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Transformation Space

It is necessary when beginning to formulate the registration problem to decide what will be the space of allowable transformations, ST. The most general class of transformations are the local, non-linear transformations. These allow each voxel to be moved separately, leading to potentially millions of Degrees Of Freedom (DOF), although in practice, some constraints are necessary such as requiring that topology be preserved.

One basic class of transformations are the linear transformations where all voxels are constrained to move according to a global, linear relationship. That is,

\begin{displaymath}\begin{bmatrix}x' \\ y' \\ z' \\ 1 \end{bmatrix} =
\begin{bma...
...1 \\ \end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1 \end{bmatrix}\end{displaymath} (2)

or X' = A X using homogeneous coordinates.

The most general of these transformations is the affine transformation with 12 DOF ( $A_{11}, \ldots, A_{34}$), which includes rigid body (6 DOF) and similarity (7 DOF) transformations as particular cases. These transformations are also interesting physically because they represent the transformations that physical, rigid objects undergo. However, even for these simple linear cases, with a maximum of 12 DOF for the entire volume, the registration problem is still difficult. Therefore this report examines these affine transformations because they are the simplest, most common transformations used and, furthermore, many non-linear methods rely on having an initial linear fit as a preprocessing step.


next up previous
Next: Interpolation Up: Mathematical Formulation Previous: Mathematical Formulation
Mark Jenkinson
2000-05-10