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3.6 Geometric transformations 151

(a) (b)

(c) (d)

Figure 3.51 Image warping alternatives (Gomes, Darsa, Costa et al. 1999) c 1999 Morgan Kaufmann: (a)
sparse control points −⇔ deformation grid; (b) denser set of control point correspondences; (c) oriented line
correspondences; (d) uniform quadrilateral grid.

meshes are used, it may be desirable to interpolate displacements down to individual pixel
values using a smooth interpolant such as a quadratic B-spline (Farin 1996; Lee, Wolberg,
Chwa et al. 1996). 20
In some cases, e.g., if a dense depth map has been estimated for an image (Shade, Gortler,
He et al. 1998), we only know the forward displacement for each pixel. As mentioned before,
drawing source pixels at their destination location, i.e., forward warping (Figure 3.46), suffers
from several potential problems, including aliasing and the appearance of small cracks. An
alternative technique in this case is to forward warp the displacement field (or depth map) to
its new location, fill small holes in the resulting map, and then use inverse warping to perform
the resampling (Shade, Gortler, He et al. 1998). The reason that this generally works better
than forward warping is that displacement fields tend to be much smoother than images, so
the aliasing introduced during the forward warping of the displacement field is much less
noticeable.
A second approach to specifying displacements for local deformations is to use corre-
sponding oriented line segments (Beier and Neely 1992), as shown in Figures 3.51c and 3.52.
Pixels along each line segment are transferred from source to destination exactly as specified,
and other pixels are warped using a smooth interpolation of these displacements. Each line
segment correspondence specifies a translation, rotation, and scaling, i.e., a similarity trans-
form (Table 3.5), for pixels in its vicinity, as shown in Figure 3.52a. Line segments influence
the overall displacement of the image using a weighting function that depends on the mini-
mum distance to the line segment (v in Figure 3.52aif u ∈ [0, 1], else the shorter of the two

20 Note that the block-based motion models used by many video compression standards (Le Gall 1991) can be
thought of as a 0th-order (piecewise-constant) displacement field.