Transformation Parameterization

A 3D Fourier series representation[32,16] was used to parameterize the forward and reverse displacement fields $u_{AB}$ and $u_{BA}$. It is assumed that the images $T_A$ and $T_B$ are represented as $N_1 \times N_2 \times N_3$ voxel volumes. The displacement fields are defined to have the form

$$u_{AB} (x) = \sum_{i=-d_1}^{d_1} \sum_{j=-d_2}^{d_2} \sum_{k=-d_3}^{d_3} \mu_{ijk} e^{\hat{j}<x,\omega_{ijk}>} .$$ (8)

where $\mu_{ijk}$ are $(3 \times 1)$, complex-valued vectors with complex conjugate symmetry and $\omega_{ijk} = [\frac {2\pi i} {N_1}, \frac {2\pi j} {N_2}, \frac {2\pi k} {N_3}]$.

In practice, we estimate the low frequency basis coefficients before the higher ones allowing the global image features to be registered before the local features. The values of $d_1$, $d_2$, and $d_3$ are initially set small and are periodically increased throughout the iterative minimization procedure. The value of $d_1$ is restricted to be less than $(N_{1}-1)/2$ because that is the maximum spatial frequency basis coefficient realizable in the FFT representation. The limits on the other two summations are changed in a similar fashion.

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