Helical boundary conditions

As discussed previously, the helix transform Claerbout (1998b) provides boundary conditions that map multi-dimensional convolution into one-dimension. In this case, the 2-D convolution operator, $\left({\bf I} + \alpha {\bf D} \right)$ can be recast as an equivalent 1-D filter. The 5-point 2-D filter becomes the sparse 1-D filter of length 2 N_x +1 that has the form,

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

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

$${\bf D} = \left[ \begin{array} {cccccccc} -4 & 1 & . & . & 1... ...4 & 1 \\ . & . & . & 1 & . & . & 1 & -4 \\ \end{array}\right]$$ (43)

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).