A recent work Claerbout (1997) proposed a helix transform for mapping multidimensional convolution operators to their one-dimensional equivalents. The helix idea proves the feasibility of multidimensional deconvolution, an issue that has been in question for more than 15 years. By mapping discrete convolution operators to one-dimensional space, the inverse filtering problem can be conveniently recast in terms of recursive filtering, a well-known part of the digital filtering theory.
In this paper, we show how recursive deconvolution can be applied for preconditioning interpolation problems. We consider a problem of filling empty bins in a regularly gridded data volume. For a given estimate of the regularization filter, the missing data problem reduces to least-square optimization. Theoretical analysis and numerical examples show that helix preconditioning can produce a significant speed-up in the convergence of the iterative optimization schemes.