An angular parameterization of parallel Radon projections referred to in this paper as ψ-parameterization is discussed in
relevance to the efficiency of reconstruction from fan data. The fact that the ψ-parameterization coincides with the
equiangular fan beam parameterization allows us to develop a simple and efficient approach useful for the reconstruction
from fan data. Within this approach parallel projections are approximated by groups of semi-parallel rays. The
reconstruction is carried out directly, i.e. without any modification of original data, at the speed which is comparable or
even higher than that of the parallel Filtered Back Projection (FBP) algorithm.
OPED is a reconstruction algorithm for Radon data based on orthogonal polynomial expansion on the
disk. The algorithm involves a sum of<i> N</i> terms, which is determined by the number of view angles in the
data. Evaluating on a rectangular grid of M×M pixels, the algorithm can be implemented with roughly
O(N<sup>3</sup>) evaluations, if we assume M ≈ N, and the constant is rather large. The new implementation
uses a particular polar grid, so that the evaluation operation is reduced to 2N<sup>3</sup> + O(N<sup>2</sup> logN), a
reduction of the evaluation time by a factor of more than 20 times. Linear interpolation on triangle is
used to reduce our particular polar grid to the rectangular grid. Numerical experiments are presented
to demonstrate the results.
A new type of X-ray CT scanning geometry is proposed. The geometry of the scanner includes a half ring
detector array and resembles the geometry of a scanner of the fourth generation. Unlike the latter, the proposed
system collects parallel projections allowing efficient collimation of the incident beam for the purpose of scatter
reduction. The geometry of the data collected with the proposed scanner is ideal for algorithms developed for
image reconstruction from parallel projections with a non-uniform sampling such as the Orthogonal Polynomial
Expansion on Disk (OPED) algorithm. This scanner can be efficiently used in applications where high precision
measurements at micrometer scales are required, e.g. in the exact quantification of the morphology of small
The sampling geometry of CT-scanners plays an important role in the reconstruction of images. We have
previously reported a test-device that directly collects the Radon data within a special scanning geometry, whose
acquired data can be efficiently treated with series expansion algorithms such as, for example, OPED (Orthogonal
Polynomial Expansion on Disc). This geometry has the potential of reducing the radiation exposure of the patient
by about a factor of two. However, a fourth of the data must be obtained by interpolation within the measured
projections. In this contribution, we show by a Monte Carlo simulation that this interpolation has no significant
influence on the quality of the reconstructions.
Refining the sampling geometry of a CT scanner is a standard approach used for reduction of aliasing artifacts in
CT images. Although this leads to reduction of the artifacts, the principal problem of aliasing streaks artifacts
remains unsolved. A different approach is proposed, which in some special cases can solve the problem very
efficiently. It is shown that under certain specific conditions, the sum of images reconstructed from the data
collected within different sampling geometries is free of aliasing. These conditions are studied and practical
situations where they can be realized are discussed.
Preliminary results for a new CT scanning device with dose-reduction potential were presented at the SPIE
Medical Imaging conference 2007. The new device acquires the Radon data after the X-ray beam is collimated
through a special mask. This mask is combined with a new and efficient data collection geometry; thus the device
has the potential of reducing the dose by a factor of two. In this work, we report the first complete proof of the
idea using the same simplified mask of 197 detectors as last year, and a clinical C-arm with a flat panel detector
to simulate the gantry. This addition enables the acquisition of two independent and complementary data sets
for reconstruction. Moreover, this clinical set-up enables the acquisition of data for clinically relevant phantoms.
Phantom data were acquired using both detector sets and were reconstructed with the robust algorithm OPED.
The independent sinograms were matched to a single one, and from this a diagnostic image was reconstructed
successfully. This image has improved resolution, as well as less noise and artifacts compared to each single
independent reconstruction. The results obtained are highly promising, even though the current device acquires
only 197 views. Dose comparisons can be carried out in the future with a more precise prototype, comparable
to current clinical devices with respect to imaging performance.
A non-standard scanning device with dose-reduction potential was proposed at the SPIE Medical Imaging conference
2006. The new device obtains the Radon data after the X-ray beam is collimated through a special
mask. This mask is combined with a new geometry that permits an efficient data collection, thus the device
has the potential of reducing the dose by a factor of two. In this work, we report a prototype of the new device
and experimental data acquisition using only the mask of the new scanning geometry. In order to obtain the
optimal parameters for the scanning device, several factors have been considered, including detector elements
and shielding shape, fan beam angle, speed of the source rotation and materials employed. The calibration of the
detector elements needs especial attention, due to the dependence of the detector response on the energy of the
X-rays. A simplfied version of the device was designed and mounted. Phantom data were acquired using this
prototype and were used to test the performance of the new design. The results obtained are highly promising,
even though the prototype developed does not make use yet of all the potential features proposed in the theory.
The tomographic method based on the orthogonal polynomial expansion on disc (OPED) was presented at SPIE
conference of Medical Imaging 2006. We could show already some advantages compared to FBP as it is commonly used
in today's CT systems. However, OPED did show for some specific cases some noise in the reconstructed images and
even artefacts, mainly an aliasing. We have found that the OPED algorithm can be essentially improved by integrating
the polynomial over the whole area belonging to the pixel instead of assigning to the whole pixel the polynomial value
calculated just for one point of this pixel (typically bottom left). This advantageous implementation is effective in view
of reduction of the aliasing artefacts and noise without affecting the resolution. This can be fulfilled effectively for
OPED due to its simple structure.
The amount of x-ray radiation currently applied in CT practice is not utilized optimally. A portion of radiation traversing the patient is either not detected at all or is used ineffectively. The reason lies partly in the reconstruction algorithms and partly in the geometry of the CT scanners designed specifically for these algorithms. In fact, the reconstruction methods widely used in CT are intended to invert the data that correspond to ideal straight lines. However, the collection of such data is often not accurate due to likely movement of the source/detector system of the scanner in the time interval during which all the detectors are read. In this paper, a new design of the scanner geometry is proposed that is immune to the movement of the CT system and will collect all radiation traversing the patient. The proposed scanning design has a potential to reduce the patient dose by a factor of two. Furthermore, it can be used with the existing reconstruction algorithm and it is particularly suitable for OPED, a new robust reconstruction algorithm.
A new reconstruction algorithm for Radon data is introduced. We call the new algorithm OPED as it is based on Orthogonal Polynomial Expansion on the Disk. OPED is fundamentally different from the filtered back projection (FBP) method. It allows one to use fan beam geometry directly without any additional procedures such as interpolation or rebinning. It reconstructs high degree polynomials exactly and works for smooth functions without the assumption that functions are band- limited. Our initial tests indicate that the algorithm is stable, provides high resolution images, and has a small global error. Working with the geometry specified by the algorithm and a new mask, OPED could also lead to a reconstruction method that works with reduced x-ray dose (see the paper by Tischenko et al in these proceedings).