Changing Orientation

Calculating the field produced by a rotated object is equivalent to calculating the field produced by rotated applied fields -- that is, applying rotated $\vec{B}^{(0)}$. Note that the latter calculation is in the reference frame of the object, not the scanner, and so calculating the scanner defined $z$ component of the perturbed field requires projection onto the scanner frame $z$ axis unit vector. Furthermore, since $\vec{B}^{(1)}$ is a linear function of $\vec{B}^{(0)}$ then rotating $\vec{B}^{(0)}$ is equivalent to rotating $\vec{B}^{(1)}$. Hence the scanner defined $z$ component of the perturbed field resulting from an applied field $B^{(0)}$ in the $z$ direction is

$$\widetilde{B}^{(1)}_{z} = [0 \; 0 \; 1] \, R^{-1} \, \left[ \begi... ...) \end{array} \right] R \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$$ (27)

where $R$ is a $3 \times 3$ matrix that represents the rotation from the scanner to the object coordinate system, that is $\vec{x_{ob}}= R \, \vec{x_{sc}}$, and $B^{(1)}_{p}(\hat{\vec{q}})$ is the field calculated in the $p$ direction ($x, y$ or $z$) from an applied field $B^{(0)} = \hat{\vec{q}}$ (being either $\hat{\vec{x}}, \hat{\vec{y}}$ or $\hat{\vec{z}}$).

In practice, the matrix of perturb fields, $[ B^{(1)}_{p}(\hat{\vec{q}}) ]$, acts as a set of 9 basis images, which can be precalculated and then combined as specified to give the desired field at any orientation. This does not involve further approximation, it is precisely the same perturbed field that would be calculated for the object in the new orientation. In addition, although the perturbed field is linear in the basis images, it is not linear in the rotation angles, since the elements of $R$ are non-linear functions of these angles.

A similar calculation can be done for the gradients of the field, $\nabla B^{(1)}$.