#### 3.4.4.Application of Motion Estimation Methods to Spatial Interpolation

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,y0) 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,y0). One model that takes into account the spatial continuity of such elements as contours and scratches is

f(x,y0)=f(x-dx,y-1)                                                                                             (3.48)

where dx is a horizontal displacement between y-1 and y0. 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 dx in (3.48). For example, under the assumption of uniform velocity, (3.48) can be expressed as

f(x,y) =f(x-vx(y-y-1), y-1),   y-1 £ y £ y0                                                               (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 dx or vx. 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). 