The precise method of sampling and sub-sampling the volumes to calculate the cost function is also important. In the method proposed here the reference volume is initially re-sampled to an isotropic grid with voxel size 1mm cubed. This is done by interpolating the values available in the original volume which usually has anisotropic voxel dimensions. Once this isotropic 1mm resolution reference volume has been obtained, the 2mm, 4mm and 8mm sub-sampled versions are created.
Sub-sampling the reference volume by a factor of two is done by first
blurring the intensities using a convolution with a discrete, 3D
Gaussian kernel where: FWHM = n mm (or
mm), with
n being the size of the required sub-sampling (that is, 2, 4 or
8). This blurring is done so that all points on the lattice contribute
equally to the sub-sampled version. The sub-sampling then simply
takes every nth point on the lattice in each direction. Therefore,
the new volume contains 1/n3 as many points as the original and so
the total storage for all four volumes (1mm, 2mm, 4mm and 8mm
resolutions) is just 14% more than for the 1mm resolution volume
alone.
To evaluate the cost function at a resolution of n mm requires the intensities at the isotropic lattice sites to be known for both the reference and floating volumes. The reference volume intensities are already known, having been calculated and stored as described above. Therefore, it is only necessary to calculate the floating volume intensities. As various transformations are applied to the floating volume during the optimisation procedure, the interpolated values are not stored but just calculated as required by the cost function. However, before the optimisation an initial blurring is applied to the stored intensities in the floating volume, as is done for the reference volume. In this case though, the volume usually has an anisotropic voxel size, and so the discrete Gaussian kernel used is also anisotropic, reflecting the unequal sampling of the continuous, isotropic Gaussian kernel.