Statistical Reconstruction Research Articles

Flexible mini gamma camera reconstructions of extended sources using step and shoot and list mode.

Hand- and robot-guided mini gamma cameras have been introduced for the acquisition of single-photon emission computed tomography (SPECT) images. Less cumbersome than whole-body scanners, they allow for a fast acquisition of the radioactivity distribution, for example, to differentiate cancerous from hormonally hyperactive lesions inside the thyroid. This work compares acquisition protocols and reconstruction algorithms in an attempt to identify the most suitable approach for fast acquisition and efficient image reconstruction, suitable for localization of extended sources, such as lesions inside the thyroid. Our setup consists of a mini gamma camera with precise tracking information provided by a robotic arm, which also provides reproducible positioning for our experiments. Based on a realistic phantom of the thyroid including hot and cold nodules as well as background radioactivity, the authors compare "step and shoot" (SAS) and continuous data (CD) acquisition protocols in combination with two different statistical reconstruction methods: maximum-likelihood expectation-maximization (ML-EM) for time-integrated count values and list-mode expectation-maximization (LM-EM) for individually detected gamma rays. In addition, the authors simulate lower uptake values by statistically subsampling the experimental data in order to study the behavior of their approach without changing other aspects of the acquired data. All compared methods yield suitable results, resolving the hot nodules and the cold nodule from the background. However, the CD acquisition is twice as fast as the SAS acquisition, while yielding better coverage of the thyroid phantom, resulting in qualitatively more accurate reconstructions of the isthmus between the lobes. For CD acquisitions, the LM-EM reconstruction method is preferable, as it yields comparable image quality to ML-EM at significantly higher speeds, on average by an order of magnitude. This work identifies CD acquisition protocols combined with LM-EM reconstruction as a prime candidate for the wider introduction of SPECT imaging with flexible mini gamma cameras in the clinical practice.

Adaptive quantum tomography

We provide a review of the experimental and theoretical research in the field of quantum tomography with an emphasis on recently developed adaptive protocols. Several statistical frameworks for adaptive experimental design are discussed. We argue in favor of the Bayesian approach, highlighting both its advantages for a statistical reconstruction of unknown quantum states and processes, and utility for adaptive experimental design. The discussion is supported by an analysis of several recent experimental implementations and numerical recipes.

Modeling and Pre-Treatment of Photon-Starved CT Data for Iterative Reconstruction

An increasing number of X-ray CT procedures are being conducted with drastically reduced dosage, due at least in part to advances in statistical reconstruction methods that can deal more effectively with noise than can traditional techniques. As data become photon-limited, more detailed models are necessary to deal with count rates that drop to the levels of system electronic noise. We present two options for sinogram pre-treatment that can improve the performance of photon-starved measurements, with the intent of following with model-based image reconstruction. Both the local linear minimum mean-squared error (LLMMSE) filter and pointwise Bayesian restoration (PBR) show promise in extracting useful, quantitative information from very low-count data by reducing local bias while maintaining the lower noise variance of statistical methods. Results from clinical data demonstrate the potential of both techniques.

Statistical reconstruction of optical quantum states based on mutually complementary quadrature quantum measurements

We describe a new method for reconstructing the quantum state of the electromagnetic field from the results of mutually complementary optical quadrature measurements. This method is based on the root approach and displaces squeezed Fock states are used as the basis. Theoretical analysis and numerical experiments demonstrate the considerable advantage of the developed tools over those described in the literature.

Detecting asymmetry in the presence of symmetry with maximum likelihood three‐dimensional reconstructions of viruses from electron microscope images

Various modes of transmission electron microscopy can provide unique structural information on nanometre-scale biological machines such as viruses and ribosomes. A maximum-likelihood statistical reconstruction algorithm using information from two such modalities, single-particle cryo-electron microscopy and computed electron tomography, is described for the problem where the machine is nearly symmetrical, but the localised regions of asymmetry are important for the functioning of the machine. The algorithm is demonstrated on experimental bacteriophage Lambda procapsid data. Key features of the algorithm are the use of probability density functions (pdfs) derived from normalised correlation rather than Gaussian pdfs and the solution of a constrained classification problem in which 11 out of 12 sites of asymmetry belong to one class while the 12th site belongs to a second class.

Three-Dimensional Photon Counting Imaging with Axially Distributed Sensing.

In this paper, we review three-dimensional (3D) photon counting imaging with axially distributed sensing. Under severely photon-starved conditions, we have proposed various imaging and algorithmic approaches to reconstruct a scene in 3D, which are not possible by using conventional imaging system due to lack of sufficient number of photons. In this paper, we present an overview of optical sensing and imaging system along with dedicated algorithms for reconstructing 3D scenes by photon counting axially distributed sensing, which may be implemented by moving a single image sensor along its optical axis. To visualize the 3D image, statistical estimation methods and computational reconstruction of axially distributed sensing is applied.

Fast Variance Prediction for Iteratively Reconstructed CT Images With Locally Quadratic Regularization

Predicting noise properties of iteratively reconstructed CT images is useful for analyzing reconstruction methods; for example, local noise power spectrum (NPS) predictions may be used to quantify the detectability of an image feature, to design regularization methods, or to determine dynamic tube current adjustment during a CT scan. This paper presents a method for fast prediction of reconstructed image variance and local NPS for statistical reconstruction methods using quadratic or locally quadratic regularization. Previous methods either require impractical computation times to generate an approximate map of the variance of each reconstructed voxel, or are restricted to specific CT geometries. Our method can produce a variance map of the entire image, for locally shift-invariant CT geometries with sufficiently fine angular sampling, using a computation time comparable to a single back-projection. The method requires only the projection data to be used in the reconstruction, not a reconstruction itself, and is reasonably accurate except near image edges where edge-preserving regularization behaves highly nonlinearly. We evaluate the accuracy of our method using reconstructions of both simulated CT data and real CT scans of a thorax phantom.

Molecular phylogeny, biogeography and chromosome evolution of Malagasy dwarf geckos of the genus Lygodactylus (Squamata, Gekkonidae)

We performed a phylogenetic analysis using nuclear (RAG‐1, RAG‐2) and mitochondrial (16S) markers, a statistical Bayesian reconstruction of ancestral distribution areas and a karyological analysis on most Malagasy species of the gekkonid genus Lygodactylus. The phylogenetic analysis largely confirms major basal branching pattern of previous molecular studies, but highlights significant differences concerning both the relationships between different species groups as well as those within groups. The biogeographic analysis supports a Malagasy origin of Lygodactylus, an oversea dispersal to continental Africa and a return to Madagascar. The L. madagascariensis group (also including a new candidate species identified herein) is the most basal clade in Lygodactylus, and the sister group of a clade with all the remaining species. The second most basal clade is the L. verticillatus group, placed as the sister group of a clade comprising African and Malagasy species. The sister lineage of the L. verticillatus group originated the African radiation through an oversea dispersal out of Madagascar. Eventually, the sister lineage of the L. capensis group originated secondary dispersals from Africa to Madagascar. In Madagascar, lineage diversification in different species groups mainly occurred from southern to northern and eastern regions. Dispersal, vicariance and paleoclimatic refugia probably played a relevant role in the evolutionary history of closely related taxa and in speciation mechanisms. The cytogenetic analysis evidenced a high karyotypic variability in Lygodactylus (from 2n = 34 to 2n = 40), which is at least partly consistent with the phylogenetic relationships and the composition of the various species group. Chromosome evolution occurred independently in different lineages, mainly through a reduction in the chromosome number and starting from a putative primitive karyotype of 2n = 40 with all telocentric elements.

A feature refinement approach for statistical interior CT reconstruction

Interior tomography is clinically desired to reduce the radiation dose rendered to patients. In this work, a new statistical interior tomography approach for computed tomography is proposed. The developed design focuses on taking into account the statistical nature of local projection data and recovering fine structures which are lost in the conventional total-variation (TV)—minimization reconstruction. The proposed method falls within the compressed sensing framework of TV minimization, which only assumes that the interior ROI is piecewise constant or polynomial and does not need any additional prior knowledge. To integrate the statistical distribution property of projection data, the objective function is built under the criteria of penalized weighed least-square (PWLS-TV). In the implementation of the proposed method, the interior projection extrapolation based FBP reconstruction is first used as the initial guess to mitigate truncation artifacts and also provide an extended field-of-view. Moreover, an interior feature refinement step, as an important processing operation is performed after each iteration of PWLS-TV to recover the desired structure information which is lost during the TV minimization. Here, a feature descriptor is specifically designed and employed to distinguish structure from noise and noise-like artifacts. A modified steepest descent algorithm is adopted to minimize the associated objective function. The proposed method is applied to both digital phantom and in vivo Micro-CT datasets, and compared to FBP, ART-TV and PWLS-TV. The reconstruction results demonstrate that the proposed method performs better than other conventional methods in suppressing noise, reducing truncated and streak artifacts, and preserving features. The proposed approach demonstrates its potential usefulness for feature preservation of interior tomography under truncated projection measurements.

Comparison of algebraic and analytical approaches to the formulation of the statistical model-based reconstruction problem for X-ray computed tomography

The main aim of this paper is to investigate properties of our originally formulated statistical model-based iterative approach applied to the image reconstruction from projections problem which are related to its conditioning, and, in this manner, to prove a superiority of this approach over ones recently used by other authors. The reconstruction algorithm based on this conception uses a maximum likelihood estimation with an objective adjusted to the probability distribution of measured signals obtained from an X-ray computed tomography system with parallel beam geometry. The analysis and experimental results presented here show that our analytical approach outperforms the referential algebraic methodology which is explored widely in the literature and exploited in various commercial implementations.

ANTS2 package: simulation and experimental data processing for Anger camera type detectors

ANTS2 is a simulation and data processing package developed for position sensitive detectors with Anger camera type readout. The simulation module of ANTS2 is based on ROOT package from CERN, which is used to store the detector geometry and to perform 3D navigation. The module is capable of simulating particle sources, performing particle tracking, generating photons of primary and secondary scintillation, tracing optical photons and generating photosensor signals. The reconstruction module features several position reconstruction methods based on the statistical reconstruction algorithms (including GPU-based implementations), artificial neural networks and k-NN searches. The module can process simulated as well as imported experimental data containing photosensor signals. A custom library for B-spline parameterization of spatial response of photosensors is implemented which can be used to calculate and parameterize the spatial response of a detector. The package includes a graphical user interface with an extensive set of configuration, visualization and analysis tools. ANTS2 is being developed with the focus on the iterative (adaptive) reconstruction of the detector response using flood field irradiation data. The package is implemented in C++ programming language and it is a multiplatform, open source project.

Elevation-layered dendroclimatic signal in eastern Mediterranean tree rings

Networks of tree-ring data are commonly applied in statistical reconstruction of spatial fields of climate variables. The importance of elevation to the climatic interpretation of tree-ring networks is addressed using 281 station precipitation records, and a network of 79 tree-ring chronologies from different species and a range of elevations in the eastern Mediterranean. Cluster analysis of chronologies identifies 6 tree-ring groups, delineated principally by site elevation. Correlation analysis suggests several of the clusters are linked to homogenous elevational moisture regimes. Results imply that climate stations close to the elevations of the tree-ring sites are essential for assessing the seasonal climatic signal in tree-ring chronologies from this region. A broader implication is that the elevations of stations contributing to gridded climate networks should be considered in the design and interpretation of field reconstructions of climate from tree rings. Finally, results suggest elevation-stratified tree-ring networks as a strategy for seasonal climate reconstruction.

Nonlinear Statistical Reconstruction for Flat-Panel Cone-Beam CT with Blur and Correlated Noise Models.

Flat-panel cone-beam CT (FP-CBCT) is a promising imaging modality, partly due to its potential for high spatial resolution reconstructions in relatively compact scanners. Despite this potential, FP-CBCT can face difficulty resolving important fine scale structures (e.g, trabecular details in dedicated extremities scanners and microcalcifications in dedicated CBCT mammography). Model-based methods offer one opportunity to improve high-resolution performance without any hardware changes. Previous work, based on a linearized forward model, demonstrated improved performance when both system blur and spatial correlations characteristics of FP-CBCT systems are modeled. Unfortunately, the linearized model relies on a staged processing approach that complicates tuning parameter selection and can limit the finest achievable spatial resolution. In this work, we present an alternative scheme that leverages a full nonlinear forward model with both system blur and spatially correlated noise. A likelihood-based objective function is derived from this forward model and we derive an iterative optimization algorithm for its solution. The proposed approach is evaluated in simulation studies using a digital extremities phantom and resolution-noise trade-offs are quantitatively evaluated. The correlated nonlinear model outperformed both the uncorrelated nonlinear model and the staged linearized technique with up to a 86% reduction in variance at matched spatial resolution. Additionally, the nonlinear models could achieve finer spatial resolution (correlated: 0.10 mm, uncorrelated: 0.11 mm) than the linear correlated model (0.15 mm), and traditional FDK (0.40 mm). This suggests the proposed nonlinear approach may be an important tool in improving performance for high-resolution clinical applications.

A limit on dose reduction possible with CT reconstruction algorithms without prior knowledge of the scan subject.

To find an upper bound on the maximum dose reduction possible for any reconstruction algorithm, analytic or iterative, that result from the inclusion of the data statistics. The authors do not analyze noise reduction possible from prior knowledge or assumptions about the object. The authors examined the task of estimating the density of a circular lesion in a cross section. Raw data were simulated by forward projection of existing images and numerical phantoms. To assess an upper bound on the achievable dose reduction by any algorithm, the authors assume that both the background and the shape of the lesion are completely known. Under these conditions, the best possible estimate of the density can be determined by solving a weighted least squares problem directly in the raw data domain. Any possible reconstruction algorithm that does not use prior knowledge or make assumptions about the object, including filtered backprojection (FBP) or iterative reconstruction methods with this constraint, must be no better than this least squares solution. The authors simulated 10,000 sets of noisy data and compared the variance in density from the least squares solution with those from FBP. Density was estimated from FBP images using either averaging within a ROI, or streak-adaptive averaging with better noise performance. The bound on the possible dose reduction depends on the degree to which the observer can read through the possibly streaky noise. For the described low contrast detection task with the signal shape and background known exactly, the average dose reduction possible compared to FBP with streak-adaptive averaging was 42% and it was 64% if only the ROI average is used with FBP. The exact amount of dose reduction also depends on the background anatomy, with statistically inhomogeneous backgrounds showing greater benefits. The dose reductions from new, statistical reconstruction methods can be bounded. Larger dose reductions in the density estimation task studied here are only possible with the introduction of prior knowledge, which can introduce bias.

The Use of Anatomical Information for Molecular Image Reconstruction Algorithms: Attenuation/Scatter Correction, Motion Compensation, and Noise Reduction

PET and SPECT are important tools for providing valuable molecular information about patients to clinicians. Advances in nuclear medicine hardware technologies and statistical image reconstruction algorithms enabled significantly improved image quality. Sequentially or simultaneously acquired anatomical images such as CT and MRI from hybrid scanners are also important ingredients for improving the image quality of PET or SPECT further. High-quality anatomical information has been used and investigated for attenuation and scatter corrections, motion compensation, and noise reduction via post-reconstruction filtering and regularization in inverse problems. In this article, we will review works using anatomical information for molecular image reconstruction algorithms for better image quality by describing mathematical models, discussing sources of anatomical information for different cases, and showing some examples.

European summer temperatures since Roman times

The spatial context is critical when assessing present-day climate anomalies, attributing them to potential forcings and making statements regarding their frequency and severity in a long-term perspective. Recent international initiatives have expanded the number of high-quality proxy-records and developed new statistical reconstruction methods. These advances allow more rigorous regional past temperature reconstructions and, in turn, the possibility of evaluating climate models on policy-relevant, spatio-temporal scales. Here we provide a new proxy-based, annually-resolved, spatial reconstruction of the European summer (June–August) temperature fields back to 755 CE based on Bayesian hierarchical modelling (BHM), together with estimates of the European mean temperature variation since 138 BCE based on BHM and composite-plus-scaling (CPS). Our reconstructions compare well with independent instrumental and proxy-based temperature estimates, but suggest a larger amplitude in summer temperature variability than previously reported. Both CPS and BHM reconstructions indicate that the mean 20th century European summer temperature was not significantly different from some earlier centuries, including the 1st, 2nd, 8th and 10th centuries CE. The 1st century (in BHM also the 10th century) may even have been slightly warmer than the 20th century, but the difference is not statistically significant. Comparing each 50 yr period with the 1951–2000 period reveals a similar pattern. Recent summers, however, have been unusually warm in the context of the last two millennia and there are no 30 yr periods in either reconstruction that exceed the mean average European summer temperature of the last 3 decades (1986–2015 CE). A comparison with an ensemble of climate model simulations suggests that the reconstructed European summer temperature variability over the period 850–2000 CE reflects changes in both internal variability and external forcing on multi-decadal time-scales. For pan-European temperatures we find slightly better agreement between the reconstruction and the model simulations with high-end estimates for total solar irradiance. Temperature differences between the medieval period, the recent period and the Little Ice Age are larger in the reconstructions than the simulations. This may indicate inflated variability of the reconstructions, a lack of sensitivity and processes to changes in external forcing on the simulated European climate and/or an underestimation of internal variability on centennial and longer time scales.

Polychromatic Iterative Statistical Material Image Reconstruction for Photon-Counting Computed Tomography.

This work proposes a dedicated statistical algorithm to perform a direct reconstruction of material-decomposed images from data acquired with photon-counting detectors (PCDs) in computed tomography. It is based on local approximations (surrogates) of the negative logarithmic Poisson probability function. Exploiting the convexity of this function allows for parallel updates of all image pixels. Parallel updates can compensate for the rather slow convergence that is intrinsic to statistical algorithms. We investigate the accuracy of the algorithm for ideal photon-counting detectors. Complementarily, we apply the algorithm to simulation data of a realistic PCD with its spectral resolution limited by K-escape, charge sharing, and pulse-pileup. For data from both an ideal and realistic PCD, the proposed algorithm is able to correct beam-hardening artifacts and quantitatively determine the material fractions of the chosen basis materials. Via regularization we were able to achieve a reduction of image noise for the realistic PCD that is up to 90% lower compared to material images form a linear, image-based material decomposition using FBP images. Additionally, we find a dependence of the algorithms convergence speed on the threshold selection within the PCD.

Statistical Reconstruction Algorithm for Restoring Broken Tooth Surface Based on Occlusion Spatial Constraint

Statistical Reconstruction Algorithm for Restoring Broken Tooth Surface Based on Occlusion Spatial Constraint

Relaxed Linearized Algorithms for Faster X-Ray CT Image Reconstruction.

Statistical image reconstruction (SIR) methods are studied extensively for X-ray computed tomography (CT) due to the potential of acquiring CT scans with reduced X-ray dose while maintaining image quality. However, the longer reconstruction time of SIR methods hinders their use in X-ray CT in practice. To accelerate statistical methods, many optimization techniques have been investigated. Over-relaxation is a common technique to speed up convergence of iterative algorithms. For instance, using a relaxation parameter that is close to two in alternating direction method of multipliers (ADMM) has been shown to speed up convergence significantly. This paper proposes a relaxed linearized augmented Lagrangian (AL) method that shows theoretical faster convergence rate with over-relaxation and applies the proposed relaxed linearized AL method to X-ray CT image reconstruction problems. Experimental results with both simulated and real CT scan data show that the proposed relaxed algorithm (with ordered-subsets [OS] acceleration) is about twice as fast as the existing unrelaxed fast algorithms, with negligible computation and memory overhead.

Accurate low-dose iterative CT reconstruction from few projections by Generalized Anisotropic Total Variation minimization for industrial CT.

Reducing the amount of time for data acquisition and reconstruction in industrial CT decreases the operation time of the X-ray machine and therefore increases the sales. This can be achieved by reducing both, the dose and the pulse length of the CT system and the number of projections for the reconstruction, respectively. In this paper, a novel generalized Anisotropic Total Variation regularization for under-sampled, low-dose iterative CT reconstruction is discussed and compared to the standard methods, Total Variation, Adaptive weighted Total Variation and Filtered Backprojection. The novel regularization function uses a priori information about the Gradient Magnitude Distribution of the scanned object for the reconstruction. We provide a general parameterization scheme and evaluate the efficiency of our new algorithm for different noise levels and different number of projection views. When noise is not present, error-free reconstructions are achievable for AwTV and GATV from 40 projections. In cases where noise is simulated, our strategy achieves a Relative Root Mean Square Error that is up to 11 times lower than Total Variation-based and up to 4 times lower than AwTV-based iterative statistical reconstruction (e.g. for a SNR of 223 and 40 projections). To obtain the same reconstruction quality as achieved by Total Variation, the projection number and the pulse length, and the acquisition time and the dose respectively can be reduced by a factor of approximately 3.5, when AwTV is used and a factor of approximately 6.7, when our proposed algorithm is used.