, to
evaluate between s 1 and s 2 , we shift the input sample
values left one place.
With uniform sampling (as in conventional digital video),
when interpolating between the two central samples
the argument x can be recast as the phase offset, or the
fractional phase (�, phi), at which a new sample is required
between two central samples. (See Equation 17.5.)
In abstract terms, � lies between 0 and 1; in hardware,
it is implemented as a binary or a rational fraction. In
video, a 1-D interpolator is usually an FIR filter whose
coefficients are functions of the phase offset. The
weighting coefficients (c i) are functions of the phase
offset; they can be considered as basis functions.
In signal processing, cubic (third-degree) interpolation
is often used; the situation is sketched in Figure 17.6
above. In linear interpolation, one neighbor to the left
and one to the right are needed. In cubic interpolation,
we ordinarily interpolate in the central interval, using
two original samples to the left and two to the right of
the desired sample instant.
178 DIGITAL VIDEO AND HDTV ALGORITHMS AND INTERFACES
s 0
~ g(�)
s 1
s 2