Recovering Missing Slices of the Discrete Fourier Transform using Ghosts

TL;DR

This paper introduces a fast, exact method for removing artefacts caused by missing Fourier slices, enabling improved non-iterative image reconstruction from limited projection data using a new Finite Ghost theory.

Contribution

It presents a novel, computationally efficient de-convolution technique based on Finite Ghost theory for reconstructing images from incomplete Fourier data.

Findings

Method has O(n log2 n) complexity

Unifies previous Ghost theories over three decades

Enables exact non-iterative image reconstruction

Abstract

The Discrete Fourier Transform (DFT) underpins the solution to many inverse problems commonly possessing missing or un-measured frequency information. This incomplete coverage of Fourier space always produces systematic artefacts called Ghosts. In this paper, a fast and exact method for de-convolving cyclic artefacts caused by missing slices of the DFT is presented. The slices discussed here originate from the exact partitioning of DFT space, under the projective 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 is constructed from a new Finite Ghost theory. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. The paper concludes with a significant application to fast, exact, non-iterative image reconstruction from sets of discrete slices obtained for a limited range of projection angles.