Interpolation

For discrete data, the intensity is normally only defined on a grid, G, of discrete locations or lattice sites. That is, the data is stored as I_ijk = I(x_i,y_j,z_k) where I_ijk represents the discrete data and $(x_i, y_j, z_k) \in G$ the coordinates of the lattice sites , with $I(\cdot)$ the underlying but unobservable continuous image. However, when the lattices for I^r and I^f are not perfectly aligned, it is necessary to evaluate the intensity at points in between the lattice sites. This is common because virtually any transformation that is applied to I^f will cause the lattices to be out of alignment.

To evaluate the intensity at intermediate locations requires interpolation. The interpolation can be viewed as reconstructing a full continuous image from the discrete points, although to evaluate the cost function it is usually only necessary to know the intensity at the corresponding lattice sites. That is, if (x_i,y_j,z_k)represent the lattice sites for I^r then it is usually only necessary to know the value of I^f(T(x_i,y_j,z_k)). Typically, interpolation methods are based on a convolution of the discrete data with some continuous kernel such as trilinear, spline and (windowed) sinc kernels.

One major effect that the choice of interpolation has is to what degree the cost function becomes continuous or discontinuous. This is also affected by the boundary conditions used, such as padding with zeros or only using the overlapping volume. Studying the precise effects of interpolation is an active research area [Hajnal et al., 1995,Pluim et al., 2000,Thacker et al., 1999] but is beyond the scope of this report. Therefore, in this report trilinear interpolation is used on the overlapping volume. These choices require no additional parameters to be set and were motivated largely by experience.