Transformations

If we can assume that the structure imaged does not change shape or size between the different images, we use a rigid body transformation. This has six degrees of freedom - namely rotations about each of the three Cartesian axes and translations along them - and the transformation is described fully by these six values. A rigid body transformation is usually assumed to be sufficient when registering head images from the same individual. It will also be sufficient when registering images of individual bones, and may even be sufficient when registering images of soft tissue structures that are securely attached to bony structure in, for example, the mandible, neck, or pelvis. The generalization of this transformation to include shears is called the affine transformation and has twelve degrees of freedom. An affine transformation transforms parallel lines to parallel lines.

If soft tissue motion is repetitive and reproducible, for example, cardiac or breathing motion, then gating techniques may be used to ensure that images of the same part of the breathing or cardiac cycle are registered. In this case, the rigid body transformation may still suffice. If soft tissue deformation is not constrained then many more parameters or degrees of freedom are required to describe the transformation. One well-known method uses approximating B-splines on a grid of control points and may require ~1000 degrees of freedom to describe tissue deformation [5].

Transformations describing the mapping of images between individuals in cohort studies or atlas registration may also require many degrees of freedom. A popular way of doing this spatial renormalization is to use Talairach space, a piecewise rigid body transformation with scaling in which brains are aligned to an axis, defined by the intersection of the interhemispheric fissure with the anterior and posterior commissure, and scaled to fit within a bounding box [6]. Often, corresponding MR images are collected in cohort PET studies. In this case, MR images can be aligned to a common reference or atlas, perhaps itself generated from an alignment of a large number of MR images [4]. A number of non-rigid registration algorithms have been proposed for this task [5, 7-11]. Alignment of the PET images is then achieved by concatenation of the non-rigid MRI to atlas with the rigid-body, patient-specific PET to MRI transformation. Each non-rigid algorithm produces slightly different transformations and validation remains a research task.

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