Multi-dimensional Fourier transforms in the helical coordinate system

Theory

For simplicity, throughout this section we refer to a two-dimensional sampled image, ${\bf b}$; however, the beauty of the helical coordinate system is that everything can be trivially extended to an arbitrary number of dimensions.

We employ two equivalent subscripting schemes for referring to an element of the two-dimensional image, ${\bf b}$. Firstly, with two subscripts, $b_{p_x,p_y}$ refers to the element that lies $p_x$ increments along the $x$-axis, and $p_y$ increments along the $y$-axis. Ranges of $p_x$and $p_y$ are given by $0 \leq p_x < N_x$, and $0 \leq p_y < N_y$ respectively. Helical coordinates suggest an alternative subscripting scheme: We can use a single subscript, $p_h=p_x + p_y N_x$, such that $b_{p_x,p_y} = b_{p_h}$ and the range of $p_h$ is given by $0 \leq p_h < N_x N_y$. Moreover, if we impose helical boundary conditions, we can treat ${\bf b}$ as a one-dimensional function of subscript $p_h$.