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Similar to homogeneous coordinates, transformation matrices should have a final element that is equal to unity. If a computation produces a matrix with a nonzero value other than unity in the final position, the entire matrix can be rescaled by the appropriate scalar value. Transformation matrices with a zero in the final position map real coordinates to vectors and are not discussed further here.
It is assumed throughout this chapter that transformations describing spatial inversions (i.e., mirror imaging) are not of interest. In two dimensions, this means that the upper left twobytwo submatrix must have a determinant greater than zero. Similarly, in three dimensions, the upper left threebythree submatrix must have a determinant greater than zero. Many of the approaches described here will fail if this assumption is incorrect.
For medical imaging, the most constrained spatial transformation model is the rigidbody model. This model asserts that distances and internal angles within the images cannot be changed during registration. As the name implies, this model assumes that the object behaves in the real world as a rigid body, susceptible to global rotations and translations, but internally immutable. This model is well suited to objects such as individual bones, which cannot be deformed. To a reasonable approximation, this model is also applicable to the brain, which is encased in bones that protect it from forces that might lead to deformations. However, it is well established that this is only an approximation, since parts of the brain, such as the brainstem, are subject to distortions induced by cardiac and respiratory cycles. For images accumulated over many cardiac and respiratory cycles, these movements may result in a blurred but highly consistent signal that follows the rigidbody assumptions quite well. However, for images acquired with a very short time frame, these movements can produce very clear violations of the rigidbody model assumptions.
Medical images often consist of voxels that differ in the realworld distances that they represent along the x, y, and zaxes. For example, it is common for the slice thickness in magnetic resonance imaging data to be larger than the size of individual pixels within each slice. If ignored, these anisotropies in voxel size will clearly lead to apparent violations of the rigidbody model, even for solid structures that accurately follow the rigidbody assumptions in the real world. Consequently, any implementation of a rigidbody model must explicitly correct for voxel sizes to ensure that the realworld distances and angles that are being represented do not change. In a worstcase scenario, six different voxel sizes may be involved: three anisotropic voxel sizes from one image, and three different anisotropic voxel sizes from the other image. A properly implemented rigidbody model for transforming such images may choose any one of these voxel sizes or may even select some other arbitrary voxel size. However, calculations must be included to rescale distances to compensate for the various voxel sizes. For the rigidbody model to be applicable, all six of the voxel sizes must be known accurately. If the voxel sizes are not known with certainty, the best strategy is to scan a phantom with known dimensions to determine the true voxel dimensions since errors in specification of the voxel dimensions will lead to unnecessary errors in registrations produced using a rigidbody model [2]. If this is not possible, the calibration errors can be estimated by adding additional parameters to augment the rigidbody model as discussed in subsequent sections of this chapter.
In three dimensions, the rigidbody model requires specification of six independent parameters. It is traditional (but not necessary) for three of these parameters to specify a threedimensional translation that is either preceded or followed by the sequential application of specified rotations around each of the three primary coordinate axes. However, before considering the threedimensional model, it is useful to consider the
(1) The application of translations before rotations:
the three primary coordinate axes. However, before considering the threedimensional model, it is useful to consider the
simpler case of two dimensions. In two dimensions, the rigid 
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