| |
Methods
90
control point under consideration, the image is deformed based on the resulting displacement
fields and the energy is calculated. Once all displacement values in angular direction have
been tested, the best displacement values in horizontal and vertical directions are set for that
control point. Then, the same is done for the radial direction for the same control point before
the next control point is tested. Once all relevant control points have been tested in this way,
the first iteration has finished. Further iterations are done if specified by the user in the options
menu. The disadvantage of this approach is that the algorithm needs longer than the previ-
ously explained algorithm because it searches in two directions and not only in one direction.
5.2.7 Constraints
The aim of the registration process is it to find displacement fields u and v that are smooth.
Besides the computational cost, another advantage is that using a spline-based approach gen-
erates smooth displacement fields.
In addition, certain constraints have been defined. For a simple case, consider the follow-
ing 1D scenario in Figure 42. There are three control points (filled large circles). The black
dots represent the locations at which the image values are interpolated in the original image.
The interpolated values end up at the corresponding control point positions in the deformed
image. For example, the value at (3) ends up at control point (3’) and the value at (1) ends up
at control point (1’). If control point (2’) got a value from outside of the shaded area (since it
Figure 42: Constraints for displacement values in 1D
See text for details
(2)
(2’)
(2)
(1)
(1’)
(3’)
(3)
x
|  |
|
| |
|
|
