Helical boundary conditions

The helix transform Claerbout (1997) provides boundary conditions that map multi-dimensional convolution into one-dimension. In this case, the 2-D convolution operator, $\left(\alpha_1 {\bf I} + {\bf D} \right)$, can be recast as an equivalent 1-D filter.

Helical boundary conditions allow the two-dimensional convolution matrix, ${\bf A}_1$, to be expressed as a one-dimensional convolution with a filter of length 2 N_x +1 that has the form

$$a_1 = (1,\,0, \; ... \; 0,\,1,\,\alpha_1-4, \,1,\,0, \; ... \; 0,\,1)$$

The structure of the finite-difference Laplacian operator, ${\bf D}$, is simplified when compared to equation (9).  

$${\bf D} = \left[ \begin{array} {cccccccc} & & & & & & & \\ ... ...& & & & \\ . & . & . & 1 & . & . & 1 & -4 \\ \end{array}\right]$$ (10)

The 1-D filter can be factored into a causal and anti-causal parts, and the matrix inverse can be computed by recursive polynomial division (1-D deconvolution).