When do the iterative reconstruction methods become worth the effort

By accurately modeling the physics of photon transport into the projection and backprojection operations, iterative SPECT reconstruction methods can reduce the degrading effects of scatter, attenuation and the non-stationary spatial resolution of the camera. Unfortunately, iterative reconstruction methods have required very long computation times, predominately due to the complexity involved in modeling these degrading effects into the projection and backprojection operations. In this study, we describe an approach which allows SPECT iterative reconstruction algorithms to be implemented with a reduction in the number of computations needed. The idea is to pre-process the measured projection data to compensate for scatter and attenuation, as well as to transform the projection data to those which would have been obtained with a stationary system resolution. Results of simulation studies indicate that preprocessing the measured projection data reduces the number of computations needed to perform the projection and backprojection operations, and yields reconstructions which differ minimally from those obtained using the slower standard iterative approach of modeling both photon attenuation and nonstationary blurring in the projection and backprojection steps.

Iterative reconstruction methods in X-ray CT

Application of a dual-resolution voxelization scheme to compressed-sensing (CS)-based iterative reconstruction in digital tomosynthesis (DTS)

An experimental comparison of four methods of wavefront reconstruction is presented. We considered two iterative and two holographic methods with differences in mathematical models and reconstruction algorithms. The first two of these methods do not use the reference wave in the recording scheme that reduces the need of setup stability. A set of spatial intensity measurements of a volume scattered field plays the main role in phase retrieval in such methods. The obtained data are sequentially used for iterative wavefront reconstruction. Iterative approach involves numerical wavefront propagation between various planes of the volume scattered fiels. Throughout this procedure the phase information of the wavefront is retained while the calculated amplitudes is replaced by the square root of the intensity distributions measured in corresponding planes. In the first compared phase retrieval method (FRIM), a two-dimensional Fresnel transform and iterative calculation in the object plane are used as a mathematical model. In the second method (SBMIR), the angular spectrum is used for numerical wavefront propagation, and iterative calculation is made only between closely spaced planes for data registration. Two methods of digital holography, which we compared, differ from each other in algorithm of a waverfont reconstruction. The first holographic method (CWR-DH) uses the conception of spatial phase steps for complex wave retrieval, and the second method (FT-DH) is a widespread Fourier transformation method. All methods provide satisfactory capacity for image reconstruction. The results of the comparison showed that FRIM produces better quality of reconstruction, but a diffraction artifacts takes place at the boundaries of the reconstructed image. Taking this into account we can conclude that the CWR-DH method is the best among considered.

BackgroundPositron Emission Tomography (PET), Computed Tomography (CT), PET/CT and Single Photon Emission Tomography (SPECT) are non-invasive imaging tools used for creating two dimensional (2D) cross section images of three dimensional (3D) objects. PET and SPECT have the potential of providing functional or biochemical information by measuring distribution and kinetics of radiolabelled molecules, whereas CT visualizes X-ray density in tissues in the body. PET/CT provides fused images representing both functional and anatomical information with better precision in localization than PET alone.Images generated by these types of techniques are generally noisy, thereby impairing the imaging potential and affecting the precision in quantitative values derived from the images. It is crucial to explore and understand the properties of noise in these imaging techniques. Here we used autocorrelation function (ACF) specifically to describe noise correlation and its non-isotropic behaviour in experimentally generated images of PET, CT, PET/CT and SPECT.MethodsExperiments were performed using phantoms with different shapes. In PET and PET/CT studies, data were acquired in 2D acquisition mode and reconstructed by both analytical filter back projection (FBP) and iterative, ordered subsets expectation maximisation (OSEM) methods. In the PET/CT studies, different magnitudes of X-ray dose in the transmission were employed by using different mA settings for the X-ray tube. In the CT studies, data were acquired using different slice thickness with and without applied dose reduction function and the images were reconstructed by FBP. SPECT studies were performed in 2D, reconstructed using FBP and OSEM, using post 3D filtering. ACF images were generated from the primary images, and profiles across the ACF images were used to describe the noise correlation in different directions. The variance of noise across the images was visualised as images and with profiles across these images.ResultsThe most important finding was that the pattern of noise correlation is rotation symmetric or isotropic, independent of object shape in PET and PET/CT images reconstructed using the iterative method. This is, however, not the case in FBP images when the shape of phantom is not circular. Also CT images reconstructed using FBP show the same non-isotropic pattern independent of slice thickness and utilization of care dose function. SPECT images show an isotropic correlation of the noise independent of object shape or applied reconstruction algorithm. Noise in PET/CT images was identical independent of the applied X-ray dose in the transmission part (CT), indicating that the noise from transmission with the applied doses does not propagate into the PET images showing that the noise from the emission part is dominant. The results indicate that in human studies it is possible to utilize a low dose in transmission part while maintaining the noise behaviour and the quality of the images.ConclusionThe combined effect of noise correlation for asymmetric objects and a varying noise variance across the image field significantly complicates the interpretation of the images when statistical methods are used, such as with statistical estimates of precision in average values, use of statistical parametric mapping methods and principal component analysis. Hence it is recommended that iterative reconstruction methods are used for such applications. However, it is possible to calculate the noise analytically in images reconstructed by FBP, while it is not possible to do the same calculation in images reconstructed by iterative methods. Therefore for performing statistical methods of analysis which depend on knowing the noise, FBP would be preferred.

CT perfusion (CTP) is obtained cerebrovascular circulation image for assessment of stroke patients; however, at the expense of increased radiation dose by dynamic scan. Iterative reconstruction (IR) method is possible to decrease image noise, it has the potential to reduce radiation dose. The purpose of this study is to assess the visual effect of IR method by using a digital perfusion phantom. The digital perfusion phantom was created by reconstructed filtered back projection (FBP) method and IR method CT images that had five exposure doses. Various exposure dose cerebral blood flow (CBF) images were derived from deconvolution algorithm. Contrast-to-noise ratio (CNR) and visual assessment were compared among the various exposure dose and each reconstructions. Result of low exposure dose with IR method showed, compared with FBP method, high CNR in severe ischemic area, and visual assessment was significantly improvement. IR method is useful for improving image quality of low-dose CTP.

Tetrahedron beam computed tomography (TBCT) performs volumetric imaging using a stack of fan beams generated by a multiple pixel X-ray source. While the TBCT system was designed to overcome the scatter and detector issues faced by cone beam computed tomography (CBCT), it still suffers the same large cone angle artifacts as CBCT due to the use of approximate reconstruction algorithms. It has been shown that iterative reconstruction algorithms are better able to model irregular system geometries and that algebraic iterative algorithms in particular have been able to reduce cone artifacts appearing at large cone angles. In this paper, the SART algorithm is modified for the use with the different TBCT geometries and is tested using both simulated projection data and data acquired using the TBCT benchtop system. The modified SART reconstruction algorithms were able to mitigate the effects of using data generated at large cone angles and were also able to reconstruct CT images without the introduction of artifacts due to either the longitudinal or transverse truncation in the data sets. Algebraic iterative reconstruction can be especially useful for dual-source dual-detector TBCT, wherein the cone angle is the largest in the center of the field of view.

The reconstruction of dynamic PET data is usually performed using filtered backprojection algorithms (FBP). This method is fast, robust, linear and therefore yields reliable quantitative results. However, the use of FBP for data with low statistics, such as dynamic PET data, generally results in poor visual image quality, exhibiting high noise, streak artifacts and low contrast. These signal to noise ratio and contrast in the reconstructed images may alter the quantification of physiological parameters, such as the regional Binding Potential (BP) obtained from kinetic modeling. Iterative reconstruction methods such as UW-OSEM, ANW-OSEM or OP-OSEM are often presented a viable alternatives to FBP reconstruction. In this study, we investigate the characteristics of the UW-OSEM and the ANW-OSEM iterative reconstruction methods in the context of ligand-receptor PET studies exhibiting low data counts. The assessment was conducted using replicates of simulated [ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">18</sup> F]MPPF data. The results show that the positivity constraint in the MLEM algorithm leads to overestimations of the activity in regions with low activity concentration, typically the cerebellum. This overestimation results in significant bias in the BP quantification with iterative reconstruction methods.

Small animal PET scanners require high spatial resolution (<1 mm) and good sensitivity. To obtain high resolution images, iterative reconstruction methods, like OSEM, applied to image reconstruction in three-dimensional (3D) positron emission tomography (PET), have superior performance over analytical reconstruction algorithms like FBP. However, the high computational cost of iterative methods remains a serious drawback to their development and clinical routine use. The increase in performance of current computers should make iterative image reconstruction fast enough to attain clinical viability. However, dealing with the large number of probability coefficients for the response of the system in high-resolution PET scanners becomes a difficult task that prevents the algorithms from reaching peak performance. Taking into account all possible axial, in-plane and other symmetries, we have reduced the storage needs what allows us to store the whole response of the system in dynamic memory of ordinary industry standard computers, so that the reconstruction algorithm can achieve near peak performance

Advanced numerical methods based on exponential propagation have been applied to magnetohydrodynamic (MHD) simulations. This recently developed numerical technique evolves the system of nonlinear equations using exponential propagation of the Jacobian matrix. The exponential of the matrix is approximated by projecting it onto the Krylov subspace using the Arnoldi algorithm. The primary advantage of the exponential propagation method is that it allows time steps exceeding the Courant-Friedrichs-Lewy (CFL) limit. Another important aspect is faster convergence of the iteration computing the Krylov subspace projection compared to solving an implicit formulation of the system with similar iterative methods. Since the time scales in the resistive MHD equations are widely separated, the exponential propagation methods are especially advantageous for computing the long term evolution of a low-beta plasma. We analyze several types of exponential propagation methods and highlight important issues in the development of such techniques. Our analysis also suggests new ways to construct schemes of this type. Implementation issues, including scalability properties of exponential propagation methods, and performance are also discussed. In the second part of this work we present numerical MHD models which are constructed using exponential propagation methods and which describe the evolution of the magnetic arcades in the solar corona. Since these numerical methods have not been used before for large evolutionary systems like resistive MHD, we first validate our approach by demonstrating application of the exponential schemes to two existing magnetohydrodynamic models. We simulate the reconnection process resulting from shearing the footpoints of two-dimensional magnetic arcades and compute the three-dimensional linear force-free states of plasma configurations. Analysis of these calculations leads us to new insights about the topology of the solutions. The final chapter of this work is dedicated to a new three-dimensional numerical model of the dynamics of coronal plasma configurations. The model is motivated by observations and laboratory experiments simulating the evolution of solar arcades. We analyze the results of numerical simulations and demonstrate that our numerical approach provides an accurate and stable way to compute the solution to the zero-resistive MHD system. Based on comparisons of the simulation results and the observational data, we offer an explanation for the observed structure of eruptive events in the corona called coronal mass ejections (CME). We argue that the diversity of the images of CMEs obtained by the observational instruments can be explained as two-dimensional projections of a unique three-dimensional plasma configuration and suggest an eruption mechanism.

The authors present the results of comparing 3D transform and iterative reconstruction methods with measured PET data from the HEAD PENN-PET scanner, which has a very large axial acceptance angle of 27 deg. The algorithms compared are the 3D-reprojection (3DRP) method, the iterative algebraic reconstruction technique (ART), and the maximum likelihood (ML-EM) algorithm. For the iterative methods, alternatives to the cubic voxel basis functions are also implemented. The comparisons are based on a series of figures of merit, including the point-spread function, the contrast recovery coefficient, and signal to noise ratios (SNR). The results show that the ART method implemented with the new basis functions requires only 1 cycle through the data to produce images with measured SNR values comparable to or better than the 3DRP method. The ML-EM method requires 6 to 8 iterations for the test phantom used and produced images with significantly lower SNR values. These are in agreement with previous simulation studies. >

The authors present the results of comparing 3-D transform and iterative reconstruction methods with measured PET data from the HEAD PENN-PET scanner, which has a very large axial acceptance angle of 27/spl deg/. The algorithms compared are the 3-D-reprojection (3DRP) method, the iterative algebraic reconstruction technique (ART), and the maximum likelihood (ML-EM) algorithm. For the iterative methods, alternatives to the cubic voxel basis functions are also implemented. The comparisons are based on a series of figures of merit, including the point-spread function, the contrast recovery coefficient, and signal-to-noise ratios (SNR). The results show that the ART method implemented with the new basis functions requires only one cycle through the data to produce images with measured SNR values comparable to or better than the 3DRP method. The ML-EM method was run for 10 iterations and produced images with lower SNR values for the test phantom used. With the exception of the ML-EM results, these are in agreement with the authors' previous simulation studies.

Dual-energy computed tomography is used increasingly in CT colonography (CTC). The combination of computer-aided detection (CAD) and dual-energy CTC has a high clinical value because it can automatically detect clinically significant colonic lesions in CTC images with higher accuracy than does single-energy CTC. While CAD has demonstrated its ability to detect small polyps, its performance is highly dependent on the quality of the input images. The presence of artifacts such as beam hardening and image noise in ultra-low-dose CTC may severely degrade detection performance for small polyps. A further limitation to the effectiveness of CAD are the weakly tagged fecal materials in the colon that may cause false-positive detections. In this work, we developed a dual-energy method for enhancing the appearance of weakly tagged fecal materials in CTC images. The proposed method consists of two stages: 1) the detection of weakly tagged fecal materials by use of sinogram-based image decomposition and 2) the enhancement of the detected tagged fecal materials in the images using an iterative reconstruction method. In the first stage, the ultra-low-dose dual-energy projection data obtained from a CT scanner are decomposed into two basis materials – soft tissue and fecal-tagged material (iodine). Virtual monochromatic projection data are calculated from the material decomposition at a pre-determined energy. The iodine-decomposed sinogram and the virtual monochromatic projection data are then used as input to an iterative reconstruction method. In the second stage, virtual monochromatic images are reconstructed iteratively while the intensity of weakly tagged iodine in the images is enhanced. The performance of the proposed method was assessed qualitatively and quantitatively. Preliminary results show that our method effectively enhances the visual appearance of weakly tagged fecal materials in the reconstructed CT images while reducing noise and improving the overall quality of the reconstructed images.

Small animal positron emission tomography (PET) scanners are being increasingly used as a basic measurement tool in modern biomedical research. The new designs and technologies of these scanners and the modern reconstruction methods have allowed to reach high spatial resolution and sensitivity. Despite their successes, some important issues remain to be addressed in high resolution PET imaging. First, iterative reconstruction methods like maximum likelihood-expectation maximization (MLEM) are known to recover resolution, but also to create noisy images and edge artifacts if some kind of regularization is not imposed. Second, the limit of resolution achievable by iterative methods on high resolution scanners is not quantitatively understood. Third, the use of regularization methods like Sieves or maximum a posteriori (MAP) requires the determination of the optimal values of several adjustable parameter that may be object-dependent. In this work we review these problems in high resolution PET and establish that the origin of them is more related with the physical effects involved in the emission and detection of the radiation during the acquisition than with the kind of iterative reconstruction method chosen. These physical effects (positron range, non-collinearity, scatter inside the object and inside the detector materials) cause that the tube of response (TOR) that connects the voxels with a line of response (LOR) is rather thick. This implies that the higher frequencies of the patient organ structures are not recorded by the scanner and therefore cannot be recovered during the reconstruction. As iterations grow, ML-EM algorithms try to recover higher frequencies in the image. Once that a certain critic frequency is reached, this only maximizes high frequency noise. Using frequency response analyses techniques, we determine the maximum achievable resolution, before edge artifacts spoil the quality of the image, for a particular scanner as a function of the thickness of the TOR, and independently of the reconstruction method employed. With the same techniques, we can deduce well defined stopping criteria for reconstructions methods. Also, criteria for the highest number of subsets which should be used and how the design of the scanners can be optimized when statistical reconstruction methods are employed, is established.

Image reconstruction problems can be viewed as energy minimization problems and can be mapped onto a Hopfield neural network. For image reconstruction problems the authors describe the Gerchberg-Papoulis iterative method and the priorized discrete Fourier transform (PDFT) algorithm (C.L. Byrne et al., 1983). Both of these can be mapped onto a Hopfield neural network architecture, with the PDFT incorporating an iterative matrix inversion. The equations describing the operation of the Hopfield neural network are formally equivalent to those used in these iterative reconstruction methods, and these iterative reconstruction algorithms are regularized. The PDFT algorithm is a closed form solution to the Gerchberg-Papoulis algorithm when image support information is used. The regularized Gerchberg-Papoulis algorithm can be implemented synchronously, from which it follows that the Hopfield neural network implementation can also converge. >