and an instantaneous translational transformation and an instantaneous translational transformation

FIGURE 2 Demonstration that the transformation illustrated in Fig. 1 can also be achieved by a single rotation around an appropriately chosen fixed point in space. The location of this point can be determined by computing the real eigenvectors of the transformation. The intermediate transformations shown in light gray can be derived using matrix logarithms and matrix exponentials.

FIGURE 2 Demonstration that the transformation illustrated in Fig. 1 can also be achieved by a single rotation around an appropriately chosen fixed point in space. The location of this point can be determined by computing the real eigenvectors of the transformation. The intermediate transformations shown in light gray can be derived using matrix logarithms and matrix exponentials.

translations (along the x-, y-, and z-axes) and three rotations (around the x-,y-, and z-axes). For brain images, the x-axis is often defined as the axis that passes from left to right, the y-axis as the axis from back to front, and the z-axis as the axis from bottom to top. In this case, rotation around the x-axis is also referred to as pitch, rotation around the y-axis as roll, and rotation around the z-axis as yaw. None of these conventions should be considered universal, but they will be the definitions used here. Readers should also be aware that defining the positive and negative ends of the x-, y-, and z-axes requires an arbitrary choice between two different coordinate systems that are effectively three-dimensional mirror images of one another. If the origin is viewed from one end of the z-axis such that the x-axis is oriented horizontally with positive values to the right and the y-axis oriented vertically with positive values to the top, a ngfo-^anded coordinate system will place positive z-values in front of the origin and negative z-values behind the origin. The opposite arrangement defines a k/f-foanded coordinate system. Knowing the handedness of the coordinate system is vital for interpreting displayed medical images (e.g., for distinguishing which side of the brain or body is right and which side is left in the real world). However, for the purposes of registration, the issue can be ignored so long as all the images are known to follow the same convention.

If a three-dimensional point (x, y, z) is to be transformed by an elementary transformation to some new point (x', y', z'), the following equations describe the elementary transformations:

Note that these equations arbitrarily define a positive yaw as the direction such that a positive 90° rotation will map the positive end of the y-axis onto the positive end of the x-axis. Similarly, a positive roll of 90° is defined to map the positive end of the x-axis onto the positive end of the z-axis. Finally, a positive pitch of 90° is defined to map the positive end of the z-axis onto the positive end of the y-axis. These relationships are illustrated in Fig. 3. Any one, two, or three of these arbitrary definitions can be reversed by simply reversing the sign of the corresponding sin( ) terms, but the definitions used here will turn out to have certain advantages when interpreting instantaneous elementary transformations. Note that none of these arbitrary definitions have any bearing on the handedness of the coordinate system, and they cannot be used to deduce the handedness without additional information.

These elementary transformations can be rewritten in a convenient matrix formulation as follows:

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