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Magnetic particle imaging (MPI) is an emerging medical imaging modality which is based on the non-linear response of magnetic nanoparticles to an applied magnetic field. It is an important feature of MPI that even fast dynamic processes can be captured for 3D volumes. The high temporal resolution in turn leads to large amounts of data which have to be handled efficiently. But as the system matrix of MPI is non-sparse, the image reconstruction gets computationally demanding. Therefore, currently only basic image reconstruction methods such as Tikhonov regularization are used. However, Tikhonov regularization is known to oversmooth edges in the reconstructed image and to have only a limited noise reducing effect. In this work, we develop an efficient edge preserving and noise reducing reconstruction method for MPI. As regularization model, we propose to use the nonnegative fused lasso model, and we devise a discretization that is adapted to the acquisition geometry of the preclinical MPI scanner considered in this work. We develop a customized solver based on a generalized forward-backward scheme which is particularly suitable for the dense and not well-structured system matrices in MPI. Already a non-optimized prototype implementation processes a 3D volume within a few seconds so that processing several frames per second seems amenable. We demonstrate the improvement in reconstruction quality over the state-of-the-art method in an experimental medical setup for an in-vitro angioplasty of a stenosis. [ABSTRACT FROM PUBLISHER]

In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup. [ABSTRACT FROM AUTHOR]

We investigate the nonsmooth and nonconvex L¹-Potts functional in discrete and continuous time. We show G-convergence of discrete L¹-Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete L¹-Potts problem, we introduce an O(n²) time and O(n) space algorithm to compute an exact minimizer. We apply L¹-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements f. It turns out that the L¹-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the L¹-Potts functional. Furthermore, for strongly blurred signals and a known blurring operator, we derive an iterative reconstruction algorithm. [ABSTRACT FROM AUTHOR]

In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, ground-based astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the one-dimensional sphere 핊¹. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration, such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so-called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for 핊¹-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in [0, π]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm. [ABSTRACT FROM AUTHOR]

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted \ell ^1 minimization (sparse signals). [ABSTRACT FROM AUTHOR]