Edge-preserving tomographic reconstruction with nonlocal regularization

PET Mapping of Receptor Occupancy Using Joint Direct Parametric Reconstruction

Receptor occupancy (RO) studies using PET neuroimaging play a critical role in the development of drugs targeting the central nervous system (CNS). The conventional approach to estimate drug receptor occupancy consists in estimation of binding potential changes between two PET scans (baseline and post-drug injection). This estimation is typically performed separately for each scan by first reconstructing dynamic PET scan data before fitting a kinetic model to time activity curves. This approach fails to properly model the noise in PET measurements, resulting in poor RO estimates, especially in low receptor density regions.

OBJECTIVE

In this work, we evaluate a novel joint direct parametric reconstruction framework to directly estimate distributions of RO and other kinetic parameters in the brain from a pair of baseline and post-drug injection dynamic PET scans.

METHODS

The proposed method combines the use of regularization on RO maps with alternating optimization to enable estimation of occupancy even in low binding regions.

RESULTS

Simulation results demonstrate the quantitative improvement of this method over conventional approaches in terms of accuracy and precision of occupancy. The proposed method is also evaluated in preclinical in-vivo experiments using 11C-MK-6884 and a muscarinic acetylcholine receptor 4 positive allosteric modulator drug, showing improved estimation of receptor occupancy as compared to traditional estimators.

CONCLUSION

The proposed joint direct estimation framework improves RO estimation compared to conventional methods, especially in intermediate to low-binding regions.

SIGNIFICANCE

This work could potentially facilitate the evaluation of new drug candidates targeting the CNS.

Enhancing linguistic research through 2-mm isotropic 3D dynamic speech MRI optimized by sparse temporal sampling and low-rank reconstruction

PURPOSE

To enable a more comprehensive view of articulations during speech through near-isotropic 3D dynamic MRI with high spatiotemporal resolution and large vocal-tract coverage.

METHODS

Using partial separability model-based low-rank reconstruction coupled with a sparse acquisition of both spatial and temporal models, we are able to achieve near-isotropic resolution 3D imaging with a high frame rate. The total acquisition time of the speech acquisition is shortened by introducing a sparse temporal sampling that interleaves one temporal navigator with four randomized phase and slice-encoded imaging samples. Memory and computation time are improved through compressing coils based on the region of interest for low-rank constrained reconstruction with an edge-preserving spatial penalty.

RESULTS

The proposed method has been evaluated through experiments on several speech samples, including a standard reading passage. A near-isotropic 1.875 × 1.875 × 2 mm³ spatial resolution, 64-mm through-plane coverage, and a 35.6-fps temporal resolution are achieved. Investigations and analysis on specific speech samples support novel insights into nonsymmetric tongue movement, velum raising, and coarticulation events with adequate visualization of rapid articulatory movements.

CONCLUSION

Three-dimensional dynamic images of the vocal tract structures during speech with high spatiotemporal resolution and axial coverage is capable of enhancing linguistic research, enabling visualization of soft tissue motions that are not possible with other modalities.

Memory-Efficient Training for Fully Unrolled Deep Learned PET Image Reconstruction with Iteration-Dependent Targets

We propose a new version of the forward-backward splitting expectation-maximisation network (FBSEM-Net) along with a new memory-efficient training method enabling the training of fully unrolled implementations of 3D FBSEM-Net. FBSEM-Net unfolds the maximum a posteriori expectation-maximisation algorithm and replaces the regularisation step by a residual convolutional neural network. Both the gradient of the prior and the regularisation strength are learned from training data. In this new implementation, three modifications of the original framework are included. First, iteration-dependent networks are used to have a customised regularisation at each iteration. Second, iteration-dependent targets and losses are introduced so that the regularised reconstruction matches the reconstruction of noise-free data at every iteration. Third, sequential training is performed, making training of large unrolled networks far more memory efficient and feasible. Since sequential training permits unrolling a high number of iterations, there is no need for artificial use of the regularisation step as a leapfrogging acceleration. The results obtained on 2D and 3D simulated data show that FBSEM-Net using iteration-dependent targets and losses improves the consistency in the optimisation of the network parameters over different training runs. We also found that using iteration-dependent targets increases the generalisation capabilities of the network. Furthermore, unrolled networks using iteration-dependent regularisation allowed a slight reduction in reconstruction error compared to using a fixed regularisation network at each iteration. Finally, we demonstrate that sequential training successfully addresses potentially serious memory issues during the training of deep unrolled networks. In particular, it enables the training of 3D fully unrolled FBSEM-Net, not previously feasible, by reducing the memory usage by up to 98% compared to a conventional end-to-end training. We also note that the truncation of the backpropagation (due to sequential training) does not notably impact the network’s performance compared to conventional training with a full backpropagation through the entire network.

Abbreviated on-treatment CBCT using roughness penalized mono-energization of kV-MV data and a multi-layer MV imager

Simultaneous acquisition of cone beam CT (CBCT) projections using both the kV and MV imagers of an image guided radiotherapy system reduces set-up scan times-a benefit to lung cancer radiation oncology patients-but increases noise in the 3D reconstruction. In this article, we present a kV-MV scan time reduction technique that uses two noise-reducing measures to achieve superior performance. The first is a high-DQE multi-layer MV imager prototype. The second is a beam hardening correction algorithm which combines poly-energetic modeling with edge-preserving, regularized smoothing of the projections. Performance was tested in real acquisitions of the Catphan 604 and a thorax phantom. Percent noise was quantified from voxel values in a soft tissue volume of interest (VOI) while edge blur was quantified from a VOI straddling a boundary between air and soft material. Comparisons in noise/resolution performance trade-off were made between our proposed approach, a dose-equivalent kV-only scan, and a kV-MV reconstruction technique previously published by Yinet al(2005Med. Phys.329). The proposed technique demonstrated lower noise as a function of spatial resolution than the baseline kV-MV method, notably a 50% noise reduction at typical edge blur levels. Our proposed method also exhibited fainter non-uniformity artifacts and in some cases superior contrast. Overall, we find that the combination of a multi-layer MV imager, acquiring at a LINAC source energy of 2.5 MV, and a denoised beam hardening correction algorithm enables noise, resolution, and dose performance comparable to standard kV-imager only set-up CBCT, but with nearly half the gantry rotation time.

PWLS-PR: low-dose computed tomography image reconstruction using a patch-based regularization method based on the penalized weighted least squares total variation approach

BACKGROUND

Radiation exposure computed tomography (CT) scans and the associated risk of cancer in patients have been major clinical concerns. Existing research can achieve low-dose CT imaging by reducing the X-ray current and the number of projections per rotation of the human body. However, this method may produce excessive noise and fringe artifacts in the traditional filtered back projection (FBP)-reconstructed image.

METHODS

To solve this problem, iterative image reconstruction is a promising option to obtain high-quality images from low-dose scans. This paper proposes a patch-based regularization method based on penalized weighted least squares total variation (PWLS-PR) for iterative image reconstruction. This method uses neighborhood patches instead of single pixels to calculate the nonquadratic penalty. The proposed regularization method is more robust than the conventional regularization method in identifying random fluctuations caused by sharp edges and noise. Each iteration of the proposed algorithm can be described in the following three steps: image updating via the total variation based on penalized weighted least squares (PWLS-TV), image smoothing, and pixel-by-pixel image fusion.

RESULTS

Simulation and real-world projection experiments show that the proposed PWLS-PR algorithm achieves a higher image reconstruction performance than similar algorithms. Through the qualitative and quantitative evaluation of simulation experiments, the effectiveness of the method is also verified.

CONCLUSIONS

Furthermore, this study shows that the PWLS-PR method reduces the amount of projection data required for repeated CT scans and has the useful potential to reduce the radiation dose in clinical medical applications.

An iterative reconstruction method for sparse-projection data for low-dose CT

Reducing X-ray radiation is beneficial for reducing the risk of cancer in patients. There are two main approaches for achieving this goal namely, one is to reduce the X-ray current, and another is to apply sparse-view protocols to do image scanning and projections. However, these techniques usually lead to degradation of the reconstructed image quality, resulting in excessive noise and severe edge artifacts, which seriously affect the diagnosis result. In order to overcome such limitation, this study proposes and tests an algorithm based on guided kernel filtering. The algorithm combines the characteristics of anisotropic edges between adjacent image voxels, expresses the relevant weights with an exponential function, and adjusts the weights adaptively through local gray gradients to better preserve the image structure while suppressing noise information. Experiments show that the proposed method can effectively suppress noise and preserve the image structure. Comparing with similar algorithms, the proposed algorithm greatly improves the peak signal-to-noise ratio (PSNR), structural similarity (SSIM), and root mean square error (RMSE) of the reconstructed image. The proposed algorithm has the best effect in quantitative analysis, which verifies the effectiveness of the proposed method and good image reconstruction performance. Overall, this study demonstrates that the proposed method can reduce the number of projections required for repeated CT scans and has potential for medical applications in reducing radiation doses.

A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.

Metric-guided regularisation parameter selection for statistical iterative reconstruction in computed tomography

As iterative reconstruction in Computed Tomography (CT) is an ill-posed problem, additional prior information has to be used to get a physically meaningful result (close to ground truth if available). However, the amount of influence of the regularisation prior is crucial to the outcome of the reconstruction. Therefore, we propose a scheme for tuning the strength of the prior via a certain image metric. In this work, the parameter is tuned for minimal histogram entropy in selected regions of the reconstruction as histogram entropy is a very basic approach to characterise the information content of data. We performed a sweep over different regularisation parameters showing that the histogram entropy is a suitable metric as it is well behaved over a wide range of parameters. The parameter determination is a feedback loop approach we applied to numerically simulated FORBILD phantom data and verified with an experimental measurement of a micro-CT device. The outcome is evaluated visually and quantitatively by means of root mean squared error (RMSE) and structural similarity (SSIM) for the simulation and visually for the measured sample (no ground truth available). The final reconstructed images exhibit noise-suppressed iterative reconstruction. For both datasets, the optimisation is robust where its initial value is concerned. The parameter tuning approach shows that the proposed metric-driven feedback loop is a promising tool for finding a suitable regularisation parameter in statistical iterative reconstruction.

Recent methodology advances in fluorescence molecular tomography

Molecular imaging (MI) is a novel imaging discipline that has been continuously developed in recent years. It combines biochemistry, multimodal imaging, biomathematics, bioinformatics, cell & molecular physiology, biophysics, and pharmacology, and it provides a new technology platform for the early diagnosis and quantitative analysis of diseases, treatment monitoring and evaluation, and the development of comprehensive physiology. Fluorescence Molecular Tomography (FMT) is a type of optical imaging modality in MI that captures the three-dimensional distribution of fluorescence within a biological tissue generated by a specific molecule of fluorescent material within a biological tissue. Compared with other optical molecular imaging methods, FMT has the characteristics of high sensitivity, low cost, and safety and reliability. It has become the research frontier and research hotspot of optical molecular imaging technology. This paper took an overview of the recent methodology advances in FMT, mainly focused on the photon propagation model of FMT based on the radiative transfer equation (RTE), and the reconstruction problem solution consist of forward problem and inverse problem. We introduce the detailed technologies utilized in reconstruction of FMT. Finally, the challenges in FMT were discussed. This survey aims at summarizing current research hotspots in methodology of FMT, from which future research may benefit.

MULTI-ENERGY CONE-BEAM CT RECONSTRUCTION WITH A SPATIAL SPECTRAL NONLOCAL MEANS ALGORITHM

Multi-energy computed tomography (CT) is an emerging medical image modality with a number of potential applications in diagnosis and therapy. However, high system cost and technical barriers obstruct its step into routine clinical practice. In this study, we propose a framework to realize multi-energy cone beam CT (ME-CBCT) on the CBCT system that is widely available and has been routinely used for radiotherapy image guidance. In our method, a kVp switching technique is realized, which acquires x-ray projections with kVp levels cycling through a number of values. For this kVp-switching based ME-CBCT acquisition, x-ray projections of each energy channel are only a subset of all the acquired projections. This leads to an undersampling issue, posing challenges to the reconstruction problem. We propose a spatial spectral non-local means (ssNLM) method to reconstruct ME-CBCT, which employs image correlations along both spatial and spectral directions to suppress noisy and streak artifacts. To address the intensity scale difference at different energy channels, a histogram matching method is incorporated. Our method is different from conventionally used NLM methods in that spectral dimension is included, which helps to effectively remove streak artifacts appearing at different directions in images with different energy channels. Convergence analysis of our algorithm is provided. A comprehensive set of simulation and real experimental studies demonstrate feasibility of our ME-CBCT scheme and the capability of achieving superior image quality compared to conventional filtered backprojection-type (FBP) and NLM reconstruction methods.

Joint camera blur and pose estimation from aliased data

A joint-estimation algorithm is presented that enables simultaneous camera blur and pose estimation from a known calibration target in the presence of aliasing. Specifically, a parametric maximum-likelihood (ML) point-spread function estimate is derived for characterizing a camera's optical imperfections through the use of a calibration target in an otherwise loosely controlled environment. The imaging perspective, ambient-light levels, target reflectance, detector gain and offset, quantum efficiency, and read-noise levels are all treated as nuisance parameters. The Cramér-Rao bound is derived, and simulations demonstrate that the proposed estimator achieves near optimal mean squared error performance. The proposed method is applied to experimental data to validate the fidelity of the forward models as well as to establish the utility of the resulting ML estimates for both system identification and subsequent image restoration.

Comparison Study of Regularizations in Spectral Computed Tomography Reconstruction

The energy-resolving photon-counting detectors in spectral computed tomography (CT) can acquire projections of an object in different energy channels. In other words, they are able to reliably distinguish the received photon energies. These detectors lead to the emerging spectral CT, which is also called multi-energy CT, energy-selective CT, color CT, etc. Spectral CT can provide additional information in comparison with the conventional CT in which energy integrating detectors are used to acquire polychromatic projections of an object being investigated. The measurements obtained by X-ray CT detectors are noisy in reality, especially in spectral CT where the photon number is low in each energy channel. Therefore, some regularization should be applied to obtain a better image quality for this ill-posed problem in spectral CT image reconstruction. Quadratic-based regularizations are not often satisfactory as they blur the edges in the reconstructed images. As a result, different edge-preserving regularization methods have been adopted for reconstructing high quality images in the last decade. In this work, we numerically evaluate the performance of different regularizers in spectral CT, including total variation, non-local means and anisotropic diffusion. The goal is to provide some practical guidance to accurately reconstruct the attenuation distribution in each energy channel of the spectral CT data.

l0 regularization based on a prior image incorporated non-local means for limited-angle X-ray CT reconstruction

PURPOSES

Restricted by the scanning environment in some CT imaging modalities, the acquired projection data are usually incomplete, which may lead to a limited-angle reconstruction problem. Thus, image quality usually suffers from the slope artifacts. The objective of this study is to first investigate the distorted domains of the reconstructed images which encounter the slope artifacts and then present a new iterative reconstruction method to address the limited-angle X-ray CT reconstruction problem.

METHODS

The presented framework of new method exploits the structural similarity between the prior image and the reconstructed image aiming to compensate the distorted edges. Specifically, the new method utilizes l0 regularization and wavelet tight framelets to suppress the slope artifacts and pursue the sparsity. New method includes following 4 steps to (1) address the data fidelity using SART; (2) compensate for the slope artifacts due to the missed projection data using the prior image and modified nonlocal means (PNLM); (3) utilize l0 regularization to suppress the slope artifacts and pursue the sparsity of wavelet coefficients of the transformed image by using iterative hard thresholding (l0W); and (4) apply an inverse wavelet transform to reconstruct image. In summary, this method is referred to as "l0W-PNLM".

RESULTS

Numerical implementations showed that the presented l0W-PNLM was superior to suppress the slope artifacts while preserving the edges of some features as compared to the commercial and other popular investigative algorithms. When the image to be reconstructed is inconsistent with the prior image, the new method can avoid or minimize the distorted edges in the reconstructed images. Quantitative assessments also showed that applying the new method obtained the highest image quality comparing to the existing algorithms.

CONCLUSIONS

This study demonstrated that the presented l0W-PNLM yielded higher image quality due to a number of unique characteristics, which include that (1) it utilizes the structural similarity between the reconstructed image and prior image to modify the distorted edges by slope artifacts; (2) it adopts wavelet tight frames to obtain the first and high derivative in several directions and levels; and (3) it takes advantage of l0 regularization to promote the sparsity of wavelet coefficients, which is effective for the inhibition of the slope artifacts. Therefore, the new method can address the limited-angle CT reconstruction problem effectively and have practical significance.

PET image reconstruction using multi-parametric anato-functional priors

In this study, we investigate the application of multi-parametric anato-functional (MR-PET) priors for the maximum a posteriori (MAP) reconstruction of brain PET data in order to address the limitations of the conventional anatomical priors in the presence of PET-MR mismatches. In addition to partial volume correction benefits, the suitability of these priors for reconstruction of low-count PET data is also introduced and demonstrated, comparing to standard maximum-likelihood (ML) reconstruction of high-count data. The conventional local Tikhonov and total variation (TV) priors and current state-of-the-art anatomical priors including the Kaipio, non-local Tikhonov prior with Bowsher and Gaussian similarity kernels are investigated and presented in a unified framework. The Gaussian kernels are calculated using both voxel- and patch-based feature vectors. To cope with PET and MR mismatches, the Bowsher and Gaussian priors are extended to multi-parametric priors. In addition, we propose a modified joint Burg entropy prior that by definition exploits all parametric information in the MAP reconstruction of PET data. The performance of the priors was extensively evaluated using 3D simulations and two clinical brain datasets of [18F]florbetaben and [18F]FDG radiotracers. For simulations, several anato-functional mismatches were intentionally introduced between the PET and MR images, and furthermore, for the FDG clinical dataset, two PET-unique active tumours were embedded in the PET data. Our simulation results showed that the joint Burg entropy prior far outperformed the conventional anatomical priors in terms of preserving PET unique lesions, while still reconstructing functional boundaries with corresponding MR boundaries. In addition, the multi-parametric extension of the Gaussian and Bowsher priors led to enhanced preservation of edge and PET unique features and also an improved bias-variance performance. In agreement with the simulation results, the clinical results also showed that the Gaussian prior with voxel-based feature vectors, the Bowsher and the joint Burg entropy priors were the best performing priors. However, for the FDG dataset with simulated tumours, the TV and proposed priors were capable of preserving the PET-unique tumours. Finally, an important outcome was the demonstration that the MAP reconstruction of a low-count FDG PET dataset using the proposed joint entropy prior can lead to comparable image quality to a conventional ML reconstruction with up to 5 times more counts. In conclusion, multi-parametric anato-functional priors provide a solution to address the pitfalls of the conventional priors and are therefore likely to increase the diagnostic confidence in MR-guided PET image reconstructions.

Classification and evaluation strategies of auto-segmentation approaches for PET: Report of AAPM task group No. 211

PURPOSE

The purpose of this educational report is to provide an overview of the present state-of-the-art PET auto-segmentation (PET-AS) algorithms and their respective validation, with an emphasis on providing the user with help in understanding the challenges and pitfalls associated with selecting and implementing a PET-AS algorithm for a particular application.

APPROACH

A brief description of the different types of PET-AS algorithms is provided using a classification based on method complexity and type. The advantages and the limitations of the current PET-AS algorithms are highlighted based on current publications and existing comparison studies. A review of the available image datasets and contour evaluation metrics in terms of their applicability for establishing a standardized evaluation of PET-AS algorithms is provided. The performance requirements for the algorithms and their dependence on the application, the radiotracer used and the evaluation criteria are described and discussed. Finally, a procedure for algorithm acceptance and implementation, as well as the complementary role of manual and auto-segmentation are addressed.

FINDINGS

A large number of PET-AS algorithms have been developed within the last 20 years. Many of the proposed algorithms are based on either fixed or adaptively selected thresholds. More recently, numerous papers have proposed the use of more advanced image analysis paradigms to perform semi-automated delineation of the PET images. However, the level of algorithm validation is variable and for most published algorithms is either insufficient or inconsistent which prevents recommending a single algorithm. This is compounded by the fact that realistic image configurations with low signal-to-noise ratios (SNR) and heterogeneous tracer distributions have rarely been used. Large variations in the evaluation methods used in the literature point to the need for a standardized evaluation protocol.

CONCLUSIONS

Available comparison studies suggest that PET-AS algorithms relying on advanced image analysis paradigms provide generally more accurate segmentation than approaches based on PET activity thresholds, particularly for realistic configurations. However, this may not be the case for simple shape lesions in situations with a narrower range of parameters, where simpler methods may also perform well. Recent algorithms which employ some type of consensus or automatic selection between several PET-AS methods have potential to overcome the limitations of the individual methods when appropriately trained. In either case, accuracy evaluation is required for each different PET scanner and scanning and image reconstruction protocol. For the simpler, less robust approaches, adaptation to scanning conditions, tumor type, and tumor location by optimization of parameters is necessary. The results from the method evaluation stage can be used to estimate the contouring uncertainty. All PET-AS contours should be critically verified by a physician. A standard test, i.e., a benchmark dedicated to evaluating both existing and future PET-AS algorithms needs to be designed, to aid clinicians in evaluating and selecting PET-AS algorithms and to establish performance limits for their acceptance for clinical use. The initial steps toward designing and building such a standard are undertaken by the task group members.

Local and Non-local Regularization Techniques in Emission (PET/SPECT) Tomographic Image Reconstruction Methods

Emission tomographic image reconstruction is an ill-posed problem due to limited and noisy data and various image-degrading effects affecting the data and leads to noisy reconstructions. Explicit regularization, through iterative reconstruction methods, is considered better to compensate for reconstruction-based noise. Local smoothing and edge-preserving regularization methods can reduce reconstruction-based noise. However, these methods produce overly smoothed images or blocky artefacts in the final image because they can only exploit local image properties. Recently, non-local regularization techniques have been introduced, to overcome these problems, by incorporating geometrical global continuity and connectivity present in the objective image. These techniques can overcome drawbacks of local regularization methods; however, they also have certain limitations, such as choice of the regularization function, neighbourhood size or calibration of several empirical parameters involved. This work compares different local and non-local regularization techniques used in emission tomographic imaging in general and emission computed tomography in specific for improved quality of the resultant images.

Maximum-Likelihood Joint Image Reconstruction/Motion Estimation in Attenuation-Corrected Respiratory Gated PET/CT Using a Single Attenuation Map

This work provides an insight into positron emission tomography (PET) joint image reconstruction/motion estimation (JRM) by maximization of the likelihood, where the probabilistic model accounts for warped attenuation. Our analysis shows that maximum-likelihood (ML) JRM returns the same reconstructed gates for any attenuation map (μ-map) that is a deformation of a given μ-map, regardless of its alignment with the PET gates. We derived a joint optimization algorithm accordingly, and applied it to simulated and patient gated PET data. We first evaluated the proposed algorithm on simulations of respiratory gated PET/CT data based on the XCAT phantom. Our results show that independently of which μ-map is used as input to JRM: (i) the warped μ-maps correspond to the gated μ-maps, (ii) JRM outperforms the traditional post-registration reconstruction and consolidation (PRRC) for hot lesion quantification and (iii) reconstructed gated PET images are similar to those obtained with gated μ-maps. This suggests that a breath-held μ-map can be used. We then applied JRM on patient data with a μ-map derived from a breath-held high resolution CT (HRCT), and compared the results with PRRC, where each reconstructed PET image was obtained with a corresponding cine-CT gated μ-map. Results show that JRM with breath-held HRCT achieves similar reconstruction to that using PRRC with cine-CT. This suggests a practical low-dose solution for implementation of motion-corrected respiratory gated PET/CT.

Sparsity-assisted solution to the twin image problem in phase retrieval

The problem of iterative phase retrieval from Fourier transform magnitude data for complex-valued objects is known to suffer from the twin image problem. In particular, when the object support is centrosymmetric, the iterative solution often stagnates such that the resultant complex image contains the features of both the desired solution and its inverted and complex-conjugated replica. In this work we make an important observation that the ideal solution without the twin image is typically more sparse in some suitable transform domain as compared to the stagnated solution. We further show that introducing a sparsity-enhancing step in the iterative algorithm can address the twin image problem without the need to change the object support throughout the iterative process even when the object support is centrosymmetric. In a simulation study, we use binary and gray-scale pure phase objects and illustrate the effectiveness of the sparsity-assisted phase recovery in the context of the twin image problem.

Iterative image reconstruction for ultra-low-dose CT with a combined low-mAs and sparse-view protocol

Ultra-low-dose x-ray computed tomography (CT) imaging is needed in CT fields. Through a scan protocol by lowering the milliampere-seconds (mAs) and reducing the number of projections per rotation around the body, we can realize low-dose CT imaging. However, the resulting noisy and insufficient measurements will unavoidably cause the degradation of desired-image. To solve this problem, iterative image reconstruction is a promising choice for achieving high-quality image with a low-dose scan. In this study, we are focusing on ultra-low-dose CT image reconstruction by using penalized weighted least-square (PWLS) criteria with a combined low-mAs and sparse-view protocol. Specifically, the sinogram data acquired with a combined low-mAs and sparse-view protocol is first restored by using a PWLS based sinogram restoration method. Then, the restored sinogram data is hereafter used to reconstruct image by using a PWLS based total variation (PWLS-TV) method. Qualitative and quantitative evaluations by simulations were carried out to validate the present method.

Compressive sensing in medical imaging

The promise of compressive sensing, exploitation of compressibility to achieve high quality image reconstructions with less data, has attracted a great deal of attention in the medical imaging community. At the Compressed Sensing Incubator meeting held in April 2014 at OSA Headquarters in Washington, DC, presentations were given summarizing some of the research efforts ongoing in compressive sensing for x-ray computed tomography and magnetic resonance imaging systems. This article provides an expanded version of these presentations. Sparsity-exploiting reconstruction algorithms that have gained popularity in the medical imaging community are studied, and examples of clinical applications that could benefit from compressive sensing ideas are provided. The current and potential future impact of compressive sensing on the medical imaging field is discussed.

A Novel Region Reconstruction Method for Fluorescence Molecular Tomography

Fluorescence molecular tomography (FMT) could exploit the distribution of fluorescent biomarkers that target tumors accurately and effectively, which enables noninvasive real-time 3-D visualization as well as quantitative analysis of small tumors in small animal studies in vivo. Due to the difficulties of reconstruction, continuous efforts are being made to find more practical and efficient approaches to accurately obtain the characteristics of fluorescent regions inside biological tissues. In this paper, we propose a region reconstruction method for FMT, which is defined as an L1-norm regularization piecewise constant level set approach. The proposed approach adopts a priori information including the sparsity of the fluorescent sources and the fluorescent contrast between the target and background. When the contrast of different fluorescent sources is low to a certain degree, our approach can simultaneously solve the detection and characterization problems for the reconstruction of FMT. To evaluate the performance of the region reconstruction method, numerical phantom experiments and in vivo bead-implanted mouse experiments were performed. The results suggested that the proposed region reconstruction method was able to reconstruct the features of the fluorescent regions accurately and effectively, and the proposed method was able to be feasibly adopted in in vivo application.

Acceleration of MAP-EM algorithm via over-relaxation

To improve the convergence rate of the effective maximum a posteriori expectation-maximization (MAP-EM) algorithm in tomographic reconstructions, this study proposes a modified MAP-EM which uses an over-relaxation factor to accelerate image reconstruction. The proposed method, called MAP-AEM, is evaluated and compared with the results for MAP-EM and for an ordered-subset algorithm, in terms of the convergence rate and noise properties. The results show that the proposed method converges numerically much faster than MAP-EM and with a speed that is comparable to that for an ordered-subset type method. The proposed method is effective in accelerating MAP-EM tomographic reconstruction.

Low-dose X-ray computed tomography image reconstruction with a combined low-mAs and sparse-view protocol

To realize low-dose imaging in X-ray computed tomography (CT) examination, lowering milliampere-seconds (low-mAs) or reducing the required number of projection views (sparse-view) per rotation around the body has been widely studied as an easy and effective approach. In this study, we are focusing on low-dose CT image reconstruction from the sinograms acquired with a combined low-mAs and sparse-view protocol and propose a two-step image reconstruction strategy. Specifically, to suppress significant statistical noise in the noisy and insufficient sinograms, an adaptive sinogram restoration (ASR) method is first proposed with consideration of the statistical property of sinogram data, and then to further acquire a high-quality image, a total variation based projection onto convex sets (TV-POCS) method is adopted with a slight modification. For simplicity, the present reconstruction strategy was termed as "ASR-TV-POCS." To evaluate the present ASR-TV-POCS method, both qualitative and quantitative studies were performed on a physical phantom. Experimental results have demonstrated that the present ASR-TV-POCS method can achieve promising gains over other existing methods in terms of the noise reduction, contrast-to-noise ratio, and edge detail preservation.

Sparse-view x-ray CT reconstruction via total generalized variation regularization

Sparse-view CT reconstruction algorithms via total variation (TV) optimize the data iteratively on the basis of a noise- and artifact-reducing model, resulting in significant radiation dose reduction while maintaining image quality. However, the piecewise constant assumption of TV minimization often leads to the appearance of noticeable patchy artifacts in reconstructed images. To obviate this drawback, we present a penalized weighted least-squares (PWLS) scheme to retain the image quality by incorporating the new concept of total generalized variation (TGV) regularization. We refer to the proposed scheme as 'PWLS-TGV' for simplicity. Specifically, TGV regularization utilizes higher order derivatives of the objective image, and the weighted least-squares term considers data-dependent variance estimation, which fully contribute to improving the image quality with sparse-view projection measurement. Subsequently, an alternating optimization algorithm was adopted to minimize the associative objective function. To evaluate the PWLS-TGV method, both qualitative and quantitative studies were conducted by using digital and physical phantoms. Experimental results show that the present PWLS-TGV method can achieve images with several noticeable gains over the original TV-based method in terms of accuracy and resolution properties.

Iterative reconstruction for x-ray computed tomography using prior-image induced nonlocal regularization

Repeated X-ray computed tomography (CT) scans are often required in several specific applications such as perfusion imaging, image-guided biopsy needle, image-guided intervention, and radiotherapy with noticeable benefits. However, the associated cumulative radiation dose significantly increases as comparison with that used in the conventional CT scan, which has raised major concerns in patients. In this study, to realize radiation dose reduction by reducing the X-ray tube current and exposure time (mAs) in repeated CT scans, we propose a prior-image induced nonlocal (PINL) regularization for statistical iterative reconstruction via the penalized weighted least-squares (PWLS) criteria, which we refer to as "PWLS-PINL". Specifically, the PINL regularization utilizes the redundant information in the prior image and the weighted least-squares term considers a data-dependent variance estimation, aiming to improve current low-dose image quality. Subsequently, a modified iterative successive overrelaxation algorithm is adopted to optimize the associative objective function. Experimental results on both phantom and patient data show that the present PWLS-PINL method can achieve promising gains over the other existing methods in terms of the noise reduction, low-contrast object detection, and edge detail preservation.

Threshold choices of Huber regularization using global- and local-edge-detecting operators for X-ray computed tomographic reconstruction

Statistical iterative reconstruction (SIR) approaches have shown great potential in x-ray computed tomographic (CT) reconstruction in the case of low-dose protocol. For yielding high quality image, an edge-preserving regularization should be incorporated into the objective function of SIR approaches. A typical example is the Huber regularization with an edge-preserving non-quadratic potential function which increases less rapidly than the quadratic potential function for sufficiently large arguments. However, a major drawback of the Huber regularization is the determining the threshold, which precludes its extensive applications. In this paper, we investigate both global- and local- edge-detecting operators for threshold choices of Huber regularization and apply them to SIR CT image reconstruction with low-dose scan protocol. Experiments were performed on XCAT phantom by using a CT simulator to obtain the low-dose projection data.

Penalized likelihood PET image reconstruction using patch-based edge-preserving regularization

Iterative image reconstruction for positron emission tomography (PET) can improve image quality by using spatial regularization that penalizes image intensity difference between neighboring pixels. The most commonly used quadratic penalty often oversmoothes edges and fine features in reconstructed images. Nonquadratic penalties can preserve edges but often introduce piece-wise constant blocky artifacts and the results are also sensitive to the hyper-parameter that controls the shape of the penalty function. This paper presents a patch-based regularization for iterative image reconstruction that uses neighborhood patches instead of individual pixels in computing the nonquadratic penalty. The new regularization is more robust than the conventional pixel-based regularization in differentiating sharp edges from random fluctuations due to noise. An optimization transfer algorithm is developed for the penalized maximum likelihood estimation. Each iteration of the algorithm can be implemented in three simple steps: an EM-like image update, an image smoothing and a pixel-by-pixel image fusion. Computer simulations show that the proposed patch-based regularization can achieve higher contrast recovery for small objects without increasing background variation compared with the quadratic regularization. The reconstruction is also more robust to the hyper-parameter than conventional pixel-based nonquadratic regularizations. The proposed regularization method has been applied to real 3-D PET data.

Reconstruction of electrical impedance tomography (EIT) images based on the expectation maximum (EM) method

Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative values in the solution. The negativity of the solution produces artifacts in reconstructed images in presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of choosing parameters. Simulation and experimental results indicate that the reconstructed images with higher quality can be obtained by the EM method, compared with the traditional Tikhonov and conjugate gradient (CG) methods, even with non-negative processing.

A parametric level-set approach to simultaneous object identification and background reconstruction for dual-energy computed tomography

Dual-energy computerized tomography has gained great interest because of its ability to characterize the chemical composition of a material rather than simply providing relative attenuation images as in conventional tomography. The purpose of this paper is to introduce a novel polychromatic dual-energy processing algorithm, with an emphasis on detection and characterization of piecewise constant objects embedded in an unknown cluttered background. Physical properties of the objects, particularly the Compton scattering and photoelectric absorption coefficients, are assumed to be known with some level of uncertainty. Our approach is based on a level-set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object and the fundamental physics-based limitations associated with our ability to jointly recover the Compton scattering and photoelectric absorption properties of the scene. In the absence of an object with appropriate physical properties, our approach returns a null characteristic function and, thus, can be viewed as simultaneously solving the detection and characterization problems. Unlike the vast majority of methods that define the level-set function nonparametrically, i.e., as a dense set of pixel values, we define our level set parametrically via radial basis functions and employ a Gauss-Newton-type algorithm for cost minimization. Numerical results show that the algorithm successfully detects objects of interest, finds their shape and location, and gives an adequate reconstruction of the background.

Noise-resolution tradeoffs in x-ray CT imaging: a comparison of penalized alternating minimization and filtered backprojection algorithms

PURPOSE

In comparison with conventional filtered backprojection (FBP) algorithms for x-ray computed tomography (CT) image reconstruction, statistical algorithms directly incorporate the random nature of the data and do not assume CT data are linear, noiseless functions of the attenuation line integral. Thus, it has been hypothesized that statistical image reconstruction may support a more favorable tradeoff than FBP between image noise and spatial resolution in dose-limited applications. The purpose of this study is to evaluate the noise-resolution tradeoff for the alternating minimization (AM) algorithm regularized using a nonquadratic penalty function.

METHODS

Idealized monoenergetic CT projection data with Poisson noise were simulated for two phantoms with inserts of varying contrast (7%-238%) and distance from the field-of-view (FOV) center (2-6.5 cm). Images were reconstructed for the simulated projection data by the FBP algorithm and two penalty function parameter values of the penalized AM algorithm. Each algorithm was run with a range of smoothing strengths to allow quantification of the noise-resolution tradeoff curve. Image noise is quantified as the standard deviation in the water background around each contrast insert. Modulation transfer functions (MTFs) were calculated from six-parameter model fits to oversampled edge-spread functions defined by the circular contrast-insert edges as a metric of local resolution. The integral of the MTF up to 0.5 1p/mm was adopted as a single-parameter measure of local spatial resolution.

RESULTS

The penalized AM algorithm noise-resolution tradeoff curve was always more favorable than that of the FBP algorithm. While resolution and noise are found to vary as a function of distance from the FOV center differently for the two algorithms, the ratio of noises when matching the resolution metric is relatively uniform over the image. The ratio of AM-to-FBP image variances, a predictor of dose-reduction potential, was strongly dependent on the shape of the AM's nonquadratic penalty function and was also strongly influenced by the contrast of the insert for which resolution is quantified. Dose-reduction potential, reported here as the fraction (%) of FBP dose necessary for AM to reconstruct an image with comparable noise and resolution, for one penalty parameter value of the AM algorithm was found to vary from 70% to 50% for low-contrast and high-contrast structures, respectively, and from 70% to 10% for the second AM penalty parameter value. However, the second penalty, AM-700, was found to suffer from poor low-contrast resolution when matching the high-contrast resolution metric with FBP.

CONCLUSIONS

The results of this simulation study imply that penalized AM has the potential to reconstruct images with similar noise and resolution using a fraction (10%-70%) of the FBP dose. However, this dose-reduction potential depends strongly on the AM penalty parameter and the contrast magnitude of the structures of interest. In addition, the authors' results imply that the advantage of AM can be maximized by optimizing the nonquadratic penalty function to the specific imaging task of interest. Future work will extend the methods used here to quantify noise and resolution in images reconstructed from real CT data.

Simultaneous segmentation and reconstruction: a level set method approach for limited view computed tomography

PURPOSE

An iterative tomographic reconstruction algorithm that simultaneously segments and reconstructs the reconstruction domain is proposed and applied to tomographic reconstructions from a sparse number of projection images.

METHODS

The proposed algorithm uses a two-phase level set method segmentation in conjunction with an iterative tomographic reconstruction to achieve simultaneous segmentation and reconstruction. The simultaneous segmentation and reconstruction is achieved by alternating between level set function evolutions and per-region intensity value updates. To deal with the limited number of projections, a priori information about the reconstruction is enforced via penalized likelihood function. Specifically, smooth function within each region (piecewise smooth function) and bounded function intensity values for each region are assumed. Such a priori information is formulated into a quadratic objective function with linear bound constraints. The level set function evolutions are achieved by artificially time evolving the level set function in the negative gradient direction; the intensity value updates are achieved by using the gradient projection conjugate gradient algorithm.

RESULTS

The proposed simultaneous segmentation and reconstruction results were compared to "conventional" iterative reconstruction (with no segmentation), iterative reconstruction followed by segmentation, and filtered backprojection. Improvements of 6%-13% in the normalized root mean square error were observed when the proposed algorithm was applied to simulated projections of a numerical phantom and to real fan-beam projections of the Catphan phantom, both of which did not satisfy the a priori assumptions.

CONCLUSIONS

The proposed simultaneous segmentation and reconstruction resulted in improved reconstruction image quality. The algorithm correctly segments the reconstruction space into regions, preserves sharp edges between different regions, and smoothes the noise within each region. The proposed algorithm framework has the flexibility to be adapted to different a priori constraints while maintaining the benefits achieved by the simultaneous segmentation and reconstruction.

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.

Bayesian statistical reconstruction for low-dose X-ray computed tomography using an adaptive-weighting nonlocal prior

How to reduce the radiation dose delivered to the patients has always been a important concern since the introduction of computed tomography (CT). Though clinically desired, low-dose CT images can be severely degraded by the excessive quantum noise under extremely low X-ray dose circumstances. Bayesian statistical reconstructions outperform the traditional filtered back-projection (FBP) reconstructions by accurately expressing the system models of physical effects and the statistical character of the measurement data. This work aims to improve the image quality of low-dose CT images using a novel AW nonlocal (adaptive-weighting nonlocal) prior statistical reconstruction approach. Compared to traditional local priors, the proposed prior can adaptively and selectively exploit the global image information. It imposes an effective resolution-preserving and noise-removing regularization for reconstructions. Experimentation validates that the reconstructions using the proposed prior have excellent performance for X-ray CT with low-dose scans.

Highly undersampled magnetic resonance image reconstruction via homotopic l(0) -minimization

In clinical magnetic resonance imaging (MRI), any reduction in scan time offers a number of potential benefits ranging from high-temporal-rate observation of physiological processes to improvements in patient comfort. Following recent developments in compressive sensing (CS) theory, several authors have demonstrated that certain classes of MR images which possess sparse representations in some transform domain can be accurately reconstructed from very highly undersampled K-space data by solving a convex l(1) -minimization problem. Although l(1)-based techniques are extremely powerful, they inherently require a degree of over-sampling above the theoretical minimum sampling rate to guarantee that exact reconstruction can be achieved. In this paper, we propose a generalization of the CS paradigm based on homotopic approximation of the l(0) quasi-norm and show how MR image reconstruction can be pushed even further below the Nyquist limit and significantly closer to the theoretical bound. Following a brief review of standard CS methods and the developed theoretical extensions, several example MRI reconstructions from highly undersampled K-space data are presented.

Regularized field map estimation in MRI

In fast magnetic resonance (MR) imaging with long readout times, such as echo-planar imaging (EPI) and spiral scans, it is important to correct for the effects of field inhomogeneity to reduce image distortion and blurring. Such corrections require an accurate field map, a map of the off-resonance frequency at each voxel. Standard field map estimation methods yield noisy field maps, particularly in image regions with low spin density. This paper describes regularized methods for field map estimation from two or more MR scans having different echo times. These methods exploit the fact that field maps are generally smooth functions. The methods use algorithms that decrease monotonically a regularized least-squares cost function, even though the problem is highly nonlinear. Results show that the proposed regularized methods significantly improve the quality of field map estimates relative to conventional unregularized methods.

Level set method for positron emission tomography

In positron emission tomography (PET), a radioactive compound is injected into the body to promote a tissue-dependent emission rate. Expectation maximization (EM) reconstruction algorithms are iterative techniques which estimate the concentration coefficients that provide the best fitted solution, for example, a maximum likelihood estimate. In this paper, we combine the EM algorithm with a level set approach. The level set method is used to capture the coarse scale information and the discontinuities of the concentration coefficients. An intrinsic advantage of the level set formulation is that anatomical information can be efficiently incorporated and used in an easy and natural way. We utilize a multiple level set formulation to represent the geometry of the objects in the scene. The proposed algorithm can be applied to any PET configuration, without major modifications.

PET Image Reconstruction Using Mumford-Shah Regularization Coupled with L¹Data Fitting

In positron emission tomography (PET) image reconstruction, regularization methods are usually considered to suppress noise effects in reconstructed images. In this paper, we model this reconstruction problem in a new variational framework where the Mumford-Shah (MS) regularization coupled with recently developed L¹ data fidelity term is adapted. In order to simplify the numerical computation, Ambrosio and Tortorelli's Gamma-convergence approximation is also employed to substitute the irregular parts (edge set) of MS functionals with the auxiliary smooth function. In numerical studies we compare our method with the "conventional" penalized least-square (PLS) algorithm with local nonquadratic Huber penalty. Results show both the feasibility and efficiency of the proposed algorithm.

Structure-from-motion without correspondence from tomographic projections by Bayesian inversion theory

In conventional tomography, the interior of an object is reconstructed from tomographic projections such as X-ray or transmission electron microscope images. All the current reconstruction methods assume that projection geometry of the imaging device is either known or solved in advance by using e.g., fiducial or nonfiducial feature points in the images. In this paper, we propose a novel approach where the imaging geometry is solved simultaneously with the volume reconstruction problem while no correspondence information is needed. Our approach is a direct application of Bayesian inversion theory and produces the maximum likelihood or maximum a posteriori estimates for the motion parameters under the selected noise and prior distributions. In this paper, the method is implemented for a two-dimensional model problem with one-dimensional affine projection data. The performance of the method is tested with simulated and measured X-ray projection data.

Level-set algorithm for the reconstruction of functional activation in near-infrared spectroscopic imaging

We introduce a new algorithm for the reconstruction of functional brain activations from near-infrared spectroscopic imaging (NIRSI) data. While NIRSI offers remarkable biochemical specificity, the attainable spatial resolution with this technique is rather limited, mainly due to the highly scattering nature of brain tissue and the low number of measurement channels. Our approach exploits the support-limited (spatially concentrated) nature of the activations to make the reconstruction problem well-posed. The new algorithm considers both the support and the function values of the activations as unknowns and estimates them from the data. The support of the activations is represented using a level-set scheme. We use a two-step alternating iterative scheme to solve for the activations. Since our approach uses the inherent nature of functional activations to make the problem well-posed, it provides reconstructions with better spatial resolution, fewer artifacts, and is more robust to noise than existing techniques. Numerical simulations and experimental data indicate a significant improvement in the quality (resolution and robustness to noise) over standard techniques such as truncated conjugate gradients (TCG) and simultaneous iterative reconstruction technique (SIRT) algorithms. Furthermore, results on experimental data obtained from simultaneous functional magnetic resonance imaging (fMRI) and optical measurements show much closer agreement of the optical reconstruction using the new approach with fMRI images than TCG and SIRT.

Bayesian algorithms for PET image reconstruction with mean curvature and Gauss curvature diffusion regularizations

The basic mathematical problem behind PET is an inverse problem. Due to the inherent ill-posedness of this inverse problem, the reconstructed images will have noise and edge artifacts. A roughness penalty is often imposed on the solution to control noise and stabilize the solution, but the difficulty is to avoid the smoothing of edges. In this paper, we propose two new types of Bayesian one-step-late reconstruction approaches which utilize two different prior regularizations: the mean curvature (MC) diffusion function and the Gauss curvature (GC) diffusion function. As they have been studied in image processing for removing noise, these two prior regularizations encourage preserving the edge while the reconstructed images are smoothed. Moreover, the GC constraint can preserve smaller structures which cannot be preserved by MC. The simulation results show that the proposed algorithms outperform the quadratic function and total variation approaches in terms of preserving the edges during emission reconstruction.

FIRST: Fast Iterative Reconstruction Software for (PET) tomography

Small animal PET scanners require high spatial resolution and good sensitivity. To reconstruct high-resolution images in 3D-PET, iterative methods, such as OSEM, are superior to analytical reconstruction algorithms, although their high computational cost is still a serious drawback. The higher performance of modern computers could make iterative image reconstruction fast enough to be viable, provided we are able to deal with the large number of probability coefficients for the system response matrix in high-resolution PET scanners, which is a difficult task that prevents the algorithms from reaching peak computing performance. Considering all possible axial and in-plane symmetries, as well as certain quasi-symmetries, we have been able to reduce the memory requirements to store the system response matrix (SRM) well below 1 GB, which allows us to keep the whole response matrix of the system inside RAM of ordinary industry-standard computers, so that the reconstruction algorithm can achieve near peak performance. The elements of the SRM are stored as cubic spline profiles and matched to voxel size during reconstruction. In this way, the advantages of 'on-the-fly' calculation and of fully stored SRM are combined. The on-the-fly part of the calculation (matching the profile functions to voxel size) of the SRM accounts for 10-30% of the reconstruction time, depending on the number of voxels chosen. We tested our approach with real data from a commercial small animal PET scanner. The results (image quality and reconstruction time) show that the proposed technique is a feasible solution.

Iterative reconstruction techniques in emission computed tomography

In emission tomography statistically based iterative methods can improve image quality relative to analytic image reconstruction through more accurate physical and statistical modelling of high-energy photon production and detection processes. Continued exponential improvements in computing power, coupled with the development of fast algorithms, have made routine use of iterative techniques practical, resulting in their increasing popularity in both clinical and research environments. Here we review recent progress in developing statistically based iterative techniques for emission computed tomography. We describe the different formulations of the emission image reconstruction problem and their properties. We then describe the numerical algorithms that are used for optimizing these functions and illustrate their behaviour using small scale simulations.