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Both orders of transformation are illustrated in Fig.1. Note that in either case, the total distance of translation (9.8489 units) remains unchanged.

One of the advantages of the matrix formulation of the two-dimensional rigid-body transformation is that it eliminates the need to specify whether rotations precede or follow translations. Indeed, for any given rigid-body transformation specified in matrix form, it is possible to use Eqs. (1) and (2) to respecify the transformation using either of the orders of elementary transformation. If you envision the transformation process as being displayed as a movie, the matrix formulation only represents the beginning and the ending of the movie and omits all the details about what route might have been taken in between. In fact, the intervening details as described by a particular sequence of elementary transformations are almost certainly irrelevant to the registration problem, since rotations are conceptualized as taking place around the origin of the chosen coordinate system. Medical applications of rigid-body transformations involve real biological subjects who will be unaware of and unconstrained by the point in space that will correspond to the origin when images are registered. Furthermore, biological subjects are not constrained to move from one position to another by applying sequences of pure translations and pure rotations in any coordinate system. Motions decomposed into such sequences would appear mechanical rather than biological.

A second advantage of the matrix formulation is that it allows the use of standard linear algebraic methods for inverting or combining transformations. Given a rigid-body transformation matrix for registering one image to a second image, simple matrix inversion derives a matrix for registering the second matrix to the first. This inverted matrix is guaranteed to describe a rigid-body transformation and, if necessary, can be decomposed into elementary transformations. Similarly, given a rigid-body transformation matrix for registering the first image to the second and another rigid-body transformation matrix for registering the second image to a third, simple matrix multiplication gives a rigid-body transformation matrix for registering the first image directly to the third. Matrix inversion and multiplication can also be usefully combined. For example, given a rigid-body transformation matrix for registering one image to a second image and another rigid-body transformation matrix for registering the first image to a third image, the product of the inverse of the first matrix and the second matrix will produce a rigid-body transformation matrix that will register the second image to the third.

One disadvantage of the matrix formulation is that it is not always easy to verify that a particular transformation matrix actually does describe a rigid-body transformation. In two dimensions, the obvious symmetries of the two-by-two submatrix in the upper left-hand corner of the transformation matrix are helpful, but it is still generally necessary to make computations to detect imposter matrices of the form

x'

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