Recovering Missing Slices of the Discrete Fourier Transform Using Ghosts

• Published in: IEEE transactions on image processing Vol. 21; no. 10; pp. 4431 - 4441
• Main Authors: Chandra, S. S., Svalbe, I. D., Guedon, J., Kingston, A. M., Normand, N.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2012; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Combinatorics; Computer Science; Convolution; Cyclic ghost theory; discrete Fourier slice theorem; Discrete Fourier transforms; Discrete Mathematics; discrete Radon transform; discrete tomography; Exact sciences and technology; Fourier Analysis; Ghosts; Humans; Image processing; Image Processing, Computer-Assisted - methods; Image reconstruction; Information, signal and communications theory; Kernel; limited angle; Mathematical Physics; Mathematics; Mojette transform; number theoretic transform; Physics; Signal processing; Telecommunications and information theory; Tomography; Tomography - methods; Vectors; Mojette transform; Image processing; Partition method; Discrete Fourier transformation; Discrete transformation; Iterative method; Artefact; Coverage; discrete Fourier slice theorem; Computational complexity; Image reconstruction; Inverse problem; Cyclic ghost theory; Ghosts; discrete Radon transform; number theoretic transform; Partitioning; discrete tomography; Tomography; Radon transformation; limited angle; Number theory; Limited Angle; Cyclic Ghost Theory; Discrete Radon Transform; Image Reconstruction; Discrete Fourier Slice Theorem; Mojette Transform; Discrete Tomography; Number Theoretic Transform
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2012.2206033

The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(nlog 2 n) (for an n=N×N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT.