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

Logarithmic total variation regularization via preconditioned conjugate gradient method for sparse reconstruction of bioluminescence tomography

BACKGROUND AND OBJECTIVE

Bioluminescence Tomography (BLT) is a powerful optical molecular imaging technique that enables the noninvasive investigation of dynamic biological phenomena. It aims to reconstruct the three-dimensional spatial distribution of bioluminescent sources from optical measurements collected on the surface of the imaged object. However, BLT reconstruction is a challenging ill-posed problem due to the scattering effect of light and the limitations in detecting surface photons, which makes it difficult for existing methods to achieve satisfactory reconstruction results. In this study, we propose a novel method for sparse reconstruction of BLT based on a preconditioned conjugate gradient with logarithmic total variation regularization (PCG-logTV).

METHOD

This PCG-logTV method incorporates the sparsity of overlapping groups and enhances the sparse structure of these groups using logarithmic functions, which can preserve edge features and achieve more stable reconstruction results in BLT. To accelerate the convergence of the algorithm solution, we use the preconditioned conjugate gradient iteration method on the objective function and obtain the reconstruction results. We demonstrate the performance of our proposed method through numerical simulations and in vivo experiment.

RESULTS AND CONCLUSIONS

The results show that the PCG-logTV method obtains the most accurate reconstruction results, and the minimum position error (LE) is 0.254mm, which is 26%, 31% and 34% of the FISTA (0.961), IVTCG (0.81) and L1-TV (0.739) methods, and the root mean square error (RMSE) and relative intensity error (RIE) are the smallest, indicating that it is closest to the real light source. In addition, compared with the other three methods, the PCG-logTV method also has the highest DICE similarity coefficient, which is 0.928, which means that this method can effectively reconstruct the three-dimensional spatial distribution of bioluminescent light sources, has higher resolution and robustness, and is beneficial to the preclinical and clinical studies of BLT.

The SNR of time-of-flight positron emission tomography data for joint reconstruction of the activity and attenuation images

Objective.Measurement of the time-of-flight (TOF) difference of each coincident pair of photons increases the effective sensitivity of positron emission tomography (PET). Many authors have analyzed the benefit of TOF for quantification and hot spot detection in the reconstructed activity images. However, TOF not only improves the effective sensitivity, it also enables the joint reconstruction of the tracer concentration and attenuation images. This can be used to correct for errors in CT- or MR-derived attenuation maps, or to apply attenuation correction without the help of a second modality. This paper presents an analysis of the effect of TOF on the variance of the jointly reconstructed attenuation and (attenuation corrected) tracer concentration images.Approach.The analysis is performed for PET systems that have a distribution of possibly non-Gaussian TOF-kernels, and includes the conventional Gaussian TOF-kernel as a special case. Non-Gaussian TOF-kernels are often observed in novel detector designs, which make use of two (or more) different mechanisms to convert the incoming 511 keV photon to optical photons. The analytical result is validated with a simple 2D simulation.Main results.We show that if two different TOF-kernels are equivalent for image reconstruction with known attenuation, then they are also equivalent for joint reconstruction of the activity and the attenuation images. The variance increase in the activity, caused by also jointly reconstructing the attenuation image, vanishes when the TOF-resolution approaches perfection.Significance.These results are of interest for PET detector development and for the development of stand-alone PET systems.

Efficient Regularized Field Map Estimation in 3D MRI

Magnetic field inhomogeneity estimation is important in some types of magnetic resonance imaging (MRI), including field-corrected reconstruction for fast MRI with long readout times, and chemical shift based water-fat imaging. Regularized field map estimation methods that account for phase wrapping and noise involve nonconvex cost functions that require iterative algorithms. Most existing minimization techniques were computationally or memory intensive for 3D datasets, and are designed for single-coil MRI. This paper considers 3D MRI with optional consideration of coil sensitivity, and addresses the multi-echo field map estimation and water-fat imaging problem. Our efficient algorithm uses a preconditioned nonlinear conjugate gradient method based on an incomplete Cholesky factorization of the Hessian of the cost function, along with a monotonic line search. Numerical experiments show the computational advantage of the proposed algorithm over state-of-the-art methods with similar memory requirements.

Review of CT image reconstruction open source toolkits

Computed tomography (CT) has been widely applied in medical diagnosis, nondestructive evaluation, homeland security, and other science and engineering applications. Image reconstruction is one of the core CT imaging technologies. In this review paper, we systematically reviewed the currently publicly available CT image reconstruction open source toolkits in the aspects of their environments, object models, imaging geometries, and algorithms. In addition to analytic and iterative algorithms, deep learning reconstruction networks and open codes are also reviewed as the third category of reconstruction algorithms. This systematic summary of the publicly available software platforms will help facilitate CT research and development.

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.

PWLS-ULTRA: An Efficient Clustering and Learning-Based Approach for Low-Dose 3D CT Image Reconstruction

The development of computed tomography (CT) image reconstruction methods that significantly reduce patient radiation exposure, while maintaining high image quality is an important area of research in low-dose CT imaging. We propose a new penalized weighted least squares (PWLS) reconstruction method that exploits regularization based on an efficient Union of Learned TRAnsforms (PWLS-ULTRA). The union of square transforms is pre-learned from numerous image patches extracted from a dataset of CT images or volumes. The proposed PWLS-based cost function is optimized by alternating between a CT image reconstruction step, and a sparse coding and clustering step. The CT image reconstruction step is accelerated by a relaxed linearized augmented Lagrangian method with ordered-subsets that reduces the number of forward and back projections. Simulations with 2-D and 3-D axial CT scans of the extended cardiac-torso phantom and 3-D helical chest and abdomen scans show that for both normal-dose and low-dose levels, the proposed method significantly improves the quality of reconstructed images compared to PWLS reconstruction with a nonadaptive edge-preserving regularizer. PWLS with regularization based on a union of learned transforms leads to better image reconstructions than using a single learned square transform. We also incorporate patch-based weights in PWLS-ULTRA that enhance image quality and help improve image resolution uniformity. The proposed approach achieves comparable or better image quality compared to learned overcomplete synthesis dictionaries, but importantly, is much faster (computationally more efficient).

Experimental study to optimize configurations of PCD Spectral CT

BACKGROUND

High dose efficiency of photon counting detector based spectral CT (PCD-SCT) and its value in some clinical diagnosis have been well acknowledged. However, it has not been widely adopted in practical use for medical diagnosis and security inspection.

OBJECTIVE

To evaluate the influence on PCD-SCT from multiple aspects including the number of energy channels, k-edge materials, energy thresholding, basis functions in spectral information decomposition, and the combined optimal setting for these parameters and configurations.

METHODS

Basis material decomposition after spatial reconstruction is applied for PCD-SCT. A "one-step" synthesis method, merging decomposition with synthesis, is proposed to obtain virtual monochromatic images. An I-RMSE is computed using the bias part of I-RMSE to describe the difference of a synthesized signal from ground truth and the standard deviation part of I-RMSE to express the noise level. In addition, virtual monochromatic images commonly used in the medical area are also synthesized. Both numerical simulations and practical experiments are conducted for validation.

RESULTS

Results indicated that the I-RMSE for matters significantly reduced with an increased number of energy channels compared with dual-energy channel. The maximum reduction is 6% for triple-, 18% for quadruple-and 24% for quintuple-energy, respectively. However, the improvement is not linear, and also slows down after the number of energy channels reaches a certain number. Contrast agents of high concentration can introduce up to 50% error to surrounding matters. Moreover, different energy partitions influence the total error, which demonstrates the necessity of energy threshold optimization. Last, the optimal basis-material combination varies according to targeted imaging matters and the interested monochromatic energies.

CONCLUSIONS

Gain from more energy channels could be significant with the increase of energy channel number. Introduction of contrast agents in scanned objects will increase overall error in spectral CT imaging. Energy thresholding optimization is beneficial for information recovery. Moreover, the choice of basis materials could also be important to obtain low noise results. With these studies of the effect from various configurations for PCD-SCT, one may optimize the configuration of PCD-SCT accordingly.

Improved Simultaneous Algebraic Reconstruction Technique Algorithm for Positron-Emission Tomography Image Reconstruction via Minimizing the Fast Total Variation

CONTEXT

There has been considerable progress in the instrumentation for data measurement and computer methods for generating images of measured PET data. These computer methods have been developed to solve the inverse problem, also known as the "image reconstruction from projections" problem.

AIM

In this paper, we propose a modified Simultaneous Algebraic Reconstruction Technique (SART) algorithm to improve the quality of image reconstruction by incorporating total variation (TV) minimization into the iterative SART algorithm.

METHODOLOGY

The SART updates the estimated image by forward projecting the initial image onto the sinogram space. Then, the difference between the estimated sinogram and the given sinogram is back-projected onto the image domain. This difference is then subtracted from the initial image to obtain a corrected image. Fast total variation (FTV) minimization is applied to the image obtained in the SART step. The second step is the result obtained from the previous FTV update. The SART and the FTV minimization steps run iteratively in an alternating manner. Fifty iterations were applied to the SART algorithm used in each of the regularization-based methods. In addition to the conventional SART algorithm, spatial smoothing was used to enhance the quality of the image. All images were sized at 128 × 128 pixels.

RESULTS

The proposed algorithm successfully accomplished edge preservation. A detailed scrutiny revealed that the reconstruction algorithms differed; for example, the SART and the proposed FTV-SART algorithm effectively preserved the hot lesion edges, whereas artifacts and deviations were more likely to occur in the ART algorithm than in the other algorithms.

CONCLUSIONS

Compared to the standard SART, the proposed algorithm is more robust in removing background noise while preserving edges to suppress the existent image artifacts. The quality measurements and visual inspections show a significant improvement in image quality compared to the conventional SART and Algebraic Reconstruction Technique (ART) algorithms.

Model-based Iterative CT Image Reconstruction on GPUs

Computed Tomography (CT) Image Reconstruction is an important technique used in a variety of domains, including medical imaging, electron microscopy, non-destructive testing and transportation security. Model-based Iterative Reconstruction (MBIR) using Iterative Coordinate Descent (ICD) is a CT algorithm that produces state-of-the-art results in terms of image quality. However, MBIR is highly computationally intensive and challenging to parallelize, and has traditionally been viewed as impractical in applications where reconstruction time is critical. We present the first GPU-based algorithm for ICD-based MBIR. The algorithm leverages the recently-proposed concept of SuperVoxels, and efficiently exploits the three levels of parallelism available in MBIR to better utilize the GPU hardware resources. We also explore data layout transformations to obtain more coalesced accesses and several GPU-specific optimizations for MBIR that boost performance. Across a suite of 3200 test cases, our GPU implementation obtains a geometric mean speedup of 4.43X over a state-of-the-art multi-core implementation on a 16-core iso-power CPU.

A Survey of the Use of Iterative Reconstruction Algorithms in Electron Microscopy

One of the key steps in Electron Microscopy is the tomographic reconstruction of a three-dimensional (3D) map of the specimen being studied from a set of two-dimensional (2D) projections acquired at the microscope. This tomographic reconstruction may be performed with different reconstruction algorithms that can be grouped into several large families: direct Fourier inversion methods, back-projection methods, Radon methods, or iterative algorithms. In this review, we focus on the latter family of algorithms, explaining the mathematical rationale behind the different algorithms in this family as they have been introduced in the field of Electron Microscopy. We cover their use in Single Particle Analysis (SPA) as well as in Electron Tomography (ET).

Fast alternating projection methods for constrained tomographic reconstruction

The alternating projection algorithms are easy to implement and effective for large-scale complex optimization problems, such as constrained reconstruction of X-ray computed tomography (CT). A typical method is to use projection onto convex sets (POCS) for data fidelity, nonnegative constraints combined with total variation (TV) minimization (so called TV-POCS) for sparse-view CT reconstruction. However, this type of method relies on empirically selected parameters for satisfactory reconstruction and is generally slow and lack of convergence analysis. In this work, we use a convex feasibility set approach to address the problems associated with TV-POCS and propose a framework using full sequential alternating projections or POCS (FS-POCS) to find the solution in the intersection of convex constraints of bounded TV function, bounded data fidelity error and non-negativity. The rationale behind FS-POCS is that the mathematically optimal solution of the constrained objective function may not be the physically optimal solution. The breakdown of constrained reconstruction into an intersection of several feasible sets can lead to faster convergence and better quantification of reconstruction parameters in a physical meaningful way than that in an empirical way of trial-and-error. In addition, for large-scale optimization problems, first order methods are usually used. Not only is the condition for convergence of gradient-based methods derived, but also a primal-dual hybrid gradient (PDHG) method is used for fast convergence of bounded TV. The newly proposed FS-POCS is evaluated and compared with TV-POCS and another convex feasibility projection method (CPTV) using both digital phantom and pseudo-real CT data to show its superior performance on reconstruction speed, image quality and quantification.

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

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

Accelerated fast iterative shrinkage thresholding algorithms for sparsity-regularized cone-beam CT image reconstruction

PURPOSE

The development of iterative image reconstruction algorithms for cone-beam computed tomography (CBCT) remains an active and important research area. Even with hardware acceleration, the overwhelming majority of the available 3D iterative algorithms that implement nonsmooth regularizers remain computationally burdensome and have not been translated for routine use in time-sensitive applications such as image-guided radiation therapy (IGRT). In this work, two variants of the fast iterative shrinkage thresholding algorithm (FISTA) are proposed and investigated for accelerated iterative image reconstruction in CBCT.

METHODS

Algorithm acceleration was achieved by replacing the original gradient-descent step in the FISTAs by a subproblem that is solved by use of the ordered subset simultaneous algebraic reconstruction technique (OS-SART). Due to the preconditioning matrix adopted in the OS-SART method, two new weighted proximal problems were introduced and corresponding fast gradient projection-type algorithms were developed for solving them. We also provided efficient numerical implementations of the proposed algorithms that exploit the massive data parallelism of multiple graphics processing units.

RESULTS

The improved rates of convergence of the proposed algorithms were quantified in computer-simulation studies and by use of clinical projection data corresponding to an IGRT study. The accelerated FISTAs were shown to possess dramatically improved convergence properties as compared to the standard FISTAs. For example, the number of iterations to achieve a specified reconstruction error could be reduced by an order of magnitude. Volumetric images reconstructed from clinical data were produced in under 4 min.

CONCLUSIONS

The FISTA achieves a quadratic convergence rate and can therefore potentially reduce the number of iterations required to produce an image of a specified image quality as compared to first-order methods. We have proposed and investigated accelerated FISTAs for use with two nonsmooth penalty functions that will lead to further reductions in image reconstruction times while preserving image quality. Moreover, with the help of a mixed sparsity-regularization, better preservation of soft-tissue structures can be potentially obtained. The algorithms were systematically evaluated by use of computer-simulated and clinical data sets.

High performance model based image reconstruction

Computed Tomography (CT) Image Reconstruction is an important technique used in a wide range of applications, ranging from explosive detection, medical imaging to scientific imaging. Among available reconstruction methods, Model Based Iterative Reconstruction (MBIR) produces higher quality images and allows for the use of more general CT scanner geometries than is possible with more commonly used methods. The high computational cost of MBIR, however, often makes it impractical in applications for which it would otherwise be ideal. This paper describes a new MBIR implementation that significantly reduces the computational cost of MBIR while retaining its benefits. It describes a novel organization of the scanner data into super-voxels (SV) that, combined with a super-voxel buffer (SVB), dramatically increase locality and prefetching, enable parallelism across SVs and lead to an average speedup of 187 on 20 cores.

Mixed Confidence Estimation for Iterative CT Reconstruction

Dynamic (4D) CT imaging is used in a variety of applications, but the two major drawbacks of the technique are its increased radiation dose and longer reconstruction time. Here we present a statistical analysis of our previously proposed Mixed Confidence Estimation (MCE) method that addresses both these issues. This method, where framed iterative reconstruction is only performed on the dynamic regions of each frame while static regions are fixed across frames to a composite image, was proposed to reduce computation time. In this work, we generalize the previous method to describe any application where a portion of the image is known with higher confidence (static, composite, lower-frequency content, etc.) and a portion of the image is known with lower confidence (dynamic, targeted, etc). We show that by splitting the image space into higher and lower confidence components, MCE can lower the estimator variance in both regions compared to conventional reconstruction. We present a theoretical argument for this reduction in estimator variance and verify this argument with proof-of-principle simulations. We also propose a fast approximation of the variance of images reconstructed with MCE and confirm that this approximation is accurate compared to analytic calculations of and multi-realization image variance. This MCE method requires less computation time and provides reduced image variance for imaging scenarios where portions of the image are known with more certainty than others allowing for potentially reduced radiation dose and/or improved dynamic imaging.

Relaxed Linearized Algorithms for Faster X-Ray CT Image Reconstruction

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

Constrained Total Generalized p-Variation Minimization for Few-View X-Ray Computed Tomography Image Reconstruction

Total generalized variation (TGV)-based computed tomography (CT) image reconstruction, which utilizes high-order image derivatives, is superior to total variation-based methods in terms of the preservation of edge information and the suppression of unfavorable staircase effects. However, conventional TGV regularization employs l1-based form, which is not the most direct method for maximizing sparsity prior. In this study, we propose a total generalized p-variation (TGpV) regularization model to improve the sparsity exploitation of TGV and offer efficient solutions to few-view CT image reconstruction problems. To solve the nonconvex optimization problem of the TGpV minimization model, we then present an efficient iterative algorithm based on the alternating minimization of augmented Lagrangian function. All of the resulting subproblems decoupled by variable splitting admit explicit solutions by applying alternating minimization method and generalized p-shrinkage mapping. In addition, approximate solutions that can be easily performed and quickly calculated through fast Fourier transform are derived using the proximal point method to reduce the cost of inner subproblems. The accuracy and efficiency of the simulated and real data are qualitatively and quantitatively evaluated to validate the efficiency and feasibility of the proposed method. Overall, the proposed method exhibits reasonable performance and outperforms the original TGV-based method when applied to few-view problems.

Fast multislice fluorescence molecular tomography using sparsity-inducing regularization

Fluorescence molecular tomography (FMT) is a rapidly growing imaging method that facilitates the recovery of small fluorescent targets within biological tissue. The major challenge facing the FMT reconstruction method is the ill-posed nature of the inverse problem. In order to overcome this problem, the acquisition of large FMT datasets and the utilization of a fast FMT reconstruction algorithm with sparsity regularization have been suggested recently. Therefore, the use of a joint L1/total-variation (TV) regularization as a means of solving the ill-posed FMT inverse problem is proposed. A comparative quantified analysis of regularization methods based on L1-norm and TV are performed using simulated datasets, and the results show that the fast composite splitting algorithm regularization method can ensure the accuracy and robustness of the FMT reconstruction. The feasibility of the proposed method is evaluated in an in vivo scenario for the subcutaneous implantation of a fluorescent-dye-filled capillary tube in a mouse, and also using hybrid FMT and x-ray computed tomography data. The results show that the proposed regularization overcomes the difficulties created by the ill-posed inverse problem.

An Efficient Augmented Lagrangian Method for Statistical X-Ray CT Image Reconstruction

Statistical iterative reconstruction (SIR) for X-ray computed tomography (CT) under the penalized weighted least-squares criteria can yield significant gains over conventional analytical reconstruction from the noisy measurement. However, due to the nonlinear expression of the objective function, most exiting algorithms related to the SIR unavoidably suffer from heavy computation load and slow convergence rate, especially when an edge-preserving or sparsity-based penalty or regularization is incorporated. In this work, to address abovementioned issues of the general algorithms related to the SIR, we propose an adaptive nonmonotone alternating direction algorithm in the framework of augmented Lagrangian multiplier method, which is termed as “ALM-ANAD”. The algorithm effectively combines an alternating direction technique with an adaptive nonmonotone line search to minimize the augmented Lagrangian function at each iteration. To evaluate the present ALM-ANAD algorithm, both qualitative and quantitative studies were conducted by using digital and physical phantoms. Experimental results show that the present ALM-ANAD algorithm can achieve noticeable gains over the classical nonlinear conjugate gradient algorithm and state-of-the-art split Bregman algorithm in terms of noise reduction, contrast-to-noise ratio, convergence rate, and universal quality index metrics.

Reconstruction for 3D PET Based on Total Variation Constrained Direct Fourier Method

This paper presents a total variation (TV) regularized reconstruction algorithm for 3D positron emission tomography (PET). The proposed method first employs the Fourier rebinning algorithm (FORE), rebinning the 3D data into a stack of ordinary 2D data sets as sinogram data. Then, the resulted 2D sinogram are ready to be reconstructed by conventional 2D reconstruction algorithms. Given the locally piece-wise constant nature of PET images, we introduce the total variation (TV) based reconstruction schemes. More specifically, we formulate the 2D PET reconstruction problem as an optimization problem, whose objective function consists of TV norm of the reconstructed image and the data fidelity term measuring the consistency between the reconstructed image and sinogram. To solve the resulting minimization problem, we apply an efficient methods called the Bregman operator splitting algorithm with variable step size (BOSVS). Experiments based on Monte Carlo simulated data and real data are conducted as validations. The experiment results show that the proposed method produces higher accuracy than conventional direct Fourier (DF) (bias in BOSVS is 70% of ones in DF, variance of BOSVS is 80% of ones in DF).

Evaluation of Fisher Information Matrix-Based Methods for Fast Assessment of Image Quality in Pinhole SPECT

The accurate determination of the local impulse response and the covariance in voxels from penalized maximum likelihood reconstructed images requires performing reconstructions from many noise realizations of the projection data. As this is usually a very time-consuming process, efficient analytical approximations based on the Fisher information matrix (FIM) have been extensively used in PET and SPECT to estimate these quantities. For 3D imaging, however, additional approximations need to be made to the FIM in order to speed up the calculations. The most common approach is to use the local shift-invariant (LSI) approximation of the FIM, but this assumes specific conditions which are not always necessarily valid. In this paper we take a single-pinhole SPECT system and compare the accuracy of the LSI approximation against two other methods that have been more recently put forward: the non-uniform object-space pixelation (NUOP) and the subsampled FIM. These methods do not assume such restrictive conditions while still increasing the speed of the calculations considerably. Our results indicate that in pinhole SPECT the NUOP and subsampled FIM approaches could be more reliable than the LSI approximation, especially when a high accuracy is required.

Edge-preserving PET image reconstruction using trust optimization transfer

Iterative image reconstruction for positron emission tomography can improve image quality by using spatial regularization. The most commonly used quadratic penalty often oversmoothes sharp edges and fine features in reconstructed images, while nonquadratic penalties can preserve edges and achieve higher contrast recovery. Existing optimization algorithms such as the expectation maximization (EM) and preconditioned conjugate gradient (PCG) algorithms work well for the quadratic penalty, but are less efficient for high-curvature or nonsmooth edge-preserving regularizations. This paper proposes a new algorithm to accelerate edge-preserving image reconstruction by using two strategies: trust surrogate and optimization transfer descent. Trust surrogate approximates the original penalty by a smoother function at each iteration, but guarantees the algorithm to descend monotonically; Optimization transfer descent accelerates a conventional optimization transfer algorithm by using conjugate gradient and line search. Results of computer simulations and real 3-D data show that the proposed algorithm converges much faster than the conventional EM and PCG for smooth edge-preserving regularization and can also be more efficient than the current state-of-art algorithms for the nonsmooth l1 regularization.

Comparison of SIRT and SQS for Regularized Weighted Least Squares Image Reconstruction

Tomographic image reconstruction is often formulated as a regularized weighted least squares (RWLS) problem optimized by iterative algorithms that are either inherently algebraic or derived from a statistical point of view. This paper compares a modified version of SIRT (Simultaneous Iterative Reconstruction Technique), which is of the former type, with a version of SQS (Separable Quadratic Surrogates), which is of the latter type. We show that the two algorithms minimize the same criterion function using similar forms of preconditioned gradient descent. We present near-optimal relaxation for both based on eigenvalue bounds and include a heuristic extension for use with ordered subsets. We provide empirical evidence that SIRT and SQS converge at the same rate for all intents and purposes. For context, we compare their performance with an implementation of preconditioned conjugate gradient. The illustrative application is X-ray CT of luggage for aviation security.

Combining ordered subsets and momentum for accelerated X-ray CT image reconstruction

Statistical X-ray computed tomography (CT) reconstruction can improve image quality from reduced dose scans, but requires very long computation time. Ordered subsets (OS) methods have been widely used for research in X-ray CT statistical image reconstruction (and are used in clinical PET and SPECT reconstruction). In particular, OS methods based on separable quadratic surrogates (OS-SQS) are massively parallelizable and are well suited to modern computing architectures, but the number of iterations required for convergence should be reduced for better practical use. This paper introduces OS-SQS-momentum algorithms that combine Nesterov's momentum techniques with OS-SQS methods, greatly improving convergence speed in early iterations. If the number of subsets is too large, the OS-SQS-momentum methods can be unstable, so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior. Experiments with simulated and real 3D CT scan data illustrate the performance of the proposed algorithms.

Alternating direction method of multiplier for tomography with nonlocal regularizers

The ordered subset expectation maximization (OSEM) algorithm approximates the gradient of a likelihood function using a subset of projections instead of using all projections so that fast image reconstruction is possible for emission and transmission tomography such as SPECT, PET, and CT. However, OSEM does not significantly accelerate reconstruction with computationally expensive regularizers such as patch-based nonlocal (NL) regularizers, because the regularizer gradient is evaluated for every subset. We propose to use variable splitting to separate the likelihood term and the regularizer term for penalized emission tomographic image reconstruction problem and to optimize it using the alternating direction method of multiplier (ADMM). We also propose a fast algorithm to optimize the ADMM parameter based on convergence rate analysis. This new scheme enables more sub-iterations related to the likelihood term. We evaluated our ADMM for 3-D SPECT image reconstruction with a patch-based NL regularizer that uses the Fair potential function. Our proposed ADMM improved the speed of convergence substantially compared to other existing methods such as gradient descent, EM, and OSEM using De Pierro's approach, and the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm.

Regularized estimation of magnitude and phase of multi-coil b1 field via Bloch-Siegert B1 mapping and coil combination optimizations

Parallel excitation requires fast and accurate B1 map estimation. Bloch-Siegert (BS) B1 mapping is very fast and accurate over a large dynamic range. When applied to multi-coil systems, however, this phase-based method may produce low signal-to-noise ratio estimates in low magnitude regions due to localized excitation patterns of parallel excitation systems. Also, the imaging time increases with the number of coils. In this work, we first propose to modify the standard BS B1 mapping sequence so that it avoids the scans required by previous B1 phase estimation methods. A regularized method is then proposed to jointly estimate the magnitude and phase of multi-coil B1 maps from BS B1 mapping data, improving estimation quality by using the prior knowledge of the smoothness of B1 magnitude and phase. Lastly, we use Cramer-Rao lower bound analysis to optimize the coil combinations, to improve the quality of the raw data for B1 estimation. The proposed methods are demonstrated by simulations and phantom experiments.

Direct reconstruction of cardiac PET kinetic parametric images using a preconditioned conjugate gradient approach

PURPOSE

Our research goal is to develop an algorithm to reconstruct cardiac positron emission tomography (PET) kinetic parametric images directly from sinograms and compare its performance with the conventional indirect approach.

METHODS

Time activity curves of a NCAT phantom were computed according to a one-tissue compartmental kinetic model with realistic kinetic parameters. The sinograms at each time frame were simulated using the activity distribution for the time frame. The authors reconstructed the parametric images directly from the sinograms by optimizing a cost function, which included the Poisson log-likelihood and a spatial regularization terms, using the preconditioned conjugate gradient (PCG) algorithm with the proposed preconditioner. The proposed preconditioner is a diagonal matrix whose diagonal entries are the ratio of the parameter and the sensitivity of the radioactivity associated with parameter. The authors compared the reconstructed parametric images using the direct approach with those reconstructed using the conventional indirect approach.

RESULTS

At the same bias, the direct approach yielded significant relative reduction in standard deviation by 12%-29% and 32%-70% for 50 × 10(6) and 10 × 10(6) detected coincidences counts, respectively. Also, the PCG method effectively reached a constant value after only 10 iterations (with numerical convergence achieved after 40-50 iterations), while more than 500 iterations were needed for CG.

CONCLUSIONS

The authors have developed a novel approach based on the PCG algorithm to directly reconstruct cardiac PET parametric images from sinograms, and yield better estimation of kinetic parameters than the conventional indirect approach, i.e., curve fitting of reconstructed images. The PCG method increases the convergence rate of reconstruction significantly as compared to the conventional CG method.

Direct estimation of kinetic parametric images for dynamic PET

Dynamic positron emission tomography (PET) can monitor spatiotemporal distribution of radiotracer in vivo. The spatiotemporal information can be used to estimate parametric images of radiotracer kinetics that are of physiological and biochemical interests. Direct estimation of parametric images from raw projection data allows accurate noise modeling and has been shown to offer better image quality than conventional indirect methods, which reconstruct a sequence of PET images first and then perform tracer kinetic modeling pixel-by-pixel. Direct reconstruction of parametric images has gained increasing interests with the advances in computing hardware. Many direct reconstruction algorithms have been developed for different kinetic models. In this paper we review the recent progress in the development of direct reconstruction algorithms for parametric image estimation. Algorithms for linear and nonlinear kinetic models are described and their properties are discussed.

A model based iterative reconstruction algorithm for high angle annular dark field-scanning transmission electron microscope (HAADF-STEM) tomography

High angle annular dark field (HAADF)-scanning transmission electron microscope (STEM) data is increasingly being used in the physical sciences to research materials in 3D because it reduces the effects of Bragg diffraction seen in bright field TEM data. Typically, tomographic reconstructions are performed by directly applying either filtered back projection (FBP) or the simultaneous iterative reconstruction technique (SIRT) to the data. Since HAADF-STEM tomography is a limited angle tomography modality with low signal to noise ratio, these methods can result in significant artifacts in the reconstructed volume. In this paper, we develop a model based iterative reconstruction algorithm for HAADF-STEM tomography. We combine a model for image formation in HAADF-STEM tomography along with a prior model to formulate the tomographic reconstruction as a maximum a posteriori probability (MAP) estimation problem. Our formulation also accounts for certain missing measurements by treating them as nuisance parameters in the MAP estimation framework. We adapt the iterative coordinate descent algorithm to develop an efficient method to minimize the corresponding MAP cost function. Reconstructions of simulated as well as experimental data sets show results that are superior to FBP and SIRT reconstructions, significantly suppressing artifacts and enhancing contrast.

Accelerating ordered subsets image reconstruction for X-ray CT using spatially nonuniform optimization transfer

Statistical image reconstruction algorithms in X-ray computed tomography (CT) provide improved image quality for reduced dose levels but require substantial computation time. Iterative algorithms that converge in few iterations and that are amenable to massive parallelization are favorable in multiprocessor implementations. The separable quadratic surrogate (SQS) algorithm is desirable as it is simple and updates all voxels simultaneously. However, the standard SQS algorithm requires many iterations to converge. This paper proposes an extension of the SQS algorithm that leads to spatially nonuniform updates. The nonuniform (NU) SQS encourages larger step sizes for the voxels that are expected to change more between the current and the final image, accelerating convergence, while the derivation of NU-SQS guarantees monotonic descent. Ordered subsets (OS) algorithms can also accelerate SQS, provided suitable "subset balance" conditions hold. These conditions can fail in 3-D helical cone-beam CT due to incomplete sampling outside the axial region-of-interest (ROI). This paper proposes a modified OS algorithm that is more stable outside the ROI in helical CT. We use CT scans to demonstrate that the proposed NU-OS-SQS algorithm handles the helical geometry better than the conventional OS methods and "converges" in less than half the time of ordinary OS-SQS.

Quantitative statistical methods for image quality assessment

Quantitative measures of image quality and reliability are critical for both qualitative interpretation and quantitative analysis of medical images. While, in theory, it is possible to analyze reconstructed images by means of Monte Carlo simulations using a large number of noise realizations, the associated computational burden makes this approach impractical. Additionally, this approach is less meaningful in clinical scenarios, where multiple noise realizations are generally unavailable. The practical alternative is to compute closed-form analytical expressions for image quality measures. The objective of this paper is to review statistical analysis techniques that enable us to compute two key metrics: resolution (determined from the local impulse response) and covariance. The underlying methods include fixed-point approaches, which compute these metrics at a fixed point (the unique and stable solution) independent of the iterative algorithm employed, and iteration-based approaches, which yield results that are dependent on the algorithm, initialization, and number of iterations. We also explore extensions of some of these methods to a range of special contexts, including dynamic and motion-compensated image reconstruction. While most of the discussed techniques were developed for emission tomography, the general methods are extensible to other imaging modalities as well. In addition to enabling image characterization, these analysis techniques allow us to control and enhance imaging system performance. We review practical applications where performance improvement is achieved by applying these ideas to the contexts of both hardware (optimizing scanner design) and image reconstruction (designing regularization functions that produce uniform resolution or maximize task-specific figures of merit).

Accelerating image reconstruction in three-dimensional optoacoustic tomography on graphics processing units

PURPOSE

Optoacoustic tomography (OAT) is inherently a three-dimensional (3D) inverse problem. However, most studies of OAT image reconstruction still employ two-dimensional imaging models. One important reason is because 3D image reconstruction is computationally burdensome. The aim of this work is to accelerate existing image reconstruction algorithms for 3D OAT by use of parallel programming techniques.

METHODS

Parallelization strategies are proposed to accelerate a filtered backprojection (FBP) algorithm and two different pairs of projection/backprojection operations that correspond to two different numerical imaging models. The algorithms are designed to fully exploit the parallel computing power of graphics processing units (GPUs). In order to evaluate the parallelization strategies for the projection/backprojection pairs, an iterative image reconstruction algorithm is implemented. Computer simulation and experimental studies are conducted to investigate the computational efficiency and numerical accuracy of the developed algorithms.

RESULTS

The GPU implementations improve the computational efficiency by factors of 1000, 125, and 250 for the FBP algorithm and the two pairs of projection/backprojection operators, respectively. Accurate images are reconstructed by use of the FBP and iterative image reconstruction algorithms from both computer-simulated and experimental data.

CONCLUSIONS

Parallelization strategies for 3D OAT image reconstruction are proposed for the first time. These GPU-based implementations significantly reduce the computational time for 3D image reconstruction, complementing our earlier work on 3D OAT iterative image reconstruction.

Non-cartesian MRI reconstruction with automatic regularization Via Monte-Carlo SURE

Magnetic resonance image (MRI) reconstruction from undersampled k-space data requires regularization to reduce noise and aliasing artifacts. Proper application of regularization however requires appropriate selection of associated regularization parameters. In this work, we develop a data-driven regularization parameter adjustment scheme that minimizes an estimate [based on the principle of Stein's unbiased risk estimate (SURE)] of a suitable weighted squared-error measure in k-space. To compute this SURE-type estimate, we propose a Monte-Carlo scheme that extends our previous approach to inverse problems (e.g., MRI reconstruction) involving complex-valued images. Our approach depends only on the output of a given reconstruction algorithm and does not require knowledge of its internal workings, so it is capable of tackling a wide variety of reconstruction algorithms and nonquadratic regularizers including total variation and those based on the l1-norm. Experiments with simulated and real MR data indicate that the proposed approach is capable of providing near mean squared-error optimal regularization parameters for single-coil undersampled non-Cartesian MRI reconstruction.

Statistical reconstruction algorithms for continuous wave electron spin resonance imaging

Electron spin resonance imaging (ESRI) is an important branch of ESR that deals with heterogeneous samples ranging from semiconductor materials to small live animals and even humans. ESRI can produce either spatial images (providing information about the spatially dependent radical concentration) or spectral-spatial images, where an extra dimension is added to describe the absorption spectrum of the sample (which can also be spatially dependent). The mapping of oxygen in biological samples, often referred to as oximetry, is a prime example of an ESRI application. ESRI suffers frequently from a low signal-to-noise ratio (SNR), which results in long acquisition times and poor image quality. A broader use of ESRI is hampered by this slow acquisition, which can also be an obstacle for many biological applications where conditions may change relatively quickly over time. The objective of this work is to develop an image reconstruction scheme for continuous wave (CW) ESRI that would make it possible to reduce the data acquisition time without degrading the reconstruction quality. This is achieved by adapting the so-called "statistical reconstruction" method, recently developed for other medical imaging modalities, to the specific case of CW ESRI. Our new algorithm accounts for unique ESRI aspects such as field modulation, spectral-spatial imaging, and possible limitation on the gradient magnitude (the so-called "limited angle" problem). The reconstruction method shows improved SNR and contrast recovery vs. commonly used back-projection-based methods, for a variety of simulated synthetic samples as well as in actual CW ESRI experiments.

Accelerated edge-preserving image restoration without boundary artifacts

To reduce blur in noisy images, regularized image restoration methods have been proposed that use nonquadratic regularizers (like l1 regularization or total-variation) that suppress noise while preserving edges in the image. Most of these methods assume a circulant blur (periodic convolution with a blurring kernel) that can lead to wraparound artifacts along the boundaries of the image due to the implied periodicity of the circulant model. Using a noncirculant model could prevent these artifacts at the cost of increased computational complexity. In this paper, we propose to use a circulant blur model combined with a masking operator that prevents wraparound artifacts. The resulting model is noncirculant, so we propose an efficient algorithm using variable splitting and augmented Lagrangian (AL) strategies. Our variable splitting scheme, when combined with the AL framework and alternating minimization, leads to simple linear systems that can be solved noniteratively using fast Fourier transforms (FFTs), eliminating the need for more expensive conjugate gradient-type solvers. The proposed method can also efficiently tackle a variety of convex regularizers, including edge-preserving (e.g., total-variation) and sparsity promoting (e.g., l1-norm) regularizers. Simulation results show fast convergence of the proposed method, along with improved image quality at the boundaries where the circulant model is inaccurate.

Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods

Regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values. We focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive (possibly complex) Gaussian noise. Generalized cross-validation (GCV) and (weighted) mean-squared error (MSE) approaches (based on Steinfs Unbiased Risk Estimate. SURE) need the Jacobian matrix of the nonlinear reconstruction operator (representative of the iterative algorithm) with respect to the data. We derive the desired Jacobian matrix for two types of nonlinear iterative algorithms: a fast variant of the standard iterative reweighted least-squares method and the contemporary split-Bregman algorithm, both of which can accommodate a wide variety of analysis- and synthesis-type regularizers. The proposed approach iteratively computes two weighted SURE-type measures: Predicted-SURE and Projected-SURE (that require knowledge of noise variance Ð2), and GCV (that does not need Ð2) for these algorithms. We apply the methods to image restoration and to magnetic resonance image (MRI) reconstruction using total variation (TV) and an analysis-type .1-regularization. We demonstrate through simulations and experiments with real data that minimizing Predicted-SURE and Projected-SURE consistently lead to near-MSE-optimal reconstructions. We also observed that minimizing GCV yields reconstruction results that are near-MSE-optimal for image restoration and slightly suboptimal for MRI. Theoretical derivations in this work related to Jacobian matrix evaluations can be extended, in principle, to other types of regularizers and reconstruction algorithms.

Robust framework for PET image reconstruction incorporating system and measurement uncertainties

In Positron Emission Tomography (PET), an optimal estimate of the radioactivity concentration is obtained from the measured emission data under certain criteria. So far, all the well-known statistical reconstruction algorithms require exactly known system probability matrix a priori, and the quality of such system model largely determines the quality of the reconstructed images. In this paper, we propose an algorithm for PET image reconstruction for the real world case where the PET system model is subject to uncertainties. The method counts PET reconstruction as a regularization problem and the image estimation is achieved by means of an uncertainty weighted least squares framework. The performance of our work is evaluated with the Shepp-Logan simulated and real phantom data, which demonstrates significant improvements in image quality over the least squares reconstruction efforts.

A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction

Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and nonsmooth sparsity-promoting ones based on the l(1)-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT.

Preconditioning for edge-preserving image super resolution

We propose a simple preconditioning method for accelerating the solution of edge-preserving image super-resolution (SR) problems in which a linear shift-invariant point spread function is employed. Our technique involves reordering the high-resolution (HR) pixels in a similar manner to what is done in preconditioning methods for quadratic SR formulations. However, due to the edge preserving requirements, the Hessian matrix of the cost function varies during the minimization process. We develop an efficient update scheme for the preconditioner in order to cope with this situation. Unlike some other acceleration strategies that round the displacement values between the low-resolution (LR) images on the HR grid, the proposed method does not sacrifice the optimality of the observation model. In addition, we describe a technique for preconditioning SR problems involving rational magnification factors. The use of such factors is motivated in part by the fact that, under certain circumstances, optimal SR zooms are nonintegers. We show that, by reordering the pixels of the LR images, the structure of the problem to solve is modified in such a way that preconditioners based on circulant operators can be used.

Image reconstruction in phase-contrast tomography exploiting the second-order statistical properties of the projection data

X-ray phase-contrast tomography (PCT) methods seek to quantitatively reconstruct separate images that depict an object's absorption and refractive contrasts. Most PCT reconstruction algorithms generally operate by explicitly or implicitly performing the decoupling of the projected absorption and phase properties at each tomographic view angle by use of a phase-retrieval formula. However, the presence of zero-frequency singularity in the Fourier-based phase retrieval formulas will lead to a strong noise amplification in the projection estimate and the subsequent refractive image obtained using conventional algorithms like filtered backprojection (FBP). Tomographic reconstruction by use of statistical methods can account for the noise model and a priori information, and thereby can produce images with better quality over conventional filtered backprojection algorithms. In this work, we demonstrate an iterative image reconstruction method that exploits the second-order statistical properties of the projection data can mitigate noise amplification in PCT. The autocovariance function of the reconstructed refractive images was empirically computed and shows smaller and shorter noise correlation compared to those obtained using the FBP and unweighted penalized least-squares methods. Concepts from statistical decision theory are applied to demonstrate that the statistical properties of images produced by our method can improve signal detectability.

SPECT IMAGE RECONSTRUCTION USING COMPOUND PRIOR MODELS

We propose a new iterative method for Maximum a Posteriori (MAP) reconstruction of SPECT (Single Photon Emission Computed Tomography) images. The method uses Compound Gauss Markov Random Fields (CGMRF) as prior model and is stochastic for the line process and deterministic for the reconstruction. Synthetic and real images are used to compare the new method with existing ones.

Adaptive Angular Sampling for SPECT Imaging

This paper presents an analytical approach for performing adaptive angular sampling in single photon emission computed tomography (SPECT) imaging. It allows for a rapid determination of the optimum sampling strategy that minimizes image variance in regions-of-interest (ROIs). The proposed method consists of three key components: (a) a set of close-form equations for evaluating image variance and resolution attainable with a given sampling strategy, (b) a gradient-based algorithm for searching through the parameter space to find the optimum sampling strategy and (c) an efficient computation approach for speeding up the search process. In this paper, we have demonstrated the use of the proposed analytical approach with a single-head SPECT system for finding the optimum distribution of imaging time across all possible sampling angles. Compared to the conventional uniform angular sampling approach, adaptive angular sampling allows the camera to spend larger fractions of imaging time at angles that are more efficient in acquiring useful imaging information. This leads to a significantly lowered image variance. In general, the analytical approach developed in this study could be used with many nuclear imaging systems (such as SPECT, PET and X-ray CT) equipped with adaptive hardware. This strategy could provide an optimized sampling efficiency and therefore an improved image quality.

Parallel MR image reconstruction using augmented Lagrangian methods

Magnetic resonance image (MRI) reconstruction using SENSitivity Encoding (SENSE) requires regularization to suppress noise and aliasing effects. Edge-preserving and sparsity-based regularization criteria can improve image quality, but they demand computation-intensive nonlinear optimization. In this paper, we present novel methods for regularized MRI reconstruction from undersampled sensitivity encoded data--SENSE-reconstruction--using the augmented Lagrangian (AL) framework for solving large-scale constrained optimization problems. We first formulate regularized SENSE-reconstruction as an unconstrained optimization task and then convert it to a set of (equivalent) constrained problems using variable splitting. We then attack these constrained versions in an AL framework using an alternating minimization method, leading to algorithms that can be implemented easily. The proposed methods are applicable to a general class of regularizers that includes popular edge-preserving (e.g., total-variation) and sparsity-promoting (e.g., l(1)-norm of wavelet coefficients) criteria and combinations thereof. Numerical experiments with synthetic and in vivo human data illustrate that the proposed AL algorithms converge faster than both general-purpose optimization algorithms such as nonlinear conjugate gradient (NCG) and state-of-the-art MFISTA.

Noise propagation for iterative penalized-likelihood image reconstruction based on Fisher information

Iterative reconstruction algorithms have been widely used in PET and SPECT emission tomography. Accurate modeling of photon noise propagation is crucial for quantitative tomography applications. Iteration-based noise propagation methods have been developed for only a few algorithms that have explicit multiplicative update equations. And there are discrepancies between the iteration-based methods and Fessler's fixed-point method because of improper approximations. In this paper, we present a unified theoretical prediction of noise propagation for any penalized expectation maximization (EM) algorithm where the EM approach incorporates a penalty term. The proposed method does not require an explicit update equation. The update equation is assumed to be implicitly defined by a differential equation of a surrogate function. We derive the expressions using the implicit function theorem, Taylor series and the chain rule from vector calculus. We also derive the fixed-point expressions when iterative algorithms converge and show the consistency between the proposed method and the fixed-point method. These expressions are solely defined in terms of the partial derivatives of the surrogate function and the Fisher information matrices. We also apply the theoretical noise predictions for iterative reconstruction algorithms in emission tomography. Finally, we validate the theoretical predictions for MAP-EM and OSEM algorithms using Monte Carlo simulations with Jaszczak-like and XCAT phantoms, respectively.