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An insight into changes of soft biological tissue ultrastructures under loading conditions is essential to understand their response to mechanical stimuli. Therefore, this study offers an approach to investigate the arrangement of collagen fibrils and proteoglycans (PGs), which are located within the mechanically loaded aortic wall. The human aortic samples were either fixed directly with glutaraldehyde in the load-free state or subjected to a planar biaxial extension test prior to fixation. The aortic ultrastructure was recorded using electron tomography. Collagen fibrils and PGs were segmented using convolutional neural networks, particularly the ESPNet model. The 3D ultrastructural reconstructions revealed a complex organization of collagen fibrils and PGs. In particular, we observed that not all PGs are attached to the collagen fibrils, but some fill the spaces between the fibrils with a clear distance to the collagen. The complex organization cannot be fully captured or can be severely misinterpreted in 2D. The approach developed opens up practical possibilities, including the quantification of the spatial relationship between collagen fibrils and PGs as a function of the mechanical load. Such quantification can also be used to compare tissues under different conditions, e.g., healthy and diseased, to improve or develop new material models. STATEMENT OF SIGNIFICANCE: The developed approach enables the 3D reconstruction of collagen fibrils and proteoglycans as they are embedded in the loaded human aortic wall. This methodological pipeline comprises the knowledge of arterial mechanics, imaging with transmission electron microscopy and electron tomography, segmentation of 3D image data sets with convolutional neural networks and finally offers a unique insight into the ultrastructural changes in the aortic tissue caused by mechanical stimuli.

We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: regularization graphs. Regularization graphs allow to construct functionals using as building blocks linear operators and convex functionals, assembled by means of operators that can be seen as generalizations of classical infimal convolution operators. This class of functionals exhaustively covers existing regularization approaches and it is flexible enough to craft new ones in a simple and constructive way. We provide well-posedness and convergence results with the proposed class of functionals in a general setting. Further, we consider a bilevel optimization approach to learn optimal weights for such regularization graphs from training data. We demonstrate that this approach is capable of optimizing the structure and the complexity of a regularization graph, allowing, for example, to automatically select a combination of regularizers that is optimal for given training data.

Purpose: The presence of dipole-inconsistent data due to substantial noise or artifacts causes streaking artifacts in quantitative susceptibility mapping (QSM) reconstructions. Often used Bayesian approaches rely on regularizers, which in turn yield reduced sharpness. To overcome this problem, we present a novel L1-norm data fidelity approach that is robust with respect to outliers, and therefore prevents streaking artifacts. Methods: QSM functionals are solved with linear and nonlinear L1-norm data fidelity terms using functional augmentation, and are compared with equivalent L2-norm methods. Algorithms were tested on synthetic data, with phase inconsistencies added to mimic lesions, QSM Challenge 2.0 data, and in vivo brain images with hemorrhages. Results: The nonlinear L1-norm-based approach achieved the best overall error metric scores and better streaking artifact suppression. Notably, L1-norm methods could reconstruct QSM images without using a brain mask, with similar regularization weights for different data fidelity weighting or masking setups. Conclusion: The proposed L1-approach provides a robust method to prevent streaking artifacts generated by dipole-inconsistent data, renders brain mask calculation unessential, and opens novel challenging clinical applications such asassessing brain hemorrhages and cortical layers.

Purpose Multiplication of FLAIR and T2-weighted MRI scans results in images (called FLAIR2) with an improved contrast-to-noise ratio (CNR) for multiple sclerosis (MS) lesions but with a reduced signal-to-noise ratio (SNR). Denoising of these images may therefore further improve FLAIR2 image quality. The purpose of this work is to present a systematic investigation of FLAIR2 image denoising methods using Gaussian, Wiener and Total Generalized Variation (TGV) filtering approaches. Materials and methods T2-weighted and FLAIR data of four MS patients were used. For CNR and SNR measurements, each scan was performed up to three times. TGV, Gaussian and Wiener filtering was applied to T2, FLAIR and the FLAIR2 data. FLAIR2 images were afterwards additionally created using all combinations of input data (native, filtered T2 and filtered FLAIR). SNR and CNR measurements were performed using the subtraction method for all FLAIR2 approaches (native and filtered input data) and for twenty MS lesions. Additionally, quantitative analysis of filtering based image blurring was performed on all data sets. Results FLAIR2 images denoised with TGV showed the highest SNR and CNR, while SNR values were similar for Gaussian and Wiener filtered images. The average CNR over 20 MS lesions within the native FLAIR2 (32.99) achieved an improvement to 91.17, 82.33 and 56.07 corresponding to TGV, Wiener and Gaussian filtering. FLAIR multiplied with T2.denoised showed no improvement, while FLAIR.denoised multiplied with T2 showed an increase by a factor of two to the native, not filtered FLAIR2. Blurring was most pronounced in Gaussian filtered images and similar in TGV and Wiener filtered images. Conclusion FLAIR images filtered with Wiener or TGV multiplied with the unfiltered T2 results in FLAIR2 images with increased SNR and CNR and with minimal edge blurring.

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in (Bredies and Fanzon, ESAIM: M2AN, 54:2351-2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou-Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou-Brenier energy (Bredies et al., arXiv:1907.11589, 2019) we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists in iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimize the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model at reconstructing heavily undersampled dynamic data, together with the presence of noise., 40 pages, 8 figures, 1 table

Multi-Delay single-shot arterial spin labeling (ASL) imaging provides accurate cerebral blood flow (CBF) and, in addition, arterial transit time (ATT) maps but the inherent low SNR can be challenging. Especially standard fitting using non-linear least squares often fails in regions with poor SNR, resulting in noisy estimates of the quantitative maps. State-of-the-art fitting techniques improve the SNR by incorporating prior knowledge in the estimation process which typically leads to spatial blurring. To this end, we propose a new estimation method with a joint spatial total generalized variation regularization on CBF and ATT. This joint regularization approach utilizes shared spatial features across maps to enhance sharpness and simultaneously improves noise suppression in the final estimates. The proposed method is evaluated at three levels, first on synthetic phantom data including pathologies, followed by in vivo acquisitions of healthy volunteers, and finally on patient data following an ischemic stroke. The quantitative estimates are compared to two reference methods, non-linear least squares fitting and a state-of-the-art ASL quantification algorithm based on Bayesian inference. The proposed joint regularization approach outperforms the reference implementations, substantially increasing the SNR in CBF and ATT while maintaining sharpness and quantitative accuracy in the estimates., Comment: 35 pages, 20 figures, 3 tables, including supplementary material; submitted to Medical Image Analysis

This paper presents a novel mathematical framework for understanding pixel-driven approaches for the parallel beam Radon transform as well as for the fanbeam transform, showing that with the correct discretization strategy, convergence - including rates - in the $L^2$ operator norm can be obtained. These rates inform about suitable strategies for discretization of the occurring domains/variables, and are first established for the Radon transform. In particular, discretizing the detector in the same magnitude as the image pixels (which is standard practice) might not be ideal and in fact, asymptotically smaller pixels than detectors lead to convergence. Possible adjustments to limited-angle and sparse-angle Radon transforms are discussed, and similar convergence results are shown. In the same vein, convergence results are readily extended to a novel pixel-driven approach to the fanbeam transform. Numerical aspects of the discretization scheme are discussed, and it is shown in particular that with the correct discretization strategy, the typical high-frequency artifacts can be avoided., 33 pages, 8 figures

Purpose Highly accelerated B 1 + -mapping based on the Bloch-Siegert shift to allow 3D acquisitions even within a brief period of a single breath-hold. Theory and methods The B 1 + dependent Bloch-Siegert phase shift is measured within a highly subsampled 3D-volume and reconstructed using a two-step variational approach, exploiting the different spatial distribution of morphology and B 1 + -field. By appropriate variable substitution the basic non-convex optimization problem is transformed in a sequential solution of two convex optimization problems with a total generalized variation (TGV) regularization for the morphology part and a smoothness constraint for the B 1 + -field. The method is evaluated on 3D in vivo data with retro- and prospective subsampling. The reconstructed B 1 + -maps are compared to a zero-padded low resolution reconstruction and a fully sampled reference. Results The reconstructed B 1 + -field maps are in high accordance to the reference for all measurements with a mean error below 1% and a maximum of about 4% for acceleration factors up to 100. The minimal error for different sampling patterns was achieved by sampling a dense region in k-space center with acquisition times of around 10-12 s for 3D-acquistions. Conclusions The proposed variational approach enables highly accelerated 3D acquisitions of Bloch-Siegert data and thus full liver coverage in a single breath hold.

We propose a new type of regularization functional for images called oscillation total generalized variation (TGV) which can represent structured textures with oscillatory character in a specified direction and scale. The infimal convolution of oscillation TGV with respect to several directions and scales is then used to model images with structured oscillatory texture. Such functionals constitute a regularizer with good texture preservation properties and can flexibly be incorporated into many imaging problems. We give a detailed theoretical analysis of the infimal-convolution-type model with oscillation TGV in function spaces. Furthermore, we consider appropriate discretizations of these functionals and introduce a first-order primal-dual algorithm for solving general variational imaging problems associated with this regularizer. Finally, numerical experiments are presented which show that our proposed models can recover textures well and are competitive in comparison to existing state-of-the-art methods.

In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data.

In arterial spin labeling (ASL) a perfusion weighted image is achieved by subtracting a label image from a control image. This perfusion weighted image has an intrinsically low signal to noise ratio and numerous measurements are required to achieve reliable image quality, especially at higher spatial resolutions. To overcome this limitation various denoising approaches have been published using the perfusion weighted image as input for denoising. In this study we propose a new spatio-temporal filtering approach based on total generalized variation (TGV) regularization which exploits the inherent information of control and label pairs simultaneously. In this way, the temporal and spatial similarities of all images are used to jointly denoise the control and label images. To assess the effect of denoising, virtual ground truth data were produced at different SNR levels. Furthermore, high-resolution in-vivo pulsed ASL data sets were acquired and processed. The results show improved image quality, quantitative accuracy and robustness against outliers compared to seven state of the art denoising approaches.

Purpose The aim of the 2016 quantitative susceptibility mapping (QSM) reconstruction challenge was to test the ability of various QSM algorithms to recover the underlying susceptibility from phase data faithfully. Methods Gradient-echo images of a healthy volunteer acquired at 3T in a single orientation with 1.06 mm isotropic resolution. A reference susceptibility map was provided, which was computed using the susceptibility tensor imaging algorithm on data acquired at 12 head orientations. Susceptibility maps calculated from the single orientation data were compared against the reference susceptibility map. Deviations were quantified using the following metrics: root mean squared error (RMSE), structure similarity index (SSIM), high-frequency error norm (HFEN), and the error in selected white and gray matter regions. Results Twenty-seven submissions were evaluated. Most of the best scoring approaches estimated the spatial frequency content in the ill-conditioned domain of the dipole kernel using compressed sensing strategies. The top 10 maps in each category had similar error metrics but substantially different visual appearance. Conclusion Because QSM algorithms were optimized to minimize error metrics, the resulting susceptibility maps suffered from over-smoothing and conspicuity loss in fine features such as vessels. As such, the challenge highlighted the need for better numerical image quality criteria. Magn Reson Med, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

We study preconditioned algorithms of alternating direction method of multipliers type for nonsmooth optimization problems. The alternating direction method of multipliers is a popular first-order method for general constrained optimization problems. However, one of its drawbacks is the need to solve implicit subproblems. In various applications, these subproblems are either easily solvable or linear, but nevertheless challenging. We derive a preconditioned version that allows for flexible and efficient preconditioning for these linear subproblems. The original and preconditioned version is written as a new kind of proximal point method for the primal problem, and the weak (strong) convergence in infinite (finite) dimensional Hilbert spaces is proved. Various efficient preconditioners with any number of inner iterations may be used in this preconditioned framework. Furthermore, connections between the preconditioned version and the recently introduced preconditioned Douglas---Rachford method for general nonsmooth problems involving quadratic---linear terms are established. The methods are applied to total variation denoising problems, and their benefits are shown in numerical experiments.

While current state of the art MR-PET scanners enable simultaneous MR and PET measurements, the acquired data sets are still usually reconstructed separately. We propose a new multi-modality reconstruction framework using second order Total Generalized Variation (TGV) as a dedicated multi-channel regularization functional that jointly reconstructs images from both modalities. In this way, information about the underlying anatomy is shared during the image reconstruction process while unique differences are preserved. Results from numerical simulations and in-vivo experiments using a range of accelerated MR acquisitions and different MR image contrasts demonstrate improved PET image quality, resolution, and quantitative accuracy.

We propose and study a reconstruction method for photoacoustic tomography (PAT) based on total generalized variation (TGV) regularization for the inversion of the slice-wise 2D-Radon transform in 3D. The latter problem occurs for recently-developed PAT imaging techniques with parallelized integrating ultrasound detection where projection data from various directions is sequentially acquired. As the imaging speed is presently limited to 20 seconds per 3D image, the reconstruction of temporally-resolved 3D sequences of, e.g., one heartbeat or breathing cycle, is very challenging and currently, the presence of motion artifacts in the reconstructions obstructs the applicability for biomedical research. In order to push these techniques forward towards real time, it thus becomes necessary to reconstruct from less measured data such as few-projection data and consequently, to employ sophisticated reconstruction methods in order to avoid typical artifacts. The proposed TGV-regularized Radon inversion is a variational method that is shown to be capable of such artifact-free inversion. It is validated by numerical simulations, compared to filtered back projection (FBP), and performance-tested on real data from phantom as well as in-vivo mouse experiments. The results indicate that a speed-up factor of four is possible without compromising reconstruction quality.

In multi-modal electron tomography, tilt series of several signals such as X-ray spectra, electron energy-loss spectra, annular dark-field, or bright-field data are acquired at the same time in a transmission electron microscope and subsequently reconstructed in three dimensions. However, the acquired data are often incomplete and suffer from noise, and generally each signal is reconstructed independently of all other signals, not taking advantage of correlation between different datasets. This severely limits both the resolution and validity of the reconstructed images. In this paper, we show how image quality in multi-modal electron tomography can be greatly improved by employing variational modeling and multi-channel regularization techniques. To achieve this aim, we employ a coupled Total Generalized Variation (TGV) regularization that exploits correlation between different channels. In contrast to other regularization methods, coupled TGV regularization allows to reconstruct both hard transitions and gradual changes inside each sample, and links different channels at the level of first and higher order derivatives. This favors similar interface positions for all reconstructions, thereby improving the image quality for all data, in particular, for 3D elemental maps. We demonstrate the joint multi-channel TGV reconstruction on tomographic energy-dispersive X-ray spectroscopy (EDXS) and high-angle annular dark field (HAADF) data, but the reconstruction method is generally applicable to all types of signals used in electron tomography, as well as all other types of projection-based tomographies.

In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase field is an $H^1$-function, shows that the new model leads to sharper phase fields., 32 pages, 4 figures, 1 table, 39 references

In this paper we propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose is based on dynamic (un-)balanced optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation, i.e., the Radon measure at time $t$ is advected by a velocity field $v$ and varies with a growth rate $g$, and (ii) are penalized with the kinetic energy induced by $v$ and a growth energy induced by $g$. We establish a functional-analytic framework for these regularized inverse problems, prove that minimizers exist and are unique in some cases, and study regularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modelling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents., 35 pages

Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.

Funder: Christian Doppler Research Association; Id: http://dx.doi.org/10.13039/501100006012, Funder: Royal Society; Id: http://dx.doi.org/10.13039/501100000288, In this paper, we characterize the extremal points of the unit ball of the Benamou–Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite‐dimensional data and optimal‐transport based regularization.

Purpose To accelerate dynamic MR applications using infimal convolution of total generalized variation functionals (ICTGV) as spatio-temporal regularization for image reconstruction. Theory and Methods ICTGV comprises a new image prior tailored to dynamic data that achieves regularization via optimal local balancing between spatial and temporal regularity. Here it is applied for the first time to the reconstruction of dynamic MRI data. CINE and perfusion scans were investigated to study the influence of time dependent morphology and temporal contrast changes. ICTGV regularized reconstruction from subsampled MR data is formulated as a convex optimization problem. Global solutions are obtained by employing a duality based non-smooth optimization algorithm. Results The reconstruction error remains on a low level with acceleration factors up to 16 for both CINE and dynamic contrast-enhanced MRI data. The GPU implementation of the algorithm suites clinical demands by reducing reconstruction times of one dataset to less than 4 min. Conclusion ICTGV based dynamic magnetic resonance imaging reconstruction allows for vast undersampling and therefore enables for very high spatial and temporal resolutions, spatial coverage and reduced scan time. With the proposed distinction of model and regularization parameters it offers a new and robust method of flexible decomposition into components with different degrees of temporal regularity. Magn Reson Med 78:142–155, 2017. © 2016 International Society for Magnetic Resonance in Medicine

In this article, we evaluate Parallel Level Sets (PLS) and Bowsher's method as segmentation-free anatomical priors for regularized brain positron emission tomography (PET) reconstruction. We derive the proximity operators for two PLS priors and use the EM-TV algorithm in combination with the first order primal-dual algorithm by Chambolle and Pock to solve the non-smooth optimization problem for PET reconstruction with PLS regularization. In addition, we compare the performance of two PLS versions against the symmetric and asymmetric Bowsher priors with quadratic and relative difference penalty function. For this aim, we first evaluate reconstructions of 30 noise realizations of simulated PET data derived from a real hybrid positron emission tomography/magnetic resonance imaging (PET/MR) acquisition in terms of regional bias and noise. Second, we evaluate reconstructions of a real brain PET/MR data set acquired on a GE Signa time-of-flight PET/MR in a similar way. The reconstructions of simulated and real 3D PET/MR data show that all priors were superior to post-smoothed maximum likelihood expectation maximization with ordered subsets (OSEM) in terms of bias-noise characteristics in different regions of interest where the PET uptake follows anatomical boundaries. Our implementation of the asymmetric Bowsher prior showed slightly superior performance compared with the two versions of PLS and the symmetric Bowsher prior. At very high regularization weights, all investigated anatomical priors suffer from the transfer of non-shared gradients. ispartof: IEEE Transactions on Medical Imaging vol:37 issue:2 pages:590-603 ispartof: location:United States status: published

In this paper we characterize sparse solutions for variational problems of the form $$\min _{u\in X} \phi (u) + F(\mathcal {A}u)$$minu∈Xϕ(u)+F(Au), where X is a locally convex space, $$\mathcal {A}$$A is a linear continuous operator that maps into a finite dimensional Hilbert space and $$\phi $$ϕ is a seminorm. More precisely, we prove that there exists a minimizer that is “sparse” in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer $$\phi $$ϕ (possibly translated by an element in the null space of $$\phi $$ϕ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in Unser et al. (SIAM Rev 59(4):769–793, 2017) under weaker hypotheses.

Our first encounter with a partial differential equation is this book was Application 3.23 on edge detection according to Canny: we obtained a smoothed image by solving the heat equation. The underlying idea was that images contain information on different spatial scales and one should not fix one scale a priori.

Like the methods covered in Chap. 3, the methods based on frequency or scale-space decompositions belong to the older methods in image processing. In this case, the basic idea is to transform an image into a different representation in order to determine its properties or carry out manipulations. In this context, the Fourier transformation plays an important role.

The recently introduced preconditioned Douglas---Rachford iteration (PDR) for convex---concave saddle-point problems is studied with respect to convergence rates and applied to variational imaging problems with total variation (TV) and total generalized variation (TGV) penalty. A rate of $${\mathcal {O}}(1/k)$$O(1/k) for restricted primal---dual gaps evaluated for ergodic sequences generated by the PDR iteration is established. Based on PDR, new fast iterative algorithms for TV-denoising, TV-deblurring, and TGV-denoising of second order with $$L^2$$L2 and $$L^1$$L1 discrepancy are proposed. While for denoising, symmetric (block) Red---Black Gauss---Seidel preconditioners are effective, fast Fourier transform-based preconditioners are employed for the deblurring problems. Finally, for the $$L^2$$L2-TGV-denoising problem, an effective modified primal---dual gap is developed which may serve as a stopping criterion. All algorithms are tested and compared in numerical experiments. In particular, for problems where strong convexity does not hold, it turns out that the proposed preconditioning techniques are beneficial and lead to competitive results.

We propose a preconditioned version of the Douglas--Rachford splitting method for solving convex-concave saddle-point problems associated with Fenchel--Rockafellar duality. Our approach makes it possible to use approximate solvers for the linear subproblem arising in this context. We prove weak convergence in Hilbert space under minimal assumptions. In particular, various efficient preconditioners are introduced in this framework for which only a few inner iterations are needed instead of computing an exact solution or controlling the error. The method is applied to a discrete total-variation denoising problem. Numerical experiments show that the proposed algorithms with appropriate preconditioners are very competitive with existing fast algorithms including the first-order primal-dual algorithm for saddle-point problems of Chambolle and Pock.

SFB-F32

A variational model for image reconstruction is introduced and analyzed in function space. Specific to the model is the data fidelity, which is realized via a basis transformation with respect to a Riesz basis followed by interval constraints. This setting in particular covers the task of reconstructing images constrained to data obtained from JPEG or JPEG 2000 compressed files. As image prior, the total generalized variation (TGV) functional of arbitrary order is employed. The present paper, the first of two works that deal with both analytical and numerical aspects of the model, provides a comprehensive analysis in function space and defines concrete instances for particular applications. A new, noncoercive existence result and optimality conditions, including a characterization of the subdifferential of the TGV functional, are obtained in the general setting.

We propose a convex, lower semicontinuous, coercive approximation of Euler's elastica energy for images, which is thus very well-suited as a regularizer in image processing. The approximation is not quite the convex relaxation, and we discuss its close relation to the exact convex relaxation as well as the difficulties associated with computing the latter. Interestingly, the convex relaxation of the elastica energy reduces to constantly zero if the total variation part of the elastica energy is neglected. Our convex approximation arises via functional lifting of the image gradient into a Radon measure on the four-dimensional space $\Omega\times S^1\times\mathbb{R}$, $\Omega\subset\mathbb{R}^2$, of which the first two coordinates represent the image domain and the last two the normal and curvature of the image level lines. It can be expressed as a linear program which also admits a predual representation. Combined with a tailored discretization of measures via linear combinations of short line measures, th...

Multiplication of FLAIR and T2-weighted MRI scans results in images (called FLAIRT2-weighted and FLAIR data of four MS patients were used. For CNR and SNR measurements, each scan was performed up to three times. TGV, Gaussian and Wiener filtering was applied to T2, FLAIR and the FLAIRFLAIRFLAIR images filtered with Wiener or TGV multiplied with the unfiltered T2 results in FLAIR

The aim of the 2016 quantitative susceptibility mapping (QSM) reconstruction challenge was to test the ability of various QSM algorithms to recover the underlying susceptibility from phase data faithfully.Gradient-echo images of a healthy volunteer acquired at 3T in a single orientation with 1.06 mm isotropic resolution. A reference susceptibility map was provided, which was computed using the susceptibility tensor imaging algorithm on data acquired at 12 head orientations. Susceptibility maps calculated from the single orientation data were compared against the reference susceptibility map. Deviations were quantified using the following metrics: root mean squared error (RMSE), structure similarity index (SSIM), high-frequency error norm (HFEN), and the error in selected white and gray matter regions.Twenty-seven submissions were evaluated. Most of the best scoring approaches estimated the spatial frequency content in the ill-conditioned domain of the dipole kernel using compressed sensing strategies. The top 10 maps in each category had similar error metrics but substantially different visual appearance.Because QSM algorithms were optimized to minimize error metrics, the resulting susceptibility maps suffered from over-smoothing and conspicuity loss in fine features such as vessels. As such, the challenge highlighted the need for better numerical image quality criteria. Magn Reson Med 79:1661-1673, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Convergence analysis is carried out for a forward-backward splitting/generalized gradient projection method for the minimization of a special class of non-smooth and genuinely non-convex minimization problems in infinite-dimensional Hilbert spaces. The functionals under consideration are the sum of a smooth, possibly non-convex and non-smooth, necessarily non-convex functional. For separable constraints in the sequence space, we show that the generalized gradient projection method amounts to a discontinuous iterative thresholding procedure, which can easily be implemented. In this case we prove strong subsequential convergence and moreover show that the limit satisfies strengthened necessary conditions for a global minimizer, i.e., it avoids a certain set of non-global minimizers. Eventually, the method is applied to problems arising in the recovery of sparse data, where strong convergence of the whole sequence is shown, and numerical tests are presented.