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.

 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.  

Most contemporary CT-sytems employ non-exact methods. This treatise reports on how these methods could be transformed from non-exact to exact reconstruction methods by means of iterative post-processing. Compared to traditional algebraic reconstruction (ART) we expect much faster convergence (in theory quadratic), due to a much improved first guess and the fact that each iteration includes the same non-exact analytical reconstruction step as the first guess.

In most, if not all fast marching methods published hitherto, the input,cost function and the output arrival time are sampled on exactly the same grid. But since the input data samples are differences of the output samples. we found it natural to separate the input and output grid half a sampling unit in all coordinates (two or three). We also employ 8-neighborhood (26-neighborhood in the 3D-case) in the basic updating step of the algorithm. Some simple numerical experiments verify that the modified method improves the accuracy considerably. However, we also feel the modified method leads itself more naturally to image processing applications like tracking and segmentation.

This paper presents a proposal (not entirely new) for combining analytical and algebraic reconstruction techniques. Such a combination bears the promise to improve the image quality of fast but non-exact, reconstruction of the filtered backprojection type. The difference between the present proposal and traditional ART is that we compute a full error image with FBP, applied to projection differences, to update the solution in each iteration step. The main road-block seems to be the same that has been an obstacle for many ART-algorithms in CT applications, namely that the forward projections are subjected to ailiasing, which tend to override the intended benefits of the updating loop. We present an analysis of this problem and indicate some possible solutions.