The general idea behind motion-compensated temporal interpolation can be used to develop new algorithms for spatial interpolation. To examine these new algorithms for special interpolation. To exmine these new algorithms, let us consider a specific spatial interpolation problem. As discussed in Section 2.4, an NTSC television system uses a 2:1 interlaced format, scanned at a rate of 30 frames/sec. A frame consists of 525 horizontal scan lines which are divided into two fields, the odd field consisting of odd-numbered lines and the even field consisting of even-numbered ones. Creating a frame at time (from a field at time (through spatial interpolation can he useful in many applications, including a 60 frames/sec television without bandwidth increase and improved vertical resolution of frozen frames.

The spatial interpolation techniques discussed in Section 3.4.1 may be used in
creating a frame from a field, but incorporating some additional knowledge
about images may improve the performance of spatial interpolation algorithms.
Many elements in the visual world, such as contours and scratches, are
spatially continuous. This information can be exploited in creating a frame
from a field. Let f(x,y_{-1})** **and f(x,y_{0}) denote
image intensities of two adjacent horizontal scan lines of a field. We wish to
creafe a new horizontal scan line between f(x,y_{-1})** **and f(x,y_{0}).
One model that takes into account the spatial continuity of such elements as
contours and scratches is

f(x,y_{0})=f(x-d_{x},y_{-1}) (3.48)

where
d_{x} is a horizontal displacement between y_{-1 }and y_{0}.
Equation (3.48) can be viewed as a special case of the uniform transinational
velocity model of (3.29). The spatial variable y in (3.48) has a function very
similar to the time variable t in (3.29), and there is only one spatial
variable x in (3.48), while there are two spatial variables x and y in (3.29).
As a result, all our discussions in Section 3.4.2 apply to the problem of
estimating d_{x} in (3.48). For example, under the assumption of
uniform velocity, (3.48) can be expressed as

f(x,y)
=f(x-v_{x}(y-y_{-1}), y_{-}1), y_{-1} £ y £ y_{0} (3.49)

which leads directly to

_{} (3.50)

Equation
(3.48) can be used to develop region matching methods and (3.50) can be used to
develop spatio-temporal constraint methods for estimating d_{x} or v_{x}.
Once the horizontal displacement (or velocity) is estimated, it** **can be
used in spatial interpolation in a manner analogous to the temporal
interpolation discussed in Section 3.4.3. Figure 3.48 illustrates the
pertormailce of a spatial interpolation algorithm based on (3.50). Figure
3.48(a) shows a frame of 256 x 256 pixels obtained by repeating each horizontal
line of a 256 x 128-pixel image. Figure 3.48(b) shows the frame obtained by
bilinear spatial interpolation. Figure 3.48(c) shows the frame obtained by
estimating the horizontal displacement based on (8.50) and then using the
estimate for spatial interpolation. Spatial continuity of lines and contours is
preserved better in the image in Figure 3.48(c) than in the other two images in
Figures 3.48(a) and (b).