Details: Oblique slice calculation using rotation matrices.

Here are some examples of using the rotation matrices to calculate the linear combinations of Gx, Gy, and Gz needed to produce oblique slices. In the first example, assume the following coordinate system with the three planes XY, XZ, and YZ.

The rotation equations for clockwise (CW) rotation of θ=90° about an axis are:

,

,

.

In this convention, X’, Y’, and Z’ refer to the original components of the gradients and X, Y, and Z refer to the final components of the gradients. To image in a tomographic slice in the following planes, the following gradients are needed.

.

To image an XZ plane knowing only the conditions for imaging an XY plane you need to take the conditions necessary to image the XY plane and transform them into an XZ plane by rotation about +X by ??= -90°. The rotation equation is

For the slice selection gradient we have

For the phase encoding gradient we have

or

.

For the frequency encoding gradient we have

or

.

Please note that these are identical to what you would think for the slice. Les anyone think we are restricted to starting from the XY plane, let’s start from the YZ plane and go to the XZ. The rotation matrix needed is a 90° CW rotation about +Z.

For the slice selection gradient we have

For the phase encoding gradient we have

or

.
For the frequency encoding gradient we have

or

.

The same procedure works for oblique planes. Consider a CW rotation of the YZ plane by +45º about +Z.

For the slice selection gradient we have

.
For the phase encoding gradient we have

,
or

.
For the frequency encoding gradient we have

,
or

.

It should be apparent that this process can be applied to any number of rotations to get a any starting plane to any oblique location. All that is needed is a repeated set of rotations to get the starting plane to the desired plane. Simple examples may be more informative so consider the following. Starting with the ZY plane we rotate about Y by -90° to get an XY plane which we rotate by -90° about X to get an XZ plane.

The rotation of a gradient vector about Y is done first followed by the rotation about X.

For the slice selection gradient we have

.
For the phase encoding gradient we have

,
or

.

For the frequency encoding gradient we have

,
or

.