MO-D-BRA-01: Limits of Dose Reduction in CT: Where are They and How Will We Know When We Get There?

Radiation Dose continues to be a concern with respect to all diagnostic imaging using ionizing radiation, but especially so with CT imaging. We have always known how to reduce radiation dose in CT - for example, simply turning down the system output (e.g. reduce mAs). What we have not been able to do is to simultaneously reduce dose and maintain "diagnostic image quality". Many recent technical developments have appeared, and will continue to appear, that will allow users to reduce radiation dose in CT while "maintaining image quality". However, this last term is ill-defined and current metrics of image quality are not very applicable to actual clinical practice. The purpose of this symposium is to: (a) describe several current and possible future radiation dose reduction methods and the magnitude of their potential for dose reduction, (b) some description of what "diagnostic image quality" means, the effects that dose reductions methods have on this property, description of some metrics that may help us assess this property quantitatively and this information can be used to guide how low radiation doses can be reduced. LEARNING OBJECTIVES: 1. Understand both conventional and emerging radiation dose reduction methods in CT. 2. Understand the implications on diagnostic image quality for each radiation dose reduction method. 3. Understand some of the issues in evaluating how much radiation dose can be reduced and still accomplish a diagnostic imaging task.

Direct determination of geometric alignment parameters for cone-beam scanners.

This paper describes a comprehensive method for determining the geometric alignment parameters for cone-beam scanners (often called calibrating the scanners or performing geometric calibration). The method is applicable to x-ray scanners using area detectors, or to SPECT systems using pinholes or cone-beam converging collimators. Images of an alignment test object (calibration phantom) fixed in the field of view of the scanner are processed to determine the nine geometric parameters for each view. The parameter values are found directly using formulae applied to the projected positions of the test object marker points onto the detector. Each view is treated independently, and no restrictions are made on the position of the cone vertex, or on the position or orientation of the detector. The proposed test object consists of 14 small point-like objects arranged with four points on each of three orthogonal lines, and two points on a diagonal line. This test object is shown to provide unique solutions for all possible scanner geometries, even when partial measurement information is lost by points superimposing in the calibration scan. For the many situations where the cone vertex stays reasonably close to a central plane (for circular, planar, or near-planar trajectories), a simpler version of the test object is appropriate. The simpler object consists of six points, two per orthogonal line, but with some restrictions on the positioning of the test object. This paper focuses on the principles and mathematical justifications for the method. Numerical simulations of the calibration process and reconstructions using estimated parameters are also presented to validate the method and to provide evidence of the robustness of the technique.

SOLVING THE INTERIOR PROBLEM OF COMPUTED TOMOGRAPHY USING A PRIORI KNOWLEDGE.

The case of incomplete tomographic data for a compactly supported attenuation function is studied. When the attenuation function is a priori known in a subregion, we show that a reduced set of measurements is enough to uniquely determine the attenuation function over all the space. Furthermore, we found stability estimates showing that reconstruction can be stable near the region where the attenuation is known. These estimates also suggest that reconstruction stability collapses quickly when approaching the set of points that are viewed under less than 180 degrees. This paper may be seen as a continuation of the work "Truncated Hilbert transform and Image reconstruction from limited tomographic data" that was published in Inverse Problems in 2006. This continuation tackles new cases of incomplete data that could be of interest in applications of computed tomography.

Redundant data and exact helical cone-beam reconstruction.

This paper is about helical cone-beam reconstruction and the use of redundant data in the framework of two reconstruction methods. The first method is the approximate wedge reconstruction formula introduced by Tuy at the 3D meeting in 1999. The second method is a (exact) hybrid implementation of the exact filtered backprojection formula of Katsevich (2004 Adv. Appl. Math. at press) that combines filtering in the native cone-beam geometry with backprojection in the wedge geometry. The similarity of the two methods is explored and their image quality performance is compared for geometries with up to 112 detector rows. Furthermore, the concept of aperture weighting is introduced to allow the handling of variable amounts of redundant data. A reduction of motion artefacts using redundant data is demonstrated for geometries with 16, 32 and 112 detector rows using a pitch factor of 1.25. For scans with up to 100 rows, utilizing 50% of the redundant data provided excellent results without any introduction of cone-beam artefacts. For larger cone angles, an alternative approach that utilizes all available redundant data, even at reduced pitch factors, is suggested.

Exact inversion of the exponential x-ray transform for rotating slant-hole (RSH) SPECT.

The RSH SPECT scanner provides parallel-beam attenuated projections for a fully 3D acquisition geometry. The geometry can be represented by circles on the unit sphere of projection directions, one circle for each position of the detector head. Unlike most other fully 3D geometries this one is particularly challenging because there are no 2D subsets in the data. When no attenuation is present, it is well known that an unmeasured projection can be synthesized if it lies inside one of the measured circles. The main result of this work is that under some assumptions on the attenuation distribution, attenuated projections within a circle can be synthesized from available attenuated projections. One consequence is that RSH SPECT projections can be rebinned into a conventional SPECT geometry for which analytic attenuation correction techniques are available.

Rebinning-based algorithms for helical cone-beam CT.

Several image reconstruction algorithms based on rebinning have been proposed recently for helical cone-beam CT. These algorithms separate the 3D reconstruction into a set of independent 2D reconstructions for a set of surfaces: planar or non-planar surfaces are defined and then reconstructed using 2D filtered backprojection from a 2D fan-beam or parallel-beam set of data estimated from the cone-beam (CB) measurements. The first part of this paper presents a unified derivation of rebinning algorithms for planar and non-planar surfaces. An integral equation is derived for the surface allowing the best rebinning and an iterative algorithm converging to the solution of that equation is given. The second part presents an efficient method to correct the residual reconstruction artefacts observed with rebinning algorithms when the cone-angle is too large for the required accuracy. This correction algorithm involves a CB backprojection and the reconstruction time is slightly longer than for the zero-boundary (ZB) method.

Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography.

Exact reconstruction from axially truncated cone-beam projections acquired with a helical vertex path is a challenging problem for which solutions are currently under investigation by some researchers. This paper deals with a difficult class of this problem called the long-object problem. Its purpose is to reconstruct a central region of interest (ROI) of a long object when the helical path extends only a little bit above and below the ROI. By extending the authors' recent approach based on the triangular decomposition of the Grangeat formula, we derive quasi-exact reconstruction algorithms whose overall structure is of filtered backprojection (FBP) style. Unlike the previous similar approaches to the long-object problem, the proposed FBP algorithms do not require additional two circular scans at the ends of the helical path. Furthermore, the algorithms require a significantly smaller detector area and achieve improved image quality even for a large pitch compared with the approximate Feldkamp algorithms. One drawback of the proposed algorithms is the computational time, which is much longer than for the Feldkamp algorithms. We show some simulation results to demonstrate the performances of the proposed algorithms.

Analytic method based on identification of ellipse parameters for scanner calibration in cone-beam tomography.

This paper is about calibration of cone-beam (CB) scanners for both x-ray computed tomography and single-photon emission computed tomography. Scanner calibration refers here to the estimation of a set of parameters which fully describe the geometry of data acquisition. Such parameters are needed for the tomographic reconstruction step. The discussion is limited to the usual case where the cone vertex and planar detector move along a circular path relative to the object. It is also assumed that the detector does not have spatial distortions. We propose a new method which requires a small set of measurements of a simple calibration object consisting of two spherical objects, that can be considered as 'point' objects. This object traces two ellipses on the detector and from the parametric description of these ellipses, the calibration geometry can be determined analytically using explicit formulae. The method is robust and easy to implement. However, it is not fully general as it is assumed that the detector is parallel to the rotation axis of the scanner. Implementation details are given for an experimental x-ray CB scanner.

A solution to the long-object problem in helical cone-beam tomography.

This paper presents a new algorithm for the long-object problem in helical cone-beam (CB) computerized tomography (CT). This problem consists in reconstructing a region-of-interest (ROI) bounded by two given transaxial slices, using axially truncated CB projections corresponding to a helix segment long enough to cover the ROI, but not long enough to cover the whole axial extent of the object. The new algorithm is based on a previously published method, referred to as CB-FBP (Kudo et al 1998 Phys. Med. Biol. 43 2885-909), which is suitable for quasi-exact reconstruction when the helix extends well beyond the support of the object. We first show that the CB-FBP algorithm simplifies dramatically, and furthermore constitutes a solution to the long-object problem, when the object under study has line integrals which vanish along all PI-lines. (A PI line is a line which connects two points of the helix separated by less than one pitch.) Exploiting a geometric property of the helix, we then show how the image can be expressed as the sum of two images, where the first image can be reconstructed from the measured CB projections by a simple backprojection procedure, and the second image has zero PI-line integrals and hence can be reconstructed using the simplified CB-FBP algorithm. The resulting method is a quasi-exact solution to the long-object problem, called the ZB method. We present its implementation and illustrate its performance using simulated CB data of the 3D Shepp phantom and of a more challenging head-like phantom.

Attenuation correction in SPECT using consistency conditions for the exponential ray transform.

Using data consistency conditions for the exponential ray transform, a method is derived to correct SPECT data for attenuation effects. No transmission measurements are required, and no operator-defined contours are needed. Furthermore, any 3D parallel-ray geometry can be considered for SPECT data acquisition, even unconventional geometries which do not lead to a set of 2D parallel-beam sinograms. The method is presented for both the 2D parallel-beam geometry and a particular 3D case, called the rotating slant hole geometry. Full details of the algorithms are given. Implementation has been carried out and results are presented in 2D and in 3D using simulated data.

Single-slice rebinning method for helical cone-beam CT.

In this paper, we present reconstruction results from helical cone-beam CT data, obtained using a simple and fast algorithm, which we call the CB-SSRB algorithm. This algorithm combines the single-slice rebinning method of PET imaging with the weighting schemes of spiral CT algorithms. The reconstruction is approximate but can be performed using 2D multislice fan-beam filtered backprojection. The quality of the results is surprisingly good, and far exceeds what one might expect, even when the pitch of the helix is large. In particular, with this algorithm comparable quality is obtained using helical cone-beam data with a normalized pitch of 10 to that obtained using standard spiral CT reconstruction with a normalized pitch of 2.

Cone-beam filtered-backprojection algorithm for truncated helical data.

This paper investigates 3D image reconstruction from truncated cone-beam (CB) projections acquired with a helical vertex path. First, we show that a rigorous derivation of Grangeat's formula for truncated projections leads to a small additional term compared with previously published similar formulations. This correction term is called the boundary term. Next, this result is used to develop a CB filtered-backprojection (FBP) algorithm for truncated helical projections. This new algorithm only requires the CB projections to be measured within the region that is bounded, in the detector, by the projections of the upper and lower turns of the helix. Finally, simulations with mathematical phantoms demonstrate that: (i) the boundary term is necessary to obtain high-quality imaging of low-contrast structures and (ii) good image quality is obtained even with large values of the pitch of the helix.

The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography.

We investigate the way data are used in the algorithm proposed by Kudo and Saito for the exact reconstruction of long objects from axially truncated cone-beam projections. Specifically, we show that the algorithm wastes a large part of the data. To overcome the problem, we propose to use a vertex path consisting of two crossing ellipses, for which we devised a new reconstruction algorithm, called the cross algorithm, which does not waste data and is still suitable to exactly handle axial truncation. Results of reconstruction are presented on simulated data and real data from an experimental scanner.

Direct reconstruction of cone-beam data acquired with a vertex path containing a circle.

Cone-beam data acquired with a vertex path satisfying the data sufficiency condition of Tuy can be reconstructed using exact filtered backprojection algorithms. These algorithms are based on the application to each cone-beam projection of a two-dimensional (2-D) filter that is nonstationary, and therefore more complex than the one-dimensional (1-D) ramp filter used in the approximate algorithm of Feldkamp, Davis, and Kress (1984) (FDK). We determine in this paper the general conditions under which the 2-D nonstationary filter reduces to a 2-D stationary filter, and also give the explicit expression of the corresponding convolution kernel. Using this result and the redundancy of the cone-beam data, a composite algorithm is derived for the class of vertex paths that consist of one circle and some complementary subpath designed to guarantee data sufficiency. In this algorithm the projections corresponding to vertex points along the circle are filtered using a 2-D stationary filter, whereas the other projections are handled with a 2-D nonstationary filter. The composite algorithm generalizes the method proposed by Kudo and Saito (1990), in which the circle data are processed with a 1-D ramp filter as in the FDK algorithm. The advantage of the 2-D filter introduced in this paper is to guarantee that the filtered cone-beam projections do not contain singularities in smooth regions of the object. Tests of the composite algorithm on simulated data are presented.