State-of-the-art reconstruction algorithms for medical helical cone-beam Computed Tomography (CT) are of type non-exact Filtered Backprojection (FBP). They are attractive because of their simplicity and low computational cost, but they produce sub-optimal images with respect to artifacts, resolution, and noise. This thesis deals with possibilities to improve the image quality by means of iterative techniques.

The first algorithm, Regularized Iterative Weighted Filtered Backprojection (RIWFBP), is an iterative algorithm employing the non-exact Weighted FilteredBackprojection (WFBP) algorithm [Stierstorfer et al., Phys. Med. Biol. 49, 2209-2218, 2004] in the update step. We have measured and compared artifact reduction as well as resolution and noise properties for RIWFBP and WFBP. The results show that artifacts originating in the non-exactness of the WFBP algorithm are suppressed within five iterations without notable degradation in terms of resolution versus noise. Our experiments also indicate that the number of required iterations can be reduced by employing a technique known as ordered subsets.

A small modification of RIWFBP leads to a new algorithm, the Weighted Least Squares Iterative Filtered Backprojection (WLS-IFBP). This algorithm has a slightly lower rate of convergence than RIWFBP, but in return it has the attractive property of converging to a solution of a certain least squares minimization problem. Hereby, theory and algorithms from optimization theory become applicable.

Besides linear regularization, we have examined edge-preserving non-linear regularization.In this case, resolution becomes contrast dependent, a fact that can be utilized for improving high contrast resolution without degrading the signal-to-noise ratio in low contrast regions. Resolution measurements at different contrast levels and anthropomorphic phantom studies confirm this property. Furthermore, an even morepronounced suppression of artifacts is observed.

Iterative reconstruction opens for more realistic modeling of the input data acquisition process than what is possible with FBP. We have examined the possibility to improve the forward projection model by (i) multiple ray models, and (ii) calculating strip integrals instead of line integrals. In both cases, for linearregularization, the experiments indicate a trade off: the resolution is improved atthe price of increased noise levels. With non-linear regularization on the other hand, the degraded signal-to-noise ratio in low contrast regions can be avoided.

Huge input data sizes make experiments on real medical CT data very demanding. To alleviate this problem, we have implemented the most time consuming parts of the algorithms on a Graphics Processing Unit (GPU). These implementations are described in some detail, and some specific problems regarding parallelism and memory access are discussed.

We report on experiments on combining iterative weighted filtered backprojection with nonlinear regularization. The resulting methods allow for contrast dependent resolution. In combination with improved geometrical modeling of the aquisition process, they have the potential to improve high contrast resolution while preserving low contrast image properties. The two investigated methods have been evaluated by visual inspection and mean square error measurements. In the reconstruction results, strong suppression of wind-mill artifacts has been observed: within 7 iterations, there is a reduction in amplitude with more than a factor of 6. Furthermore, there is a clear reduction of pixel noise without any increase of errors at high contrast structures. The proposed methods can be used to efficiently improve image quality for certain applications. Future investigations will include analysis of spatial resolution properties with respect to contrast, and choice of regularization parameters depending on noise levels and spatial resolution requirements.  

Contemporary reconstruction methods employed for clinical helical cone-beam computed tomography (CT) are analytical (noniterative) but mathematically nonexact, i.e., the reconstructed image contains so called cone-beam artifacts, especially for higher cone angles. Besides cone artifacts, these methods also suffer from windmill artifacts: alternating dark and bright regions creating spiral-like patterns occurring in the vicinity of high z-direction derivatives. In this article, the authors examine the possibility to suppress cone and windmill artifacts by means of iterative application of nonexact three-dimensional filtered backprojection, where the analytical part of the reconstruction brings about accelerated convergence. Specifically, they base their investigations on the weighted filtered backprojection method [Stierstorfer et al., Phys. Med. Biol. 49, 2209-2218 (2004)]. Enhancement of high frequencies and amplification of noise is a common but unwanted side effect in many acceleration attempts. They have employed linear regularization to avoid these effects and to improve the convergence properties of the iterative scheme. Artifacts and noise, as well as spatial resolution in terms of modulation transfer functions and slice sensitivity profiles have been measured. The results show that for cone angles up to ±2.78°, cone artifacts are suppressed and windmill artifacts are alleviated within three iterations. Furthermore, regularization parameters controlling spatial resolution can be tuned so that image quality in terms of spatial resolution and noise is preserved. Simulations with higher number of iterations and long objects (exceeding the measured region) verify that the size of the reconstructible region is not reduced, and that the regularization greatly improves the convergence properties of the iterative scheme. Taking these results into account, and the possibilities to extend the proposed method with more accurate modeling of the acquisition process, the authors believe that iterative improvement with non-exact methods is a promising technique for medical CT applications.

A new projection operator is presented and evaluated. This operator has been designed to suppress aliasing artifacts due to (i) false high frequencies contained in the footprint function, and (ii) high frequencies caused by a divergent beam geometry. It is easy to implement and allows for efficient computer implementations. Instead of sampling the footprint as done in most projection operators, the footprint is integrated. This integration suppresses false high frequencies, frequency components that cause aliasing and approximately takes into account the finite size of focus and detector. Two-dimensional parallel beam experiments are presented. These experiments confirm that artifacts due to false high frequencies can be suppressed by the proposed technique. In order to investigate the advantages for divergent beam geometries, current experiments must be complemented with cone-beam experiments.

Although not so often expressed as a modeling problem neither projection nor back-projection can be designed without certain insights in the physics of CT. However, most of this insight is left aside, since it is generally believed that only the most simplified models can be included in the innermost timeconsuming loop in projection and back-projection. We propose that any linear projection procedure should model three functions: The irradiation function, the footprint/basis function, and the gantry rotation function. We demonstrate how a moderately advanced modeling of these three functions can be brought together in an interpolation procedure and yield a surprisingly efficient inner loop interpolation. To this end we i) carefully select a locus of interpolation path through image and projection data spaces and ii) execute multiple convolution as integration by parts implemented by table-look-up.

Contemporary algorithms employed for reconstruction of 3D volumes from helical cone beam projections are so called non-exact algorithms. This means that the reconstructed volumes contain artifacts irrespective of the detector resolution and number of projection angles employed in the process. In this thesis, three iterative schemes for suppression of these so called cone artifacts are investigated.

The first scheme, iterative weighted filtered backprojection (IWFBP), is based on iterative application of a non-exact algorithm. For this method, artifact reduction, as well as spatial resolution and noise properties are measured. During the first five iterations, cone artifacts are clearly reduced. As a side effect, spatial resolution and noise are increased. To avoid this side effect and improve the convergence properties, a regularization procedure is proposed and evaluated.

In order to reduce the cost of the IWBP scheme, a second scheme is created by combining IWFBP with the so called ordered subsets technique, which we call OSIWFBP. This method divides the projection data set into subsets, and operates sequentially on each of these in a certain order, hence the name “ordered subsets”. We investigate two different ordering schemes and number of subsets, as well as the possibility to accelerate cone artifact suppression. The main conclusion is that the ordered subsets technique indeed reduces the number of iterations needed, but that it suffers from the drawback of noise amplification.

The third scheme starts by dividing input data into high- and low-frequency data, followed by non-iterative reconstruction of the high-frequency part and IWFBP reconstruction of the low-frequency part. This could open for acceleration by reduction of data in the iterative part. The results show that a suppression of artifacts similar to that of the IWFBP method can be obtained, even if a significant part of high-frequency data is non-iteratively reconstructed.

 In medical helical cone-beam CT, it is common that the region-of-interest (ROI) is contained inside the helix cylinder, while the complete object is long and extends outside the top and the bottom of the cylinder. This is the Long Object Problem. Analytical reconstruction methods for helical cone-beam CT have been designed to handle this problem. It has been shown that a moderate amount of over-scanning is sufficient for reconstruction of a certain ROI. The over-scanning projection rays travel both through the ROI as well as outside the ROI. This is unfortunate for iterative methods since it seems impossible to compute accurate values for the projection rays which travel partly inside and partly outside the ROI. Therefore, it seems that the useful ROI will diminish for every iteration step. We propose the following solution to the problem. Firstly, we reconstruct volume regions also outside the ROI. These volume regions will certainly be incompletely reconstructed, but our experimental results show that they serve well for projection generation. This is rather counter-intuitive and contradictory to our initial assumptions. Secondly, we use careful extrapolation and masking of projection data. This is not a general necessity, but needed for the chosen iterative algorithm, which includes rebinning and iterative filtered backprojection. Our idea here was to use an approximate reconstruction method which gives cone-beam artifacts and then improve the reconstructed result by iterative filtered backprojection. The experimental results seem very encouraging. The cone-beam artifacts can indeed be removed. Even voxels close to the boundary of the ROI are as well enhanced by the iterative loop as those in the middle of the ROI.

This largely tutorial treatise presents a Fourier based model for 2D-projection, the latter being a most important ingredient in any iterative reconstruction method. For sampled images the model requires an assumed basis function, which implicitly defines the necessary window and interpolation functions. We unravel the basis and window functions for some projection techniques described as procedures. Circular symmetric basis functions make it simple to find interpolation coefficients but require well tuned interpolation functions to avoid aliasing. We find it unnecessary to distinguish between voxel and ray driven projection. These two techniques concern only the innermost loop and both can be applied to any interpolation function, and to projection and back-projection alike. 

Contemporary analytical reconstruction methods for helical cone-beam CT have to be designed to handle the Long Object Problem. Normally, a moderate amount of over-scanning is sufficient for reconstruction of a certain Region-of-interest (ROI). Unfortunately, for iterative methods, it seems that the useful ROI will diminish for every iteration step. The remedies proposed here are twofold. Firstly, we use careful extrapolation and masking of projection data. Secondly, we generate and utilize projection data from incompletely reconstructed volume parts, which is rather counter-intuitive and contradictory to our initial assumptions. The results seem very encouraging. Even voxels close to the boundary in the original ROI are as well enhanced by the iterative loop as the middle part.

Contemporary reconstruction for helical cone-beam CT is mostly based on non-exact algorithms, which produce more or less unacceptable artifacts for cone angles above a certain limit. We report on attempts to extend the applicability of these algorithms to higher cone angles by suppressing artifacts by means of iterative post-processing. The iterative loop includes a ramp-filtering step before back-projection, which promotes fast convergence. The scheme has been applied to the original PI-method as well as to Siemens' AMPR and WFBP methods. Using ordered subsets in the iterative loop for WFBP, we achieved almost spotless images in one single iteration for cone angles \pm 9 degrees.