Recovering Missing Slices of the Discrete Fourier Transform using Ghosts

摘要

The Discrete Fourier Transform (DFT) underpins the solution to many inverseproblems commonly possessing missing or un-measured frequency information. Thisincomplete coverage of Fourier space always produces systematic artefactscalled Ghosts. In this paper, a fast and exact method for de-convolving cyclicartefacts caused by missing slices of the DFT is presented. The slicesdiscussed here originate from the exact partitioning of DFT space, under theprojective Discrete Radon Transform, called the Discrete Fourier Slice Theorem.The method has a computational complexity of O(n log2 n) (where n = N^2) and isconstructed from a new Finite Ghost theory. This theory is also shown to unifyseveral aspects of work done on Ghosts over the past three decades. The paperconcludes with a significant application to fast, exact, non-iterative imagereconstruction from sets of discrete slices obtained for a limited range ofprojection angles.