1. Recovering Missing Slices of the Discrete Fourier Transform Using Ghosts
- Author
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Jean-Pierre Guédon, Shekhar S. Chandra, Nicolas Normand, Andrew Kingston, Imants D. Svalbe, School of Physics, Monash University [Clayton], Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN), Mines Nantes (Mines Nantes)-École Centrale de Nantes (ECN)-Ecole Polytechnique de l'Université de Nantes (EPUN), and Université de Nantes (UN)-Université de Nantes (UN)-PRES Université Nantes Angers Le Mans (UNAM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
FOS: Computer and information sciences ,Discrete Fourier Slice Theorem ,Ghosts ,Discrete Mathematics (cs.DM) ,Iterative method ,Computer science ,Discrete Tomography ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,FOS: Physical sciences ,Cyclic Ghost Theory ,Image processing ,02 engineering and technology ,Iterative reconstruction ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,Discrete Radon Transform ,Discrete Fourier transform ,Limited Angle ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Projection-slice theorem ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Image Processing, Computer-Assisted ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Humans ,Mathematics - Combinatorics ,Image Reconstruction ,Mojette Transform ,Tomography ,Mathematical Physics ,Fourier Analysis ,020206 networking & telecommunications ,Mathematical Physics (math-ph) ,Inverse problem ,Computer Graphics and Computer-Aided Design ,Frequency domain ,020201 artificial intelligence & image processing ,Combinatorics (math.CO) ,Deconvolution ,Number Theoretic Transform ,Artifacts ,Algorithm ,Discrete tomography ,Software ,Computer Science - Discrete Mathematics - 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., 10 pages, 18 figures (submitted to IEEE Image Proc.)
- Published
- 2012
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