1 Recovering Missing Slices of the Discrete Fourier Transform using Ghosts Shekhar Chandra, Imants Svalbe, Jeanpierre Guédon, Andrew Kingston and Nicolas Normand Abstract —The Discrete Fourier Transform (DFT) underpins the solution to many inverse problems com- monly 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 com- plexity of O(n log 2 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 im- age reconstruction from sets of discrete slices obtained for a limited range of projection angles. Index Terms—Discrete Radon Transform, Mojette Transform, Discrete Tomography, Image Reconstruc- tion, Discrete Fourier Slice Theorem, Ghosts, Number Theoretic Transform, Limited Angle, Finite Ghost The- ory I. Introduction The Discrete Fourier Transform (DFT) is an important tool for inverse problems, where the Fourier representation of an object is used as a mechanism to recover that object. For example, in Computed Tomography (CT), the internal structure of an object is recovered or reconstructed from its projected “views” or projections [1]. The Fourier Transform (FT) of the projections can be placed into Fourier space and the inverse FT is used to reconstruct the object. This is especially advantageous given the efficient and low computationally complex algorithm of the DFT known as the Fast Fourier Transform (FFT) [2]. In such cases, the acquired data cannot fully cover Fourier space as the problem is ill-posed [3, 4]. Thus, there are missing Fourier coefficients and it is common practice to interpolate the space from the known Fourier coefficients. The choice of interpolation method is a major factor in determining the quality of the reconstruction [5, 6]. A. Ghosts Incomplete Fourier coverage leads to the introduction of reconstruction artefacts known as “Ghosts” or “invisible Shekhar Chandra and Imants Svalbe are with the School of Physics, Monash University, Australia. Email: Shekhar.Chandra@monash.edu or Imants.Svalbe@monash.edu Andrew Kingston is with the Department of Applied Mathematics, Australian National University. Jeanpierre Guédon and Nicolas Normand are with IRCCyN-IVC, École polytechnique de l’Université de Nantes, France. (a) (b) Figure 1. An example of discrete and finite Ghosts. (a) shows a Ghost, as constructed by Katz [8], which is invisible when projected at any of the four rational angles shown. (b) shows a finite (periodic) Ghost, which is invisible in three finite angles (m =1, 2, 3) shown. distributions” [7]. Ghosts are effectively formed from an under-determined set of projections, i.e. from the “missing” or unmeasured projections. Therefore, Ghosts are always present within CT reconstructions and are the main reason for the filtering and interpolation of projection data [3]. The initial work on Ghosts in continuous space was pioneered by Bracewell and Roberts [7], Logan [3], Katz [8] and Louis [9]. Recent work by Candès et al. [10] has also addressed this issue using more modern methods. In contrast, projection sets in Discrete Tomography (DT), which have practically well defined reconstruction processes [11], are often deliberately or naturally under- determined for various applications. These applications range from image encoding [12], network transmission [13] to tomography [14]. Discrete Ghosts were first proposed by Katz [8] as a way to describe zero-valued discrete projections taken at rational angles θpq , i.e. θpq = tan −1 ( q /p) (or simply the vector [q,p]) where p, q ∈ Z. A simple example of these discrete Ghosts is shown in Fig. 1(a). Katz [8] determined that an N × N image can be reconstructed exactly from a set of µ rational angle projections if and only if N  1 + max   μ−1  j=0 |p j |, μ−1  j=0 |q j |   . (1) This is now known as the Katz Criterion. It is a state- ment that the information contained in the projection set needs to be one-to-one with the image data. Thus, knowing whether the projections of a sub-region in the reconstruction meets the Katz Criterion will allow one to determine if the sub-region is exactly reconstructable. Chandra et al. [14, 15] showed that Ghosts also exist in the DFT as cyclic artefacts when DT projection data is missing. These cyclic Ghosts will be referred to as “Finite arXiv:1101.0076v1 [math-ph] 30 Dec 2010