Multi-dimensional spectral factorization on a helix

Claerbout (1998b) describes the isomorphic process by which multi-dimensional functions can be mapped into equivalent one-dimensional functions. The process depends on the concept of the helical boundary conditions, and is best illustrated by Figure , which shows a small five-point filter on a two-dimensional space, being mapped into an equivalent one-dimensional filter.

 

helix
Figure 1 Illustration of helical boundary conditions mapping a two-dimensional function (a) onto a helix (b), and then unwrapping the helix (c) into an equivalent one-dimensional function (d). Figure by Sergey Fomel.

Under such a transformation, the concepts of causality and minimum-phase become clear. One-dimensional spectral factorization algorithms can be directly applied to the multi-dimensional helical functions.