Joint reconstruction via coupled Bregman iterations with applications to PET-MR imaging

Rasch, Julian; Brinkmann, Eva-Maria; Burger, Martin

doi:10.1088/1361-6420/aa9425

Cited by 30 publications

(51 citation statements)

References 49 publications

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“…It should be mentioned that the (technical) assumption of a continuous subgradient is crucial for this illustration, but not necessarily for the result, meaning that a zero Bregman distance with respect to TV is still well-defined if q is not continuous. However, then it is hard to say anything about the behavior, and we refer to [28] for further information. Moreover, it is possible that |q| = 1 on a |Du|-zero set, i.e.…”

Section: Model Manifoldsmentioning

confidence: 99%

Convergence rates and structure of solutions of inverse problems with imperfect forward models

2019

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The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the regularised solutions and allow us to obtain convergence rates in terms of Bregman distances -as usual in inverse problems, under an additional assumption on the exact solution called the source condition. These results are obtained for general absolutely one-homogeneous functionals. In the special case of TV-based regularisation we also study the structure of regularised solutions and prove convergence of their level sets to those of an exact solution. Finally, using the developed theory, we adapt the concept of debiasing to inverse problems with imperfect operators and propose an approach to pointwise error estimation in TV-based regularisation.Keywords: inverse problems, imperfect forward models, total variation, extended support, Bregman distances, convergence rates, error estimation, debiasing where A : L 1 (Ω) → L ∞ (Ω) is a linear operator and Ω ⊂ R m is a bounded domain. We assume that there exists a non-negative solution of (1.1).For an appropriate functional J (·) : L 1 → R + ∪{∞} we consider non-negative J -minimising solutions, which solve the following problem:We assume that the feasible set in (1.2) has at least one point with a finite value of J and denote a (possibly non-unique) solution of (1.2) byū J . Throughout this paper it is assumed that the regularisation functional J (·) is convex, proper and absolutely one-homogeneous.In practice the data f are not known precisely and only their perturbed versionf is available. In this case, we cannot simply replace the constraint Au = f in (1.2) with Au =f , since the solutions of the original problem (1.1) would no longer be feasible in this case. Therefore, we need to relax the equality in (1.2) to guarantee the feasibility of solutions of the original problem (1.1). This is the idea of the residual method [20,23]. If the error in the data is bounded by some known constant δ, the residual method accounts to solving the following constrained problem: min( 1.3)The fidelity function becomes in this case the characteristic function of the convex set {u : Au− f δ}. In the linear case, the residual method is equivalent to Tikhonov regularisation min u∈L 1with the regularisation parameter α = α(f , δ) chosen according to Morozov's discrepancy principle [23]. In many practical situations not only the data contain errors, but also the forward operator, that generated the data, are not perfectly known. In order to guarantee the feasibility of solutions of the original problem (1.1) in the constrained problem (1.3), one needs to account for the errors in the operator in the feasible set. If the errors in the operator are bounded by a known constant h (in the operator norm), the feasible set can be amended as follows in order to guarantee feasibility of the solutions of the original problem (1.1):whereÃ is the noisy operator. This optimi...

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Section: Model Manifoldsmentioning

confidence: 99%

Convergence rates and structure of solutions of inverse problems with imperfect forward models

2019

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“…The structural similarity index (SSIM) 64 was used to determine the similarity between the reconstructed images and the ground truth, as is generally perceived to provide a closer numerical representation of visual perception. SSIM has previously been used in the PET and image processing literature to evaluate novel methodologies, 54,65,66 and to assess structural similarity between multimodality images as part of a proposed methodology, 17 among other uses. The simplified formulation of SSIM employed in this work is…”

Section: B1 Simulation Studies Parameter Selectionmentioning

confidence: 99%

Intercomparison of MR-Informed Methods for PET Image Reconstruction

Bland

Mehranian

Belzunce

et al. 2018

2018 IEEE Nuclear Science Symposium and Medical Imaging Conference Proceedings (NSS/MIC)

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Purpose: Numerous image reconstruction methodologies for positron emission tomography (PET) have been developed that incorporate magnetic resonance (MR) imaging structural information, producing reconstructed images with improved suppression of noise and reduced partial volume effects. However, the influence of MR structural information also increases the possibility of suppression or bias of structures present only in the PET data (PET-unique regions). To address this, further developments for MR-informed methods have been proposed, for example, through inclusion of the current reconstructed PET image, alongside the MR image, in the iterative reconstruction process. In this present work, a number of kernel and maximum a posteriori (MAP) methodologies are compared, with the aim of identifying methods that enable a favorable trade-off between the suppression of noise and the retention of unique features present in the PET data. Methods: The reconstruction methods investigated were: the MR-informed conventional and spatially compact kernel methods, referred to as KEM and KEM largest value sparsification (LVS) respectively; the MR-informed Bowsher and Gaussian MR-guided MAP methods; and the PET-MR-informed hybrid kernel and anato-functional MAP methods. The trade-off between improving the reconstruction of the whole brain region and the PET-unique regions was investigated for all methods in comparison with postsmoothed maximum likelihood expectation maximization (MLEM), evaluated in terms of structural similarity index (SSIM), normalized root mean square error (NRMSE), bias, and standard deviation. Both simulated BrainWeb (10 noise realizations) and real [ 18 F] fluorodeoxyglucose (FDG) threedimensional datasets were used. The real [ 18 F]FDG dataset was augmented with simulated tumors to allow comparison of the reconstruction methodologies for the case of known regions of PET-MR discrepancy and evaluated at full counts (100%) and at a reduced (10%) count level. Results: For the high-count simulated and real data studies, the anato-functional MAP method performed better than the other methods under investigation (MR-informed, PET-MR-informed and postsmoothed MLEM), in terms of achieving the best trade-off for the reconstruction of the whole brain and PET-unique regions, assessed in terms of the SSIM, NRMSE, and bias vs standard deviation. The inclusion of PET information in the anato-functional MAP method enables the reconstruction of PET-unique regions to attain similarly low levels of bias as unsmoothed MLEM, while moderately improving the whole brain image quality for low levels of regularization. However, for low count simulated datasets the anato-functional MAP method performs poorly, due to the inclusion of noisy PET information in the regularization term. For the low counts simulated dataset, KEM LVS and to a lesser extent, HKEM performed better than the other methods under investigation in terms of achieving the best trade-off for the reconstruction of the whole brain and PET-unique regions, assessed in te...

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“…In addition to PET reconstruction, SPDHG may help easing the computational effort in multi-modal medical imaging as well, where the PDHG has been used to jointly reconstruct from simultaneously acquired PET-MRI data with a nuclear norm variant of TGV 52 or coupled Bregman iterations. 53 Similarly, a comparable speed up is expected in other imaging modalities that involve line integrals like computerized tomography (CT) where PDHG also found its applications.…”

Section: Discussionmentioning

confidence: 85%

Faster PET reconstruction with a stochastic primal-dual hybrid gradient method

Ehrhardt

Markiewicz

Chambolle

et al. 2017

Wavelets and Sparsity XVII

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Image reconstruction in positron emission tomography (PET) is computationally challenging due to Poisson noise, constraints and potentially non-smooth priors-let alone the sheer size of the problem. An algorithm that can cope well with the first three of the aforementioned challenges is the primal-dual hybrid gradient algorithm (PDHG) studied by Chambolle and Pock in 2011. However, PDHG updates all variables in parallel and is therefore computationally demanding on the large problem sizes encountered with modern PET scanners where the number of dual variables easily exceeds 100 million. In this work, we numerically study the usage of SPDHG-a stochastic extension of PDHG-but is still guaranteed to converge to a solution of the deterministic optimization problem with similar rates as PDHG. Numerical results on a clinical data set show that by introducing randomization into PDHG, similar results as the deterministic algorithm can be achieved using only around 10 % of operator evaluations. Thus, making significant progress towards the feasibility of sophisticated mathematical models in a clinical setting.

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Joint reconstruction via coupled Bregman iterations with applications to PET-MR imaging

Cited by 30 publications

References 49 publications

Convergence rates and structure of solutions of inverse problems with imperfect forward models

Convergence rates and structure of solutions of inverse problems with imperfect forward models

Intercomparison of MR-Informed Methods for PET Image Reconstruction

Faster PET reconstruction with a stochastic primal-dual hybrid gradient method

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