Hybrid Poisson/polynomial objective functions for tomographic image reconstruction from transmission scans

Setup and characterisation according to NEMA NU 4 of thephenoPET scanner, a PET system dedicated for plant sciences

Objective.ThephenoPET system is a plant dedicated positron emission tomography (PET) scanner consisting of fully digital photo multipliers with lutetium-yttrium oxyorthosilicate crystals and located inside a custom climate chamber. Here, we present the setup ofphenoPET, its data processing and image reconstruction together with its performance.Approach.The performance characterization follows the national electrical manufacturers association (NEMA) standard for small animal PET systems with a number of adoptions due to the vertical oriented bore of a PET for plant sciences. In addition temperature stability and spatial resolution with a hot rod phantom are addressed.Main results.The spatial resolution for a22Na point source at a radial distance of 5 mm to the center of the field-of-view (FOV) is 1.45 mm, 0.82 mm and 1.88 mm with filtered back projection in radial, tangential and axial direction, respectively. A hot rod phantom with18F gives a spatial resolution of up to 1.6 mm. The peak noise-equivalent count rates are 550 kcps @ 35.08 MBq, 308 kcps @ 33 MBq and 45 kcps @ 40.60 MBq for the mouse, rat and monkey size scatter phantoms, respectively. The scatter fractions for these phantoms are 12.63%, 22.64% and 55.90%. We observe a peak sensitivity of up to 3.6% and a total sensitivity of up toSA,tot= 2.17%. For the NEMA image quality phantom we observe a uniformity of %STD= 4.22% with ordinary Poisson maximum likelihood expectation-maximization with 52 iterations. Here, recovery coefficients of 0.12, 0.64, 0.89, 0.93 and 0.91 for 1 mm, 2 mm, 3 mm, 4 mm and 5 mm rods are obtained and spill-over ratios of 0.08 and 0.14 for the water-filled and air-filled inserts, respectively.Significance.ThephenoPET and its laboratory are now in routine operation for the administration of [11C]CO2and non-invasive measurement of transport and allocation of11C-labelled photoassimilates in plants.

Correcting statistical CT number biases without access to raw detector counts: Applications to high spatial resolution photon counting CT imaging

BACKGROUND

Due to the nonlinear nature of the logarithmic operation and the stochastic nature of photon counts (N), sinogram data of photon counting detector CT (PCD-CT) are intrinsically biased, which leads to statistical CT number biases. When raw counts are available, nearly unbiased statistical estimators for projection data were developed recently to address the CT number bias issue. However, for most clinical PCD-CT systems, users' access to raw detector counts is limited. Therefore, it remains a challenge for end users to address the CT number bias issue in clinical applications.

PURPOSE

To develop methods to correct statistical biases in PCD-CT without requiring access to raw PCD counts.

METHODS

(1) The sample variance of air-only post-log sinograms was used to estimate air-only detector counts,N ¯ 0 $\bar{N}_0$ . (2) If the post-log sinogram data, y, is available, then N of each detector pixel was estimated usingN = N ¯ 0 e - y $N = \bar{N}_0 \, \mathrm{e}^{-y}$ . Once N was estimated, a closed-form analytical bias correction was applied to the sinogram. (3) If a patient's post-log sinogram data are not archived, a forward projection of the bias-contaminated CT image was used to perform a first-order bias correction. Both the proposed sinogram domain- and image domain-based bias correction methods were validated using experimental PCD-CT data.

RESULTS

Experimental results demonstrated that both sinogram domain- and image domain-based bias correction methods enabled reduced-dose PCD-CT images to match the CT numbers of reference-standard images within [-5, 5] HU. In contrast, uncorrected reduced-dose PCD-CT images demonstrated biases ranging from -25 to 55 HU, depending on the material. No increase in image noise or spatial resolution degradation was observed using the proposed methods.

CONCLUSIONS

CT number bias issues can be effectively addressed using the proposed sinogram or image domain method in PCD-CT, allowing PCD-CT acquired at different radiation dose levels to have consistent CT numbers desired for quantitative imaging.

Reconstruction of three-dimensional tomographic patient models for radiation dose modulation in CT from two scout views using deep learning

BACKGROUND

A tomographic patient model is essential for radiation dose modulation in x-ray computed tomography (CT). Currently, two-view scout images (also known as topograms) are used to estimate patient models with relatively uniform attenuation coefficients. These patient models do not account for the detailed anatomical variations of human subjects, and thus, may limit the accuracy of intraview or organ-specific dose modulations in emerging CT technologies.

PURPOSE

The purpose of this work was to show that 3D tomographic patient models can be generated from two-view scout images using deep learning strategies, and the reconstructed 3D patient models indeed enable accurate prescriptions of fluence-field modulated or organ-specific dose delivery in the subsequent CT scans.

METHODS

CT images and the corresponding two-view scout images were retrospectively collected from 4214 individual CT exams. The collected data were curated for the training of a deep neural network architecture termed ScoutCT-NET to generate 3D tomographic attenuation models from two-view scout images. The trained network was validated using a cohort of 55 136 images from 212 individual patients. To evaluate the accuracy of the reconstructed 3D patient models, radiation delivery plans were generated using ScoutCT-NET 3D patient models and compared with plans prescribed based on true CT images (gold standard) for both fluence-field-modulated CT and organ-specific CT. Radiation dose distributions were estimated using Monte Carlo simulations and were quantitatively evaluated using the Gamma analysis method. Modulated dose profiles were compared against state-of-the-art tube current modulation schemes. Impacts of ScoutCT-NET patient model-based dose modulation schemes on universal-purpose CT acquisitions and organ-specific acquisitions were also compared in terms of overall image appearance, noise magnitude, and noise uniformity.

RESULTS

The results demonstrate that (1) The end-to-end trained ScoutCT-NET can be used to generate 3D patient attenuation models and demonstrate empirical generalizability. (2) The 3D patient models can be used to accurately estimate the spatial distribution of radiation dose delivered by standard helical CTs prior to the actual CT acquisition; compared to the gold-standard dose distribution, 95.0% of the voxels in the ScoutCT-NET based dose maps have acceptable gamma values for 5 mm distance-to-agreement and 10% dose difference. (3) The 3D patient models also enabled accurate prescription of fluence-field modulated CT to generate a more uniform noise distribution across the patient body compared to tube current-modulated CT. (4) ScoutCT-NET 3D patient models enabled accurate prescription of organ-specific CT to boost image quality for a given body region-of-interest under a given radiation dose constraint.

CONCLUSION

3D tomographic attenuation models generated by ScoutCT-NET from two-view scout images can be used to prescribe fluence-field-modulated or organ-specific CT scans with high accuracy for the overall objective of radiation dose reduction or image quality improvement for a given imaging task.

Reducing residual-motion artifacts in iterative 3D CBCT reconstruction in image-guided radiation therapy

PURPOSE

Recent evaluations of a 3D iterative cone-beam computed tomography (iCBCT) reconstruction method available on Varian radiation treatment devices demonstrated that iCBCT provides superior image quality when compared to analytical Feldkamp-Davis-Kress (FDK) method. However, iCBCT employs statistical penalized likelihood (PL) that is known to be highly sensitive to inconsistencies due to physiological motion occurring during the acquisition. We propose a computationally inexpensive extension of iCBCT addressing this deficiency.

METHODS

During the iterative process, the gradients of PL are modified to avoid the generation of motion-related artifacts. To assess the impact of this modification, we propose a motion simulation generating CBCT projections of a moving anatomy together with artifact-free images used as ground truth. Contrast-to-noise ratio and power spectra of difference images are computed to quantify the impact of the motion on reconstructed CBCT volumes as well as the effect of the proposed modification.

RESULTS

Using both simulated and clinical data, it is shown that the motion of patient's abdominal wall during the acquisition results in artifacts that can be quantified as low-frequency components in volumes reconstructed with iCBCT. Further, a quantitative evaluation demonstrates that the proposed modification of PL reduces these low-frequency components. While preserving the advantages of PL, it effectively suppresses the propagation of motion-related artifacts into clinically important regions, thus increasing the motion resiliency of iCBCT.

CONCLUSIONS

The proposed modified iterative reconstruction method significantly improves the quality of CBCT images of anatomies suffering from residual motion.

SPULTRA: Low-Dose CT Image Reconstruction With Joint Statistical and Learned Image Models

Low-dose CT image reconstruction has been a popular research topic in recent years. A typical reconstruction method based on post-log measurements is called penalized weighted-least squares (PWLS). Due to the underlying limitations of the post-log statistical model, the PWLS reconstruction quality is often degraded in low-dose scans. This paper investigates a shifted-Poisson (SP) model based likelihood function that uses the pre-log raw measurements that better represents the measurement statistics, together with a data-driven regularizer exploiting a Union of Learned TRAnsforms (SPULTRA). Both the SP induced data-fidelity term and the regularizer in the proposed framework are nonconvex. The proposed SPULTRA algorithm uses quadratic surrogate functions for the SP induced data-fidelity term. Each iteration involves a quadratic subproblem for updating the image, and a sparse coding and clustering subproblem that has a closed-form solution. The SPULTRA algorithm has a similar computational cost per iteration as its recent counterpart PWLS-ULTRA that uses post-log measurements, and it provides better image reconstruction quality than PWLS-ULTRA, especially in low-dose scans.

Quantitative accuracy of CT numbers: Theoretical analyses and experimental studies

PURPOSE

The CT number accuracy, that is, CT number bias, plays an important role in clinical diagnosis. When strategies to reduce radiation dose are discussed, it is important to make sure that the CT number bias is controlled within an acceptable range. The purpose of this paper was to investigate the dependence of CT number bias on radiation dose level and on image contrast (i.e., the difference in CT number between the ROI and the background) in Computed Tomography (CT).

METHODS

A lesion-background model was introduced to theoretically study how the CT number bias changes with radiation exposure level and with CT number contrast when a simple linear reconstruction algorithm such as filtered backprojection (FBP) is used. The theoretical results were validated with experimental studies using a benchtop CT system equipped with a photon-counting detector (XC-HYDRA FX50, XCounter AB, Sweden) and a clinical diagnostic MDCT scanner (Discovery CT750 HD, GE Healthcare, Waukesha, WI, USA) equipped with an energy-integrating detector. The Catphan phantom (Catphan 600, the Phantom Laboratory, Salem, NY, USA) was scanned at different mAs levels and 50 scans were performed for each mAs. The bias of CT number was evaluated for each combination of mAs and ROIs with different contrast levels. An anthropomorphic phantom (ATOM 10-year-old phantom, Model 706, CIRS Inc. Norfolk, VA, USA) with much more heterogeneous object content was used to test the applicability of the theory to the more general image object cases.

RESULTS

Both theoretical and experimental studies showed that the CT number bias is inversely proportional to the radiation exposure level yet linearly dependent on the CT number contrast between the lesion and the background, that is, Bias ( μ ^ 1 FBP ) = α mAs ( 1 + β Δ H U ) .

CONCLUSIONS

The quantitative accuracy of CT numbers can be problematic and thus needs some extra attention when radiation dose is reduced. In this work, we showed that the bias of the FBP reconstruction increases as mAs is reduced; both positive and negative bias can be observed depending on the contrast difference between a targeted ROI and its surrounding background tissues.

Hybrid Pre-Log and Post-Log Image Reconstruction for Computed Tomography

Tomographic image reconstruction for low-dose computed tomography (CT) is increasingly challenging as dose continues to reduce in clinical applications. Pre-log domain methods and post-log domain methods have been proposed individually and each method has its own disadvantage. While having the potential to improve image quality for low-dose data by using an accurate imaging model, pre-log domain methods suffer slow convergence in practice due to the nonlinear transformation from the image to measurements. In contrast, post-log domain methods have fast convergence speed but the resulting image quality is suboptimal for low dose CT data because the log transformation is extremely unreliable for low-count measurements and undefined for negative values. This paper proposes a hybrid method that integrates the pre-log model and post-log model together to overcome the disadvantages of individual pre-log and post-log methods. We divide a set of CT data into high-count and low-count regions. The post-log weighted least squares model is used for measurements in the high-count region and the pre-log shifted Poisson model for measurements in the low-count region. The hybrid likelihood function can be optimized using an existing iterative algorithm. Computer simulations and phantom experiments show that the proposed hybrid method can achieve faster early convergence than the pre-log shifted Poisson likelihood method and better signal-to-noise performance than the post-log weighted least squares method.

Comparison Between Pre-Log and Post-Log Statistical Models in Ultra-Low-Dose CT Reconstruction

X-ray detectors in clinical computed tomography (CT) usually operate in current-integrating mode. Their complicated signal statistics often lead to intractable likelihood functions for practical use in model-based image reconstruction (MBIR). It is therefore desirable to design simplified statistical models without losing the essential factors. Depending on whether the CT transmission data are logarithmically transformed, pre-log and post-log models are two major categories of choices in CT MBIR. Both being approximations, it remains an open question whether one model can notably improve image quality over the other on real scanners. In this study, we develop and compare several pre-log and post-log MBIR algorithms under a unified framework. Their reconstruction accuracy based on simulation and clinical datasets are evaluated. The results show that pre-log MBIR can achieve notably better quantitative accuracy than post-log MBIR in ultra-low-dose CT, although in less extreme cases, post-log MBIR with handcrafted pre-processing remains a competitive alternative. Pre-log MBIR could play a growing role in emerging ultra-low-dose CT applications.

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.

Model-based iterative reconstruction for flat-panel cone-beam CT with focal spot blur, detector blur, and correlated noise

While model-based reconstruction methods have been successfully applied to flat-panel cone-beam CT (FP-CBCT) systems, typical implementations ignore both spatial correlations in the projection data as well as system blurs due to the detector and focal spot in the x-ray source. In this work, we develop a forward model for flat-panel-based systems that includes blur and noise correlation associated with finite focal spot size and an indirect detector (e.g. scintillator). This forward model is used to develop a staged reconstruction framework where projection data are deconvolved and log-transformed, followed by a generalized least-squares reconstruction that utilizes a non-diagonal statistical weighting to account for the correlation that arises from the acquisition and data processing chain. We investigate the performance of this novel reconstruction approach in both simulated data and in CBCT test-bench data. In comparison to traditional filtered backprojection and model-based methods that ignore noise correlation, the proposed approach yields a superior noise-resolution tradeoff. For example, for a system with 0.34 mm FWHM scintillator blur and 0.70 FWHM focal spot blur, using the correlated noise model instead of an uncorrelated noise model increased resolution by 42% (with variance matched at 6.9  ×  10(-8) mm(-2)). While this advantage holds across a wide range of systems with differing blur characteristics, the improvements are greatest for systems where source blur is larger than detector blur.

Blind inpainting using l0 and total variation regularization

In this paper, we address the problem of image reconstruction with missing pixels or corrupted with impulse noise, when the locations of the corrupted pixels are not known. A logarithmic transformation is applied to convert the multiplication between the image and binary mask into an additive problem. The image and mask terms are then estimated iteratively with total variation regularization applied on the image, and l0 regularization on the mask term which imposes sparseness on the support set of the missing pixels. The resulting alternating minimization scheme simultaneously estimates the image and mask, in the same iterative process. The logarithmic transformation also allows the method to be extended to the Rayleigh multiplicative and Poisson observation models. The method can also be extended to impulse noise removal by relaxing the regularizer from the l0 norm to the l1 norm. Experimental results show that the proposed method can deal with a larger fraction of missing pixels than two phase methods, which first estimate the mask and then reconstruct the image.

Regularization designs for uniform spatial resolution and noise properties in statistical image reconstruction for 3-D X-ray CT

Statistical image reconstruction methods for X-ray computed tomography (CT) provide improved spatial resolution and noise properties over conventional filtered back-projection (FBP) reconstruction, along with other potential advantages such as reduced patient dose and artifacts. Conventional regularized image reconstruction leads to spatially variant spatial resolution and noise characteristics because of interactions between the system models and the regularization. Previous regularization design methods aiming to solve such issues mostly rely on circulant approximations of the Fisher information matrix that are very inaccurate for undersampled geometries like short-scan cone-beam CT. This paper extends the regularization method proposed in to 3-D cone-beam CT by introducing a hypothetical scanning geometry that helps address the sampling properties. The proposed regularization designs were compared with the original method in with both phantom simulation and clinical reconstruction in 3-D axial X-ray CT. The proposed regularization methods yield improved spatial resolution or noise uniformity in statistical image reconstruction for short-scan axial cone-beam CT.

Motion based X-ray imaging modality

A new X-ray imaging method (patent pending) was developed to visualize function-related motion information. We modify existing X-ray imaging methods to provide four images without increasing the necessary measurement time or radiation dose. The most important of these images is a new "kinetic" image that represents motions inside the object or living body. The motion-based contrast of the kinetic image can help visualize details that were not accessible before. The broad range of the movements and high sensitivity of the method are illustrated by imaging the mechanics of a working clock and the chest of a living African clawed frog (Xenopus laevis). The heart, valves, aorta, and lungs of the frog are clearly visualized in spite of the low soft tissue contrast of the animal. The new technology also reconstructs a "static" image similar to the existing conventional X-ray image. The static image shows practically the same information as the conventional image. The new technology presents two more images which show the point-wise errors of the static and kinetic images. This technique gives a better estimation of errors than present methods because it is based entirely on measured data. The new technology could be used in imaging cardiopulmonary movements, nondestructive testing, or port security screening.

Ordered subsets convex algorithm for 3D terahertz transmission tomography

We investigate in this paper a new reconstruction method in order to perform 3D Terahertz (THz) tomography using a continuous wave acquisition setup in transmission mode. This method is based on the Maximum Likelihood for TRansmission tomography (ML-TR) first developed for X-ray imaging. We optimize the Ordered Subsets Convex (OSC) implementation of the ML-TR by including the Gaussian propagation model of THz waves and take into account the intensity distributions of both blank calibration scan and dark-field measured on THz detectors. THz ML-TR reconstruction quality and accuracy are discussed and compared to other tomographic reconstructions.

LSO background radiation as a transmission source using time of flight

LSO scintillators (Lu2Sio5:Ce) have a background radiation which originates from the isotope Lu-176 that is present in natural occurring lutetium. The decay that occurs in this isotope is a beta decay that is in coincidence with cascade gamma emissions with energies of 307,202 and 88 keV. The coincidental nature of the beta decay with the gamma emissions allow for separation of emission data originating from a positron annihilation event from transmission type data from the Lu-176 beta decay. By using the time of flight information, and information of the chord length between two LSO pixels in coincidence as a result of a beta emission and emitted gamma, a second time window can be set to observe transmission events simultaneously to emission events. Using the time when the PET scanner is not actively acquiring positron emission data, a continuous blank can be acquired and used to reconstruct a transmission image. With this blank and the measured transmission data, a transmission image can be reconstructed. This reconstructed transmission image can be used to perform emission data corrections such as attenuation correction and scatter corrections or starting images for algorithms that estimate emission and attenuation simultaneously. It is observed that the flux of the background activity is high enough to create useful transmission images with an acquisition time of 10 min.

A Fourier-based compressed sensing technique for accelerated CT image reconstruction using first-order methods

As a solution to iterative CT image reconstruction, first-order methods are prominent for the large-scale capability and the fast convergence rate [Formula: see text]. In practice, the CT system matrix with a large condition number may lead to slow convergence speed despite the theoretically promising upper bound. The aim of this study is to develop a Fourier-based scaling technique to enhance the convergence speed of first-order methods applied to CT image reconstruction. Instead of working in the projection domain, we transform the projection data and construct a data fidelity model in Fourier space. Inspired by the filtered backprojection formalism, the data are appropriately weighted in Fourier space. We formulate an optimization problem based on weighted least-squares in the Fourier space and total-variation (TV) regularization in image space for parallel-beam, fan-beam and cone-beam CT geometry. To achieve the maximum computational speed, the optimization problem is solved using a fast iterative shrinkage-thresholding algorithm with backtracking line search and GPU implementation of projection/backprojection. The performance of the proposed algorithm is demonstrated through a series of digital simulation and experimental phantom studies. The results are compared with the existing TV regularized techniques based on statistics-based weighted least-squares as well as basic algebraic reconstruction technique. The proposed Fourier-based compressed sensing (CS) method significantly improves both the image quality and the convergence rate compared to the existing CS techniques.

A low dose simulation tool for CT systems with energy integrating detectors

PURPOSE

This paper introduces a new strategy for simulating low-dose computed tomography (CT) scans using real scans of a higher dose as an input. The tool is verified against simulations and real scans and compared to other approaches found in the literature.

METHODS

The conditional variance identity is used to properly account for the variance of the input high-dose data, and a formula is derived for generating a new Poisson noise realization which has the same mean and variance as the true low-dose data. The authors also derive a formula for the inclusion of real samples of detector noise, properly scaled according to the level of the simulated x-ray signals.

RESULTS

The proposed method is shown to match real scans in number of experiments. Noise standard deviation measurements in simulated low-dose reconstructions of a 35 cm water phantom match real scans in a range from 500 to 10 mA with less than 5% error. Mean and variance of individual detector channels are shown to match closely across the detector array. Finally, the visual appearance of noise and streak artifacts is shown to match in real scans even under conditions of photon-starvation (with tube currents as low as 10 and 80 mA). Additionally, the proposed method is shown to be more accurate than previous approaches (1) in achieving the correct mean and variance in reconstructed images from pure-Poisson noise simulations (with no detector noise) under photon-starvation conditions, and (2) in simulating the correct noise level and detector noise artifacts in real low-dose scans.

CONCLUSIONS

The proposed method can accurately simulate low-dose CT data starting from high-dose data, including effects from photon starvation and detector noise. This is potentially a very useful tool in helping to determine minimum dose requirements for a wide range of clinical protocols and advanced reconstruction algorithms.

Simultaneous reconstruction of emission activity and attenuation coefficient distribution from TOF data, acquired with external transmission source

The simultaneous PET data reconstruction of emission activity and attenuation coefficient distribution is presented, where the attenuation image is constrained by exploiting an external transmission source. Data are acquired in time-of-flight (TOF) mode, allowing in principle for separation of emission and transmission data. Nevertheless, here all data are reconstructed at once, eliminating the need to trace the position of the transmission source in sinogram space. Contamination of emission data by the transmission source and vice versa is naturally modeled. Attenuated emission activity data also provide additional information about object attenuation coefficient values. The algorithm alternates between attenuation and emission activity image updates. We also proposed a method of estimation of spatial scatter distribution from the transmission source by incorporating knowledge about the expected range of attenuation map values. The reconstruction of experimental data from the Siemens mCT scanner suggests that simultaneous reconstruction improves attenuation map image quality, as compared to when data are separated. In the presented example, the attenuation map image noise was reduced and non-uniformity artifacts that occurred due to scatter estimation were suppressed. On the other hand, the use of transmission data stabilizes attenuation coefficient distribution reconstruction from TOF emission data alone. The example of improving emission images by refining a CT-based patient attenuation map is presented, revealing potential benefits of simultaneous CT and PET data reconstruction.

Automatic alignment and reconstruction of images for soft X-ray tomography

Soft X-ray tomography (SXT) is a powerful imaging technique that generates quantitative, 3D images of the structural organization of whole cells in a near-native state. SXT is also a high-throughput imaging technique. At the National Center for X-ray Tomography (NCXT), specimen preparation and image collection for tomographic reconstruction of a whole cell require only minutes. Aligning and reconstructing the data, however, take significantly longer. Here we describe a new component of the high throughput computational pipeline used for processing data at the NCXT. We have developed a new method for automatic alignment of projection images that does not require fiducial markers or manual interaction with the software. This method has been optimized for SXT data sets, which routinely involve full rotation of the specimen. This software gives users of the NCXT SXT instrument a new capability - virtually real-time initial 3D results during an imaging experiment, which can later be further refined. The new code, Automatic Reconstruction 3D (AREC3D), is also fast, reliable, and robust. The fundamental architecture of the code is also adaptable to high performance GPU processing, which enables significant improvements in speed and fidelity.

Parallel proximal algorithm for image restoration using hybrid regularization

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet-domain regularization brings other artefacts, e.g., ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain (e.g., spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques.

Statistical sinogram restoration in dual-energy CT for PET attenuation correction

Dual-energy (DE) X-ray computed tomography (CT) has been found useful in various applications. In medical imaging, one promising application is using low-dose DECT for attenuation correction in positron emission tomography (PET). Existing approaches to sinogram material decomposition ignore noise characteristics and are based on logarithmic transforms, producing noisy component sinogram estimates for low-dose DECT. In this paper, we propose two novel sinogram restoration methods based on statistical models: penalized weighted least square (PWLS) and penalized likelihood (PL), yielding less noisy component sinogram estimates for low-dose DECT than classical methods. The proposed methods consequently provide more precise attenuation correction of the PET emission images than do previous methods for sinogram material decomposition with DECT. We report simulations that compare the proposed techniques and existing approaches.

Radiation dose reduction in computed tomography: techniques and future perspective

Despite universal consensus that computed tomography (CT) overwhelmingly benefits patients when used for appropriate indications, concerns have been raised regarding the potential risk of cancer induction from CT due to the exponentially increased use of CT in medicine. Keeping radiation dose as low as reasonably achievable, consistent with the diagnostic task, remains the most important strategy for decreasing this potential risk. This article summarizes the general technical strategies that are commonly used for radiation dose management in CT. Dose-management strategies for pediatric CT, cardiac CT, dual-energy CT, CT perfusion and interventional CT are specifically discussed, and future perspectives on CT dose reduction are presented.

PET transmission tomography using a novel nonlocal MRF prior

In positron emission tomography, transmission scans can be performed to estimate attenuation correction factors (ACFs) which are in turn used to correct the emission scans. And such an attenuation correction is crucial for quantitatively accurate PET reconstructions. The prior model used in this work was based on our assumption that the attenuation values vary smoothly, with occasional discontinuities at anatomical borders. And on the other hand, long acquisition or scan times, although alleviating the noise effect of the count-limited scans, are blamed for patient uncomfortableness and movements. So, transmission tomography often suffers from the noise effect because of the short scan time. Thus reconstruction which is capable of overcoming the noise effect is highly needed. In this article, we apply the nonlocal prior Bayesian reconstruction method in PET transmission tomography. Resulting experimentations validate that the reconstructions using the nonlocal prior can reconstruct better transmission images and overcome noise effect even when the scan time is relatively short.

A three-dimensional statistical approach to improved image quality for multislice helical CT

Multislice helical computed tomography scanning offers the advantages of faster acquisition and wide organ coverage for routine clinical diagnostic purposes. However, image reconstruction is faced with the challenges of three-dimensional cone-beam geometry, data completeness issues, and low dosage. Of all available reconstruction methods, statistical iterative reconstruction (IR) techniques appear particularly promising since they provide the flexibility of accurate physical noise modeling and geometric system description. In this paper, we present the application of Bayesian iterative algorithms to real 3D multislice helical data to demonstrate significant image quality improvement over conventional techniques. We also introduce a novel prior distribution designed to provide flexibility in its parameters to fine-tune image quality. Specifically, enhanced image resolution and lower noise have been achieved, concurrently with the reduction of helical cone-beam artifacts, as demonstrated by phantom studies. Clinical results also illustrate the capabilities of the algorithm on real patient data. Although computational load remains a significant challenge for practical development, superior image quality combined with advancements in computing technology make IR techniques a legitimate candidate for future clinical applications.

Dynamic PET reconstruction using wavelet regularization with adapted basis functions

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an l(1) -regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the "optimal" E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality. We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-LemariE wavelets in a series of experiments: the 1-D fitting of TACs, and the tomographic reconstruction of both simulated and clinical data. We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or equivalently a reduced exposure to the patient for the same image quality.

Statistical modeling and reconstruction of randoms precorrected PET data

Randoms precorrected positron emission tomography (PET) data is formed as the difference of two Poisson random variables. Its exact probability mass function (PMF) is inconvenient for use in likelihood-based iterative image reconstruction as it contains an infinite summation. The shifted Poisson model is a tractable approximation to this PMF but requires that negative values are truncated, resulting in positively biased reconstructions in low count studies. Here we analyze the properties of the exact PMF and propose a simple but accurate approximation that allows negative valued data. We investigate the properties of this approximation and demonstrate its application to penalized maximum likelihood image reconstruction.

Penalized-likelihood sinogram smoothing for low-dose CT

We have developed a sinogram smoothing approach for low-dose computed tomography (CT) that seeks to estimate the line integrals needed for reconstruction from the noisy measurements by maximizing a penalized-likelihood objective function. The maximization is performed by an algorithm derived by use of the separable paraboloidal surrogates framework. The approach overcomes some of the computational limitations of a previously proposed spline-based penalized-likelihood sinogram smoothing approach, and it is found to yield better resolution-variance tradeoffs than this spline-based approach as well an existing adaptive filtering approach. Such sinogram smoothing approaches could be valuable when applied to the low-dose data acquired in CT screening exams, such as those being considered for lung-nodule detection.

Quantitative image reconstruction in PET from emission data only using cluster analysis

Quantitative image reconstruction in positron emission tomography requires attenuation correction. In case the attenuation correction is not measured separately, under certain conditions this can be determined from the emission data alone. We present a method based on cluster analysis that assumes only 3 empirical attenuation coefficients, i.e., 0.095 cm-1 for soft tissue, 0.02 cm-1 for lung, and 0 cm-1 for air. The subsequent image reconstruction takes place in an iterative fashion, through maximization of image likelihood. For the mathematical thorax phantom used in the present study, the results are comparable to those obtained after separate measurement of the attenuation correction.

Simultaneous iterative reconstruction of emission and attenuation images in positron emission tomography from emission data only

For quantitative image reconstruction in positron emission tomography attenuation correction is mandatory. In case that no data are available for the calculation of the attenuation correction factors one can try to determine them from the emission data alone. However, it is not clear if the information content is sufficient to yield an adequate attenuation correction together with a satisfactory activity distribution. Therefore, we determined the log likelihood distribution for a thorax phantom depending on the choice of attenuation and activity pixel values to measure the crosstalk between both. In addition an iterative image reconstruction (one-dimensional Newton-type algorithm with a maximum likelihood estimator), which simultaneously reconstructs the images of the activity distribution and the attenuation coefficients is used to demonstrate the problems and possibilities of such a reconstruction. As result we show that for a change of the log likelihood in the range of statistical noise, the associated change in the activity value of a structure is between 6% and 263%. In addition, we show that it is not possible to choose the best maximum on the basis of the log likelihood when a regularization is used, because the coupling between different structures mediated by the (smoothing) regularization prevents an adequate solution due to crosstalk. We conclude that taking into account the attenuation information in the emission data improves the performance of image reconstruction with respect to the bias of the activities, however, the reconstruction still is not quantitative.

Statistical image reconstruction for polyenergetic X-ray computed tomography

This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energy-dependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energy-dependent mass attenuation coefficient. We formulate a penalized-likelihood function for this polyenergetic model and develop an ordered-subsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated X-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts.

Simultaneous iterative reconstruction for emission and attenuation images in positron emission tomography

The quality of the attenuation correction strongly influences the outcome of the reconstructed emission scan in positron emission tomography. Usually the attenuation correction factors are calculated from the transmission and blank scan and thereafter applied during the reconstruction on the emission data. However, this is not an optimal treatment of the available data, because the emission data themselves contain additional information about attenuation: The optimal treatment must use this information for the determination of the attenuation correction factors. Therefore, our purpose is to investigate a simultaneous emission and attenuation image reconstruction using a maximum likelihood estimator, which takes the attenuation information in the emission data into account. The total maximum likelihood function for emission and transmission is used to derive a one-dimensional Newton-like algorithm for the calculation of the emission and attenuation image. Log-likelihood convergence, mean differences, and the mean of squared differences for the emission image and the attenuation correction factors of a mathematical thorax phantom were determined and compared. As a result we obtain images improved with respect to log likelihood in all cases and with respect to our figures of merit in most cases. We conclude that the simultaneous reconstruction can improve the performance of image reconstruction.

Monotonic algorithms for transmission tomography

We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log likelihood. Due to the form of the log-likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events, such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity, and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.

Iterative reconstruction for attenuation correction in positron emission tomography: maximum likelihood for transmission and blank scan

The quality of the attenuation correction strongly influences the outcome of the reconstructed emission scan in positron emission tomography. The calculation of the attenuation correction factors must take into account the Poisson nature of the radioactive decay process, because-for a reasonable scan duration-the transmission measurements contain lines of response with low count numbers in the case of large attenuation factors. Our purpose in this study is to investigate a maximum likelihood estimator for attenuation correction factor calculation in positron emission tomography, which incorporates the Poisson nature of the radioactive decay into transmission and blank measurement. Therefore, the correct maximum likelihood function is used to derive two estimators for the calculation of the attenuation coefficient image and the corresponding attenuation correction factors depending on the measured blank and transmission data. Log likelihood convergence, mean differences, and the mean of squared differences for the attenuation correction factors of a mathematical thorax phantom were determined and compared. The algorithms yield adequate attenuation correction factors, however, the algorithm taking the noise in the blank scan into account can perform better for noisy blank scans. We conclude that maximum likelihood-including blank likelihood-is advantageous to reconstruct attenuation correction factors for low statistic blank and good statistic transmission data. For normal blank and transmission statistics the implementation of the statistical nature of the blank is not mandatory.

Penalized-likelihood estimators and noise analysis for randoms-precorrected PET transmission scans

This paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms-precorrected positron emission tomography (PET) measurements. The methods apply to both emission and transmission tomography, however, in this paper we focus on transmission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the shifted Poisson (SP) method avoids the systematic bias of the conventional data-weighted least squares (WLS) method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary Poisson (OP) model. We develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors (ACF's). Empirical pixel and region variances from real transmission data agree closely with the analytical predictions. Both the approximations and the empirical results show that the performance differences between the OP model and SP model are even larger, when considering noise propagation from the transmission images into the final emission images, than the differences in the attenuation maps themselves.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Accelerated iterative transmission CT reconstruction using an ordered subsets convex algorithm

Iterative maximum likelihood (ML) transmission computed tomography algorithms have distinct advantages over Fourier-based reconstruction, but unfortunately require increased computation time. The convex algorithm [1] is a relatively fast iterative ML algorithm but it is nevertheless too slow for many applications. Therefore, an acceleration of this algorithm by using ordered subsets of projections is proposed [ordered subsets convex algorithm (OSC)]. OSC applies the convex algorithm sequentially to subsets of projections. OSC was compared with the convex algorithm using simulated and physical thorax phantom data. Reconstructions were performed for OSC using eight and 16 subsets (eight and four projections/subset, respectively). Global errors, image noise, contrast recovery, and likelihood increase were calculated. Results show that OSC is faster than the convex algorithm, the amount of acceleration being approximately proportional to the number of subsets in OSC, and it causes only a slight increase of noise and global errors in the reconstructions. Images and image profiles of the reconstructions were in good agreement. In conclusion, OSC and the convex algorithm result in similar image quality but OSC is more than an order of magnitude faster.

An iterative algorithm for X-ray CT fluoroscopy

X-ray computed tomography fluoroscopy (CTF) enables image guidance of interventions, synchronization of scanning with contrast bolus arrival, and motion analysis. However, filtered backprojection (FB), the current method for CTF image reconstruction, is subject to motion and metal artifacts from implants, needles, or other surgical instruments. Reduced target lesion conspicuity may result from increased image noise associated with reduced tube current. In this report, we adapt the row-action expectation-maximization (EM) algorithm for CTF. Because time-dependent variation in images is localized during CTF, the row-action EM-like algorithm allows rapid convergence. More importantly, this iterative CTF algorithm has fewer metal artifacts and better low-contrast performance than FB.

Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction

This paper presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from low-count transmission scans. We derive the algorithms by applying to the transmission log-likelihood a version of the convexity technique developed by De Pierro for emission tomography. The new class includes the single-coordinate ascent (SCA) algorithm and Lange's convex algorithm for transmission tomography as special cases. The new grouped-coordinate ascent (GCA) algorithms in the class overcome several limitations associated with previous algorithms. 1) Fewer exponentiations are required than in the transmission maximum likelihood-expectation maximization (ML-EM) algorithm or in the SCA algorithm. 2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based methods. 3) The algorithms are easily parallelizable, unlike the SCA algorithm and perhaps line-search algorithms. We show that the GCA algorithms converge faster than the SCA algorithm, even on conventional workstations. An example from a low-count positron emission tomography (PET) transmission scan illustrates the method.

A unified approach to statistical tomography using coordinate descent optimization

Over the past years there has been considerable interest in statistically optimal reconstruction of cross-sectional images from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. While MAP estimation requires the solution of an optimization problem, most existing reconstruction algorithms take an indirect approach based on the expectation maximization (EM) algorithm. We propose a new approach to statistically optimal image reconstruction based on direct optimization of the MAP criterion. The key to this direct optimization approach is greedy pixel-wise computations known as iterative coordinate decent (ICD). We propose a novel method for computing the ICD updates, which we call ICD/Newton-Raphson. We show that ICD/Newton-Raphson requires approximately the same amount of computation per iteration as EM-based approaches, but the new method converges much more rapidly (in our experiments, typically five to ten iterations). Other advantages of the ICD/Newton-Raphson method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are easily incorporated.

Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography

Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a "plug-in" approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems.

Globally convergent algorithms for maximum a posteriori transmission tomography

This paper reviews and compares three maximum likelihood algorithms for transmission tomography. One of these algorithms is the EM algorithm, one is based on a convexity argument devised by De Pierro (see IEEE Trans. Med. Imaging, vol.12, p.328-333, 1993) in the context of emission tomography, and one is an ad hoc gradient algorithm. The algorithms enjoy desirable local and global convergence properties and combine gracefully with Bayesian smoothing priors. Preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the EM algorithm. This superiority stems from the larger number of exponentiations required by the EM algorithm. The convex and gradient algorithms are well adapted to parallel computing.