Other Special Constrained Affine Transformations

Erroneous calibration of equipment may occasionally warrant the use of other special spatial transformation models in which additional degrees of freedom are added to the rigid-body model to account for certain inaccuracies. For example, if the voxel sizes are uncertain along all three axes and subject to session-to-session variations, one dimension can be kept fixed and the other five dimensions (two from the same session and three from the other session) can be rescaled, producing an 11-parameter model. However, it should be noted that models that involve rescaling both before and after rotation can present problems with minimization because of ambiguities that arise when rotations are absent, causing certain parameters to effectively play identical roles. In scanners that have gantries that tilt with respect to the bed, erroneous calibration of the tilt angle can cause individual planes of an image to be skewed with respect to one another. This error can be modeled with by adding degrees of freedom to alter the potentially inaccurate value. Because of the variety of different errors of this nature that can occur, it may be expedient to simply use the general affine model described in the next section, rather than trying to implement a tailored spatial transformation model. It should be noted that overparameterization may lead to an increase in errors by allowing the model to have an unnecessary mode of variation, but in practice, this may be unimportant. Analysis of transformations obtained with a general affine model may also lead to a better understanding of the source of calibration errors. For example, given one image from a well-calibrated scanner and another image of the same subject from a scanner with uncertain calibration, singular value decomposition of the transformation that registers these two images will give an estimate of the accuracy of the unknown scanner. If pure distance calibration errors are present, the U matrix or the V matrix will be close to an identity matrix and the magnitude of the inaccuracies will be revealed by the S matrix.

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