The Discrete radon transform: A more efficient approach to image reconstruction

The Radon transform and its inversion are the mathematical keys that enable tomography. Radon transforms are defined for continuous objects with continuous projections at all angles in [0,π). In practice, however, we pre-filter discrete projections take

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

Optimizing illumination patterns for classical ghost imaging

Classical ghost imaging is a new paradigm in imaging where the image of an object is not measured directly with a pixelated detector. Rather, the object is subject to a set of illumination patterns and the total interaction of the object, e.g., reflected or transmitted photons or particles, is measured for each pattern with a single-pixel or bucket detector. An image of the object is then computed through the correlation of each pattern and the corresponding bucket value. Assuming no prior knowledge of the object, the set of patterns used to compute the ghost image dictates the image quality. In the visible-light regime, programmable spatial light modulators can generate the illumination patterns. In many other regimes -- such as x rays, electrons, and neutrons -- no such dynamically configurable modulators exist, and patterns are commonly produced by employing a transversely-translated mask. In this paper we explore some of the properties of masks or speckle that should be considered to maximize ghost-image quality, given a certain experimental classical ghost-imaging setup employing a transversely-displaced but otherwise non-configurable mask.Comment: 28 pages, 17 figure

Geometric shape effects in redundant keys used to encrypt data transformed by finite discrete radon projections

The Finite discrete Radon Transform (FRT) represents digital data exactly and without redundancy. Redundancy can however be injected into the FRT by reserving part of the image area to be replaced by a key that contains pixels of known, fixed values. Th

On Correcting the Uneveness of Angle Distributions Arising from Integer Ratios Lying in Restricted Portions of the Farey Plane

In 2D discrete projcctive transforms, projection angles correspond to lines linking pixels at integer multiples of the x and y image grid spacing. To make the projection angle set non-redundant, the integer ratios are chosen from the set of relatively prime fractions given by the Farcy sequence. To sample objects uniformly, the set of projection angles should be uniformly distributed. The unevenness function measures the deviation of an angle distribution from a uniformly increasing sequence of angles. The allowed integer multiples are restricted by the size of the discrete image array or by functional limits imposed on the range of x and y increments for a particular transform. This paper outlines a method to compensate the unevenness function for the geometric effects of different restrictions on the ranges of integers selected to form these ratios. This geometric correction enables a direct comparison to be made of the effective uniformity of an angle set formed over selected portions of the Farey Plane. This result has direct application in comparing the smoothness of digital angle sets

Intertwined Digital Rays in Discrete Radon Projections Pooled over Adjacent Prime Sized Arrays

Digital projections are image intensity sums taken along directed rays that sample whole pixel values at periodic locations along the ray. For 2D square arrays with sides of prime length, the Discrete Radon Transform (DRT) is very efficient at reconstructing digital images from their digital projections. The periodic gaps in digital rays complicate the use of the DRT for efficient reconstruction of tomographic images from real projection data, where there are no gaps along the projection direction. A new approach to bridge this gap problem is to pool DRT digital projections obtained over a variety of prime sized arrays. The digital gaps are then partially filled by a staggered overlap of discrete sample positions to better approximate a continuous projection ray. This paper identifies primes that have similar and distinct DRT pixel sampling patterns for the rays in digital projections. The projections are effectively pooled by combining several images, each reconstructed at a fixed scale, but using projections that are interpolated over different prime sized arrays. The basis for the pooled image reconstruction approach is outlined and we demonstrate the principle of this mechanism works

Farey Sequences and Discrete Radon Transform Projection Angles

This paper examines how the minimal set of digital projection angles for the Discrete Radon Transform (DRT) is selected from the known sequence of Farey fractions. A square array of prime size p defines a unique direction for each digital projection, m, through an integer ratio xm/ym. Here xm and ym define the nearest neighbour distance, dm, between projection samples under a modulus p sampling rule. We show the maximum gap length, dmax, on square and hexagonal lattices is < √(2p/√3) and ≤ √p respectively. The DRT angles are shown to replicate the entire Farey fraction sequence over the interval 1 ≤ dm ≤ √p for a square lattice. For the interval √p < dm < dmax, the DRT set skips some Farey angles. The skipping of Farey angles is the result of a geometric restriction on the distance minimisation process. The DRT angle set varies significantly with p for those projections with dm near dmax. This complicates the comparison of DRT projections over similar sized arrays. The detailed angle distributions of the DRT and the Farey sequences reflect, in different ways, the variable gaps between prime numbers. Thanks are due to Graham Farr, Computer Science, Monash University, for pointing us to the literature on finding minimal vectors in lattices. IS acknowledges support from the Centre for X-Ray Physics and Imaging for this project. AK is a Monash University postgraduate student in receipt of an Australian Postgraduate Award scholarship, provided through the Australian Government