Recovering missing slices of the discrete fourier transform using ghosts

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(n\log-{2}n) (for an n=N\times 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

On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems

On Constructing Minimal Ghosts

International audienceGhosts are digital images that contain highly constrained patterns of signed pixel values. The pixels are located so as to create zero-sums when discrete projections are taken across the image at a pre-determined set of angles. Ghosts can be applied to create image/anti-image pairs. An image that is entangled with its anti-image can be used to achieve forward error-correction in redundant data transmission schemes. Ghosts can also be used to help reconstruct images from asymmetric sets of real, noisy tomographic projection data. Minimal ghosts do these tasks most efficiently. We present here new methods to construct minimal ghost images that employ just $2N$ pixels to obtain zero-sum projections for $N$ angles. Construction of an Nth order ghost had previously required $O(2^N)$ pixels