Rolf Clack

A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection

An exact inversion formula written in the form of shift-variant filtered-backprojection (FBP) is given for reconstruction from cone-beam data taken from any orbit satisfying H.K. Tuys (1983) sufficiency conditions. The method is based on a result of P. Grangeat (1987), involving the derivative of the three-dimensional (3D) Radon transform, but unlike Grangeats algorithm, no 3D rebinning step is required. Data redundancy, which occurs when several cone-beam projections supply the same values in the Radon domain, is handled using an elegant weighting function and without discarding data. The algorithm is expressed in a convenient cone-beam detector reference frame, and a specific example for the case of a dual orthogonal circular orbit is presented. When the method is applied to a single circular orbit (even though Tuys condition is not satisfied), it is shown to be equivalent to the well-known algorithm of L.A. Feldkamp et al. (1984).

Toward accurate attenuation correction in SPECT without transmission measurements

The current trend in attenuation correction for single photon emission computed tomography (SPECT) is to measure and reconstruct the attenuation coefficient map using a transmission scan, performed either sequentially or simultaneously with the emission scan. This approach requires dedicated hardware and increases the cost (and in some cases the scanning time) required to produce a clinical SPECT image. Furthermore, if short focal-length fan-beam collimators are used for transmission imaging, the projection data may be truncated, leading to errors in the attenuation coefficient map. Our goal is to obtain information about the attenuation distribution from only the measured emission data by exploiting the fact that only certain attenuation distributions are consistent with a given emission dataset. Ultimately this consistency information will either be used directly to compensate for attenuation or combined with the incomplete information from fan-beam transmission measurements to produce a more accurate attenuation coefficient map. In this manuscript the consistency conditions (which relate the measured SPECT data to the sinogram of the attenuation distribution) are used to find the uniform elliptical attenuation object which is most consistent with the measured emission data. This object is then used for attenuation correction during the reconstruction of the emission data. The method is tested using both simulated and experimentally acquired data from uniformly and nonuniformly attenuating objects. The results show that, for uniform elliptical attenuators, the consistency conditions of the SPECT data can be used to produce an accurate estimate of the attenuation map without performing any transmission measurements. The results also show that, in certain circumstances, the consistency conditions can prove useful for attenuation compensation with nonuniform attenuators.

Three-dimensional image reconstruction from complete projections

Three-dimensional medical image reconstruction for both transmission and emission tomography has traditionally decomposed the problem into a set of two-dimensional reconstructions on parallel transverse sections. There is, however, increasing interest in reconstructing projection data directly in three dimensions. For emission tomography in particular, such a reconstruction procedure would clearly make more efficient use of the available photon flux. In the past few years, a number of authors have studied the problems associated with full three-dimensional reconstruction, especially in the case of positron tomography where three-dimensional reconstruction is likely to offer the greatest benefits. While most approaches follow that of filtered backprojection, the relationship between the various filters that have been proposed is far from evident. This paper clarifies this relationship by analysing and generalising the different classes of published filters and establishes the properties and characteristics of a general solution to the three-dimensional reconstruction problem. Some guidelines are suggested for the choice of an appropriate filter in a given situation.

Implementation of Tuy's cone-beam inversion formula

Tuys cone-beam inversion formula was modified to develop a cone-beam reconstruction algorithm. The algorithm was implemented for a cone-beam vertex orbit consisting of a circle and two orthogonal lines. This orbit geometry satisfies the cone-beam data sufficiency condition and is easy to implement on commercial single photon emission computed tomography (PECT) systems. The algorithm, which consists of two derivative steps, one rebinning step, and one three-dimensional backprojection step, was verified by computer simulations and by reconstructing physical phantom data collected on a clinical SPECT system. The proposed algorithm gives equivalent results and is as efficient as other analytical cone-beam reconstruction algorithms.

Favor: a fast reconstruction algorithm for volume imaging in PET

A novel 3-D reconstruction algorithm for volume imaging in positron emission tomography (PET) is presented. This algorithm obviates the need to forward-project the data which have not been measured due to the finite length of the scanner. This results in a significant improvement in the reconstruction speed with respect to the algorithm of N.J. Pelc (1979) and P.E. Kinahan and J.G. Rogers (1990). The partially measured oblique projections are filtered with a 1-D ramp filter, whereas the untruncated direct projections are filtered with a 2-D filter, the Favor filter, calculated to ensure an exact reconstruction. A simpler, approximate, version of this algorithm is also discussed in which all projections are filtered with a 1-D ramp function. These two algorithms have been implemented, and data from a volume PET scanner are presented.<<ETX>>

Image reconstruction for a novel SPECT system with rotating slant-hole collimators

We are investigating the use of rotating slant-hole (RSH) collimators on conventional SPECT machines. The main interest in this configuration is the potential for increased photon sensitivity over standard parallel-hole collimator systems. Projection data from an RSH-SPECT system presents a novel but tractable image reconstruction problem. Special features of RSH-SPECT, such as the dimensions of the field-of-view and the rotation requirements of the detector head, are discussed, and image reconstruction is presented in some detail. Data from a Jaszczak cardiac phantom were acquired on an experimental RSH system and reconstructed. Limited-angle artifacts were clearly seen on images reconstructed from a single set of data (classical RSH). The reconstructions from multi-view datasets verify the full tomographic capability of the RSH-SPECT system.<<ETX>>

Cone Beam Single Photon Emission Computed Tomography Using Two Orbits

It is known that cone-beam projection measurements from a single planar orbit of the focal point do not satisfy Tuys sufficiency condition for exact reconstruction. It is also known that two such orbits, oriented orthogonally, do satisfy the condition. In this paper, we present a fast convolution-and-backprojection algorithm to perform reconstructions from two orbits of cone-beam data. The algorithm has been applied to simulated data, and phantom data taken on a clinical SPECT system.

Image reconstruction from truncated, two-dimensional, parallel projections

Full three-dimensional scanning allows a significant improvement in image quality in X-ray transmission computerized tomography (CT), in single-photon emission computerized tomography (SPECT) and in positron emission tomography (PET). Increased detection efficiency is obtained by increasing the solid angle seen by the detectors and by detecting photons which are no longer confined to a set of parallel slices as in the standard 2D scanning mode. Consequently, 3D image reconstruction cannot be factored as usual into a set of independent 2D reconstructions, and hence one has to invert the 3D X-ray transform with limited data. Assuming a basic knowledge of standard 2D tomography, this paper presents a review of analytic methods for the reconstruction of a 3D image from a set of 2D parallel projections along some limited set of directions. This inverse problem is overdetermined, i.e., the projection data are redundant, and the consequences of this property are analysed. Redundancy is used to generate classes of exact filters for 3D filtered-backprojection, thereby allowing considerable versatility in the design of inversion algorithms tailored to specific applications. The review also covers the inversion of the 3D X-ray transform when the 2D parallel projections are incompletely measured (truncated), a situation which arises for example in PET, In view of data redundancy, it is possible to build convolution kernels for filtered-backprojection which have a limited support and are not, therefore, affected by truncation. A similar analysis and utilization of data redundancy could be proposed in any application where, instead of trying to define the smallest data set from which the problem can be solved, one attempts to optimize the signal-to-noise ratio by measuring and by incorporating into the reconstruction as much data as practically feasible.

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.

Cone-beam reconstruction from general discrete vertex sets using Radon rebinning algorithms

Addresses image reconstruction in cone-beam tomography from an arbitrary discrete set of positions of the cone vertex. As a first step in the analysis of the problem, the authors define some measures of how close a discrete vertex set comes to satisfying Tuys condition (1983). Next, they propose 3 rebinning algorithms which use Grangeats formula (1991) and Marrs algorithm (1980), and are capable of accurate reconstructions. The first algorithm is designed to accurately process cone-beam data finely sampled along a vertex path satisfying Tuys condition. The second algorithm applies to pair-complete vertex sets. The third algorithm is suited to process any discrete vertex set. The efficacy of the algorithms is illustrated with reconstructions from computer-simulated data using several vertex sets, including a set of randomly placed vertices.