Centered Transforms in examples

The set of transformed prefix with “Centered” have the center of rotation parameter as an optimizable parameter, where the normal transforms have the center of rotation as a fixed parameter. It is my understanding that the “Centered” transform make the optimization more difficult and generally are not recommended for use. Is this correct?

The Centered Transforms are used in a number of ITK examples. If it is correct that these transforms should not be recommend, then we need to update the example to to use them.

Yes, this is correct.

The issue is that the parameters associated with a rotation, R, a translation T, and a center of rotation C, have an influence on mapping a point x to the new position of a point x’, according to this relationship:
x’ = R(x - C) + T
= Rx -RC + T

Consequently, a change in the center of rotation during the optimization has an impact on the estimated rotation and translation. The location of the center of rotation also may have an influence on whether data is mapped inside or outside of the output image grid.

Often times in practice, it is better to fix the center of rotation e.g. to the coordinates of the center of an image, or to the center of mass of an image, and then to estimate only R and T.
Given R, C and T, it is always possible to compute an equivalent mapping to represent the transform about a changed center of rotation.