Recovering Missing Slices of the Discrete Fourier Transform using Ghosts

Shekhar Chandra (1) , Imants Svalbe (1) , Jeanpierre Guédon (2)

1 School of Physics
2 IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes

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.

Mots clés: Discrete Radon Transform; Mojette Transform; Discrete Tomography; Image Reconstruction; Discrete Fourier Slice Theorem; Ghosts; Number Theoretic Transform; Limited Angle; Cyclic Ghost Theory

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Shekhar Chandra, Imants Svalbe, Jeanpierre Guédon. Recovering Missing Slices of the Discrete Fourier Transform using Ghosts. IEEE Transactions on Image Processing, 2012, 21 (10), pp.4431-4441. ⟨10.1109/TIP.2012.2206033⟩. ⟨hal-00717810⟩