In this case, one line in the source image is mapped to a corresponding line
in the destination image. The other parts of the image are moved appropriately
to maintain their relative position from the specified line. Each of these
lines is directed. This could be used to rotate and scale images.
A few examples are shown below. The original image and three different
rotated and scaled versions of the image are shown. The line chosen in the
original image was the edge to the left of the image. It was mapped to a line
along the bottom of the image. The rotated version is obtained by using two
lines of the same length. The scaling is done using the lengths of the two
lines. The scaling could be done in two different ways. They are:
Scaling along the line and perpendicular to the line
Scaling along the line only
The two other scaled images are examples of these two types of scaling.
Original Image
Rotated and Scaled in both directions
Rotated Image
Rotated and Scaled only along the line
The rotation and scaling can be restricted to a region around the line
by introducing a weighting function.
Transformation with Multiple Pairs of Lines
Transformations performed using more than one pair of lines involve a weighted
combination of the transformations performed by each line. Since reverse
mapping is used, for each pixel in the destination image we find a pixel in
the source image which would be used. The displacement corresponding to each
line is calculated for each pixel. A weigted average of these displacements
is used to determine how much each pixel needs to be moved.
The weight used depends on the distance of the point under consideration
from the line. The weight can also depend on the length of the line.
This transformation, based on multiple lines, can be used effectively for
morphing.
The details of these transformations can be found here.
