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In this paper, we present a novel image reconstruction method based on weighted least squares (WLS) objective function for positron emission tomography (PET). Unlike a usual WLS algorithm, the proposed method, which we call it SA-WLS, combines the SAGE algorithm with WLS algorithm. It minimized the WLS objective function using single coordinate descent (SCD) method in a sequence of small “hidden” data spaces (HDS). Although SA-WLS used a strategy to update parameter sequentially just like common SCD method, the use of these small HDS makes it converge much faster and produce the reconstructed images with greater contrast and detail than the usual WLS method. In order to decrease further the actual CPU time per iteration, the adaptive variable index sets were introduced to modify SA-WLS (MSA-WLS). Instead of optimizing each pixel, this MSA-WLS method sequentially optimizes many pixels located in an index set at one time. The index sets were automatically modified during each iteration step. MSA-WLS gathers the virtue of simultaneously and sequentially updating the parameters so that it achieves a good compromise between the convergence rate and the computational cost in PET reconstruction problem. Details of these algorithms were presented and the performances were evaluated by a simulated head phantom.

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... The corresponding estimation is called the penalized WLS estimation (PWLS). Many investigators approach the PWLS estimate using iterative algorithms, which include, for example, the expectation-maximization (EM) algorithm [2][3], the gradient-based algorithm [4]–[8] and the coordinate-descent based algorithm [1][9]–[11] (the latter two are similar to some extent). All of them start with an initial guess and then perform image updating iteratively until reaching some specified stopping criterions. ...

... The coordinate-descent algorithm usually offers more convergent rate than the EM or gradient-based algorithms. However, as pointed out in [3][11], this is likely at the cost of computational time. ...

This paper proposes a new sequential weighted least squares (SWLS) method for positron emission tomography (PET) reconstruction. The SWLS algorithm is noniterative and can be considered as equivalent to the penalized WLS (PWLS) method under certain initial conditions. However, a full implementation of SWLS is computationally intensive. To overcome this problem, we propose a simplified SWLS as a reasonable alternative to the SWLS. The performance of this SWLS method is evaluated in experiments using both simulated and clinical data. The results show that the method can be advantageously compared with the original SWLS both in computation time and reconstruction quality.

... Other ML-algorithms have also been utilized. These are, for example, ordered subsets separable surrogates (OSSPS) [165], paraboloidal surrogates coordinate ascent [166], several different weighted least-squares (WLS) methods [167][168][169][170], coordinate-ascent algorithms [171,172] and conjugate-gradient (CG) algorithms [161,173,174]. ...

In this thesis, the image reconstruction of fMRI and PET was studied. In fMRI, blood oxygenation level dependent (BOLD) and dynamic contrast enhanced (DCE) MRI data were used. For fMRI both measured and simulated data were utilized, while for DCE-MRI only experimental data were utilized. The goal was to improve the spatial and temporal accuracy of BOLD and DCE-MRI image reconstruction with the use of state estimation methods. The state estimation methods used in this thesis were based on Kalman filter (KF). For PET image reconstruction, a new open-source software called open-source MATLAB emission tomography software (OMEGA) was developed for MATLAB and GNU Octave. Evaluation of the OMEGA software was conducted by using both experimental preclinical PET data and simulated GATE Monte Carlo data.

... These include SAGE (Fessler and Hero 1994), OSSPS (Ahn andFessler 2003, Fessler andErdogan 1998) and RASAGE (Hongqing et al. 2004). Weighted least-squares (WLS) algorithms have also been utilized to solve the ML-problem (Fessler 1994), of these the ATP-WLS (Anderson et al. 1997) and SA-WLS (Zhu et al. 2006) have been implemented in OMEGA as example scripts utilizing the special forward/backward projection class. ...

In this paper we present OMEGA, an open-source software, for efficient and fast image reconstruction in positron emission tomography (PET). OMEGA uses the scripting language of MATLAB and GNU Octave allowing reconstruction of PET data with a MATLAB or GNU Octave interface. The goal of OMEGA is to allow easy and fast reconstruction of any PET data, and to provide a computationally efficient, easy-access platform for development of new PET algorithms with built-in forward and backward projection operations available to the user as a MATLAB/Octave class. OMEGA also includes direct support for GATE simulated data, facilitating easy evaluation of the new algorithms using Monte Carlo simulated PET data. OMEGA supports parallel computing by utilizing OpenMP for CPU implementations and OpenCL for GPU allowing any hardware to be used. OMEGA includes built-in function for the computation of normalization correction and allows several other corrections to be applied such as attenuation, randoms or scatter. OMEGA includes several different maximum-likelihood and maximum a posteriori (MAP) algorithms with several different priors. The user can also input their own priors to the built-in MAP functions. The image reconstruction in OMEGA can be computed either by using an explicitly computed system matrix or with a matrix-free formalism, where the latter can be accelerated with OpenCL. We provide an overview on the software and present some examples utilizing the different features of the software.

... In WLS, the Coordinate Descent (CD) algorithm is one of the recommended methods. Using a WLS approach to reconstruct PET images was to account for errors due to accidental coincidences and detector inefficiency (Zhu et al., 2004;Zhu et al., 2006). Recent challenges in PET image reconstruction include additional exact quantitation, TOF imaging, system modelling, motion rectification and dynamic reconstruction. ...

In this paper, reconstruction of the Positron Emission Tomography (PET) images, a CD algorithm was instigated with NN based image segmentation techniques called Neural Network Segmentation based Coordinate Descent-Weighted Least Square (NNCD-WLS). Thus, NNCD-WLS of the function is not quadratic, but natural. The iterative algorithm achieve a fashion equivalent to an analytic derivation of the Maximum Likelihood-Expectation Maximisation (ML-EM) algorithm, which gives a different minimisation process between two convex sets of matrices. Conversely the distance metric is quite distinct, and more intricate to analyse. This algorithm is similar type, shares many properties acquainted with the ML-EM algorithm. Unlike WLS algorithm, NNCD-WLS method minimises the WLS objective function. The NNCD-WLS algorithm instigates via NN based segmentation process in image reconstruction. Image quality parameter of the PSNR value, NNCD-WLS algorithm and the denoising algorithm is compared. The PET input image is reconstructed and simulated in the MATLAB/Simulink package.

An iterative reconstruction algorithm was developed based on the weighted least squares cost function for positron emission tomography. Unlike the conventional gradient-based algorithm, this algorithm made use of an auxiliary function in the current iterative point to form iteration process, by which the optimal solution was given instead of that by objective function, thus giving the new iterative points. Furthermore, the algorithm automatically satisfied the non-negative constraints of pixels without step size factor required, then ensuring the monotonous decreasing of objective function with global convergence provided. The results of the experiments based on both simulative and real clinical data showed that although the proposed algorithm requires the operating time that is much the same to the SA-WLS and ML-EM, it has higher convergence rate and better imaging quality than the latter.

In positron emission tomography (PET) image reconstruction, maximum likelihood expectation maximization(MLEM) algorithm has poor convergent rate and lots of computing time. And the noise of the reconstructed image is deteriorated with the iterations increasing. A new algorithm, named as MOS-PML, is proposed by using MLEM algorithm combining the regularizing penalty with the modified subsets technology. Simulation results for PET image reconstruction show that the MOS-PML algorithm has better performance than MLEM and PML in convergence rate and image quality.

We provide a general form for many reconstruction estimators of emission tomography. These estimators include Shepp and Vardi's maximum likelihood (ML) estimator, the quadratic weighted least squares (WLS) estimator, Anderson's WLS estimator, and Liu and Wang's multi-objective estimator, and others. We derive a generic update rule by constructing a surrogate function. This work is inspired by the ML-EM (EM, expectation maximization), where the latter naturally arises as a special case. A regularization with a specific form can also be incorporated by De Pierro's trick. We provide a general and quite different convergence proof compared with the proofs of the ML-EM and De Pierro. Theoretical analysis shows that the proposed algorithm monotonically decreases the cost function and automatically meets nonnegativity constraints. We have introduced a mechanism to provide monotonic, self-constraining, and convergent algorithms, from which some interesting existing and new algorithms can be derived. Simulation results illustrate the behavior of these algorithms in term of image quality and resolution-noise tradeoff.

This paper presents a class of image reconstruction algorithms based on Amari's α-divergence for position emission tomography. The α-divergence is actually a family of divergences indexed by α∈(-∞, +∞) that can measure discrepancy between two distributions. We consider it to model the discrepancy between projections and their estimates. By iteratively minimizing the α-divergence, a multiplicative updating algorithm is derived using an auxiliary function. The well-known ML-EM algorithm and the SA-WLS algorithm suggested by Zhu et al. arise as two special cases of our method. We prove the monotonic convergence of the algorithm, which Zhu et al. has not provided. The experiments were performed on both simulated and clinical data to study the interesting and useful behavior of the algorithm in cases where different parameters (α) were used. The results showed that some chosen algorithms exhibited much better performance than the prevalent ones.

Streak artifacts have been one of the major classes of image artifacts resulting from excessive quantum noise in low-dose X-ray CT. It has been shown that, to treat the noise in low-dose CT more accurately, both the analysis of the noise properties of the projection data and the development of a corresponding efficient filtering method are necessary. From our previous analysis of the calibrated low-dose CT projection data, it was clearly seen that the data could be regarded as approximately Gaussian distributed with nonlinear signal-dependent variance. Based on this observation, a penalized weighted least square (WLS) statistic framework was chosen for an optimal solution. In this work, we further incorporated a novel a priori idea into the framework, which can accurately preserve more information in high-noise regions with a significant reduction of the streak artifacts. This new penalty term was directly calculated from the sinogram. The method was tested by experimental data acquired at 120 kVp and 10 mA protocols, demonstrating a significant reduction on streak artifacts and noise suppression without sacrificing the spatial resolution.

The classical expectation-maximization (EM) algorithm for image
reconstruction suffers from particularly slow convergence when additive
background effects such as accidental coincidences and scatter are
included. In addition, when smoothness penalties are included in the
objective function, the M-step of the EM algorithm becomes intractable
due to parameter coupling. The authors describe the space-alternating
generalized EM (SAGE) algorithm, in which the parameters are updated
sequentially using a sequence of small “hidden” data spaces
rather than one large complete-data space. The sequential update
decouples the M-step, so the maximization can typically be performed
analytically. By choosing hidden-data spaces with considerably less
Fisher information than the conventional complete-data space for Poisson
data, the authors obtain significant improvements in convergence rate.
This acceleration is due to statistical considerations, not to numerical
overrelaxation methods, so monotonic increases in the objective function
and global convergence are guaranteed. Due to the space constraints, the
authors focus on the unpenalized case in this summary, and they
eliminate derivations that are similar to those in Lange and Carson, J.
Comput. Assist. Tomography, vol. 8, no. 2, p.306-16 (1984)

Most expectation-maximization (EM) type algorithms for penalized
maximum-likelihood image reconstruction converge slowly, particularly
when one incorporates additive background effects such as scatter,
random coincidences, dark current, or cosmic radiation. In addition,
regularizing smoothness penalties (or priors) introduce parameter
coupling, rendering intractable the M-steps of most EM-type algorithms.
This paper presents space-alternating generalized EM (SAGE) algorithms
for image reconstruction, which update the parameters sequentially using
a sequence of small “hidden” data spaces, rather than
simultaneously using one large complete-data space. The sequential
update decouples the M-step, so the maximization can typically be
performed analytically. We introduce new hidden-data spaces that are
less informative than the conventional complete-data space for Poisson
data and that yield significant improvements in convergence rate. This
acceleration is due to statistical considerations, not numerical
overrelaxation methods, so monotonic increases in the objective function
are guaranteed. We provide a general global convergence proof for SAGE
methods with nonnegativity constraints

The expectation-maximization (EM) method can facilitate maximizing
likelihood functions that arise in statistical estimation problems. In
the classical EM paradigm, one iteratively maximizes the conditional
log-likelihood of a single unobservable complete data space, rather than
maximizing the intractable likelihood function for the measured or
incomplete data. EM algorithms update all parameters simultaneously,
which has two drawbacks: 1) slow convergence, and 2) difficult
maximization steps due to coupling when smoothness penalties are used.
The paper describes the space-alternating generalized EM (SAGE) method,
which updates the parameters sequentially by alternating between several
small hidden-data spaces defined by the algorithm designer. The authors
prove that the sequence of estimates monotonically increases the
penalized-likelihood objective, derive asymptotic convergence rates, and
provide sufficient conditions for monotone convergence in norm. Two
signal processing applications illustrate the method: estimation of
superimposed signals in Gaussian noise, and image reconstruction from
Poisson measurements. In both applications, the SAGE algorithms easily
accommodate smoothness penalties and converge faster than the EM
algorithms

We have recently proposed a regularized least square criterion for adaptive regularization of single photon emission computed tomography (SPECT) reconstruction with nonuniform attenuation correction. In the present study, we show that this regularization is closely related to a diffusion scheme used for Gaussian filtering. For a given value of the regularization parameter, the amount of smoothing is independent from the patient attenuation map, and it is mathematically related to the full-width at half-maximum (FWHM) of a Gaussian filter. A second regularized least square criterion is then derived for which regularization also behaves as a diffusion scheme. The new penalty is then shown to be also applicable to the weighted least square criterion, and to the Poisson maximum-likelihood criterion for positron emission tomography (PET) data (i.e., without attenuation) solved by the expectation maximization (EM) algorithm. For all these criteria, the regularization level can thus be set as the FWHM of a Gaussian filter.

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

We present unpenalized and penalized weighted least-squares (WLS) reconstruction methods for positron emission tomography (PET), where the weights are based on the covariance of a model error and depend on the unknown parameters. The penalty function for the latter method is chosen so that certain a priori information is incorporated. The algorithms used to minimize the WLS objective functions guarantee nonnegative estimates and, experimentally, they converged faster than the maximum likelihood expectation-maximization (ML-EM) algorithm and produced images that had significantly better resolution and contrast. Although simulations suggest that the proposed algorithms are globally convergent, a proof of convergence has not yet been found. Nevertheless, we are able to show that the unpenalized method produces estimates that decrease the objective function monotonically with increasing iterations.

We present two types of globally convergent relaxed ordered subsets (OS) algorithms for penalized-likelihood image reconstruction in emission tomography: modified block sequential regularized expectation-maximization (BSREM) and relaxed OS separable paraboloidal surrogates (OS-SPS). The global convergence proof of the existing BSREM (De Pierro and Yamagishi, 2001) required a few a posteriori assumptions. By modifying the scaling functions of BSREM, we are able to prove the convergence of the modified BSREM under realistic assumptions. Our modification also makes stepsize selection more convenient. In addition, we introduce relaxation into the OS-SPS algorithm (Erdoğan and Fessler, 1999) that otherwise would converge to a limit cycle. We prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed OS-SPS is a special case; main results of the proofs are from (Nedić and Bertsekas, 2001) and (Correa and Lemaréchal, 1993). Simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms.

The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained.

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n(*)(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n(*)|lambda) of observing the actual detector count data n(*) over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n(*) = n(*)(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n(*) has probability or likelihood p(n(*)|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n(*)p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.

Computed tomography (CT) has been extensively studied for years and widely used in the modern society. Although the filtered back-projection algorithm is the method of choice by manufacturers, efforts are being made to revisit iterative methods due to their unique advantages, such as superior performance with incomplete noisy data. In 1984, the simultaneous algebraic reconstruction technique (SART) was developed as a major refinement of the algebraic reconstruction technique (ART). However, the convergence of the SART has never been established since then. In this paper, the convergence is proved under the condition that coefficients of the linear imaging system are nonnegative. It is shown that from any initial guess the sequence generated by the SART converges to a weighted least square solution.

Subspace analysis is a popular method for multivariate data analysis and is closely related to factor analysis and principal component analysis (PCA). In the context of image processing (especially positron emission tomography), all data points are nonnegative and it is expected that both basis images and factors are nonnegative in order to obtain reasonable results. Recently the nonnegative matrix factorization (NMF) was introduced (D.D. Lee and H.S. Seung, 1999). It was demonstrated that the NMF gave parts-based representation and was useful in dynamic PET image analysis. We present a sequential EM algorithm for rectified subspace analysis (subspace in nonnegativity constraint) and apply it to dynamic PET image analysis. Experimental results show that our proposed method is useful in dynamic PET image analysis.

In this paper, the authors introduce a new iterative algorithm for
reconstructing positron emission tomography (PET) images. This algorithm
seeks to minimize an objective function of weighted least squares (WLS)
type. However, unlike conventional WLS methods, the weights do not need
to be estimated from the data, but are incorporated in the objective
function and relies heavily on the Poisson nature of the data. As a
result, the objective function is not quadratic, but is convex. The
iterative algorithm is obtained in a manner similar to an analytic
derivation of the ML-EM (maximum likelihood-expectation maximization)
algorithm which employs an alternating minimization procedure between
two convex sets of matrices. However, the distance metric is quite
different in the authors' case, and much more difficult to analyze. This
algorithm is similar in form to, and shares many properties in common
with, the ML-EM algorithm. The mathematical proof of the global
convergence of the algorithm remains an open problem

In this paper, the authors present a reconstruction algorithm for
positron emission tomography that minimizes a weighted least-squares
(WLS) objective function. The weights are based on the covariance matrix
of the model error and depend on the unknown parameters. The algorithm
guarantees nonnegative estimates, and in simulation studies it converged
faster and had significantly better resolution and contrast than the
ML-EM algorithm. Although simulations suggest that the proposed
algorithm is globally convergent, a proof of convergence has not been
found yet. Nevertheless, the authors are able to show that it produces
estimates that decrease the objective function monotonically with
increasing iterations

A method for Bayesian reconstruction which relies on updates of
single pixel values, rather than the entire image, at each iteration is
presented. The technique is similar to Gauss-Seidel (GS) iteration for
the solution of differential equations on finite grids. The
computational cost per iteration of the GS approach is found to be
approximately equal to that of gradient methods. For continuously valued
images, GS is found to have significantly better convergence at modes
representing high spatial frequencies. In addition, GS is well suited to
segmentation when the image is constrained to be discretely valued. It
is shown that Bayesian segmentation using GS iteration produces useful
estimates at much lower signal-to-noise ratios than required for
continuously valued reconstruction. The convergence properties of
gradient ascent and GS for reconstruction from integral projections are
analyzed, and simulations of both maximum-likelihood and maximum a
posteriori cases are included

We develop algorithms for obtaining regularized estimates of emission means in positron emission tomography. The first algorithm iteratively minimizes a penalized maximum-likelihood (PML) objective function. It is based on standard de-coupled surrogate functions for the ML objective function and de-coupled surrogate functions for a certain class of penalty functions. As desired, the PML algorithm guarantees nonnegative estimates and monotonically decreases the PML objective function with increasing iterations. The second algorithm is based on an iteration dependent, de-coupled penalty function that introduces smoothing while preserving edges. For the purpose of making comparisons, the MLEM algorithm and a penalized weighted least-squares algorithm were implemented. In experiments using synthetic data and real phantom data, it was found that, for a fixed level of background noise, the contrast in the images produced by the proposed algorithms was the most accurate.

Presents an image reconstruction method for positron-emission
tomography (PET) based on a penalized, weighted least-squares (PWLS)
objective. For PET measurements that are precorrected for accidental
coincidences, the author argues statistically that a least-squares
objective function is as appropriate, if not more so, than the popular
Poisson likelihood objective. The author proposes a simple data-based
method for determining the weights that accounts for attenuation and
detector efficiency. A nonnegative successive over-relaxation (+SOR)
algorithm converges rapidly to the global minimum of the PWLS objective.
Quantitative simulation results demonstrate that the bias/variance
tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood
expectation-maximization (ML-EM) method (but with fewer iterations), and
is improved relative to the conventional filtered backprojection (FBP)
method. Qualitative results suggest that the streak artifacts common to
the FBP method are nearly eliminated by the PWLS+SOR method, and
indicate that the proposed method for weighting the measurements is a
significant factor in the improvement over FBP

Iterative algorithms such as maximum likelihood expectation maximization (ML-EM) algorithm are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to conventional filtered backprojection algorithm. A major drawback of the iterative image reconstruction methods is their high computational cost. In this paper, we develop a new algorithm called the improved ordered subset expectation maximization (IOS-EM) algorithm. This algorithm modifies the number of projections in each subset and the step size (i.e., the relaxation factor) for each iteration in order to recover various frequency components in early iteration steps. In the method presented in this paper, the number of projections in a subset increases and the step size decreases after each iteration. In addition, pixel data are grouped into subdivisions to accelerate image reconstruction. Experimental results show that the IOS-EM algorithm can provide high quality reconstructed images at a small number of iterations.

Diffusion regularization for iterative reconstruction in emission tomography

- Riddell