Rolf Clackdoyle

A two-step Hilbert transform method for 2D image reconstruction

The paper describes a new accurate two-dimensional (2D) image reconstruction method consisting of two steps. In the first step, the backprojected image is formed after taking the derivative of the parallel projection data. In the second step, a Hilbert filtering is applied along certain lines in the differentiated backprojection (DBP) image. Formulae for performing the DBP step in fanbeam geometry are also presented. The advantage of this two-step Hilbert transform approach is that in certain situations, regions of interest (ROIs) can be reconstructed from truncated projection data. Simulation results are presented that illustrate very similar reconstructed image quality using the new method compared to standard filtered backprojection, and that show the capability to correctly handle truncated projections. In particular, a simulation is presented of a wide patient whose projections are truncated laterally yet for which highly accurate ROI reconstruction is obtained.

Cone-beam reconstruction using the backprojection of locally filtered projections

This paper describes a flexible new methodology for accurate cone beam reconstruction with source positions on a curve (or set of curves). The inversion formulas employed by this methodology are based on first backprojecting a simple derivative in the projection space and then applying a Hilbert transform inversion in the image space. The local nature of the projection space filtering distinguishes this approach from conventional filtered-backprojection methods. This characteristic together with a degree of flexibility in choosing the direction of the Hilbert transform used for inversion offers two important features for the design of data acquisition geometries and reconstruction algorithms. First, the size of the detector necessary to acquire sufficient data for accurate reconstruction of a given region is often smaller than that required by previously documented approaches. In other words, more data truncation is allowed. Second, redundant data can be incorporated for the purpose of noise reduction. The validity of the inversion formulas along with the application of these two properties are illustrated with reconstructions from computer simulated data. In particular, in the helical cone beam geometry, it is shown that 1) intermittent transaxial truncation has no effect on the reconstruction in a central region which means that wider patients can be accommodated on existing scanners, and more importantly that radiation exposure can be reduced for region of interest imaging and 2) at maximum pitch the data outside the Tam-Danielsson window can be used to reduce image noise and thereby improve dose utilization. Furthermore, the degree of axial truncation tolerated by our approach for saddle trajectories is shown to be larger than that of previous methods.

Truncated Hilbert transform and image reconstruction from limited tomographic data

A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection of a derivative of the data and the Hilbert transform of the image along certain segments of lines covering the ROI. This paper generalizes this sufficiency condition by showing that unique and stable reconstruction can be achieved from an even more restricted family of data sets, or, conversely, that even larger ROIs can be reconstructed from a given data set. The condition is derived by analysing the inversion of the truncated Hilbert transform, here defined as the problem of recovering a function of one real variable from the knowledge of its Hilbert transform along a segment which only partially covers the support of the function but has at least one end point outside that support. A proof of uniqueness and a stability estimate are given for this problem. Numerical simulations of a 2D thorax phantom are presented to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction in the presence of noise.

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.

Attenuation correction in PET using consistency information

In this study the authors use the consistency conditions of the Radon transform to aid attenuation correction in PET. The conditions are used both for estimating the parameters of a uniform elliptical attenuation distribution (without any transmission measurements) and for correcting for patient motion between the transmission and emission acquisitions. The results show that, for a uniform elliptical attenuation distribution, the reconstructed count densities obtained using attenuation correction based on the consistency conditions are within 1% of the true values. The method is shown to be fairly tolerant to the effects of photon counting statistics and to small non-uniformities in the attenuation distribution (such as skull attenuation). The results also show that the consistency conditions may be useful in correcting for patient motion. The method is shown to effectively compensate for shifts in two dimensions using both simulated and experimental data.

New super-short-scan algorithms for fan-beam and cone-beam reconstruction

This paper deals with image reconstruction from fan-beam and cone-beam projections when only a region of interest (ROI) is to be reconstructed. We propose a new fan-beam FBP algorithm which can achieve exact ROI reconstruction if and only if all lines passing through the ROI intersect the source trajectory. This requirement on the scanning range is significantly weaker compared to that required by the conventional fan-beam FBP algorithm. Furthermore, this algorithm is extended to cone-beam tomography when the source trajectory is a short-arc on the single circle.

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.

A large class of inversion formulae for the 2D Radon transform of functions of compact support

We demonstrate that there are many direct inversion formulae for the two-dimensional (2D) Radon transform that are not equivalent to Radons original inversion formula. Our explicit formulae have a form similar to classical filtered backprojection but do require some information about the support of the object function. We present some numerical results to illustrate significant differences in reconstructions as alternate formulae are applied to noisy data.

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.

Blind estimation of compartmental model parameters

Computation of physiologically relevant kinetic parameters from dynamic PET or SPECT imaging requires knowledge of the blood input function. This work is concerned with developing methods to accurately estimate these kinetic parameters blindly; that is, without use of a directly measured blood input function. Instead, only measurements of the output functions--the tissue time-activity curves--are used. The blind estimation method employed here minimizes a set of cross-relation equations, from which the blood term has been factored out, to determine compartmental model parameters. The method was tested with simulated data appropriate for dynamic SPECT cardiac perfusion imaging with 99mTc-teboroxime and for dynamic PET cerebral blood flow imaging with 15O water. The simulations did not model the tomographic process. Noise levels typical of the respective modalities were employed. From three to eight different regions were simulated, each with different time-activity curves. The time-activity curve (24 or 70 time points) for each region was simulated with a compartment model. The simulation used a biexponential blood input function and washin rates between 0.2 and 1.3 min(-1) and washout rates between 0.2 and 1.0 min(-1). The system of equations was solved numerically and included constraints to bound the range of possible solutions. From the cardiac simulations, washin was determined to within a scale factor of the true washin parameters with less than 6% bias and 12% variability. 99mTc-teboroxime washout results had less than 5% bias, but variability ranged from 14% to 43%. The cerebral blood flow washin parameters were determined with less than 5% bias and 4% variability. The washout parameters were determined with less than 4% bias, but had 15-30% variability. Since washin is often the parameter of most use in clinical studies, the blind estimation approach may eliminate the current necessity of measuring the input function when performing certain dynamic studies.