Martin Benning

Phase reconstruction from velocity-encoded MRI measurements--a survey of sparsity-promoting variational approaches.

In recent years there has been significant developments in the reconstruction of magnetic resonance velocity images from sub-sampled k-space data. While showing a strong improvement in reconstruction quality compared to classical approaches, the vast number of different methods, and the challenges in setting them up, often leaves the user with the difficult task of choosing the correct approach, or more importantly, not selecting a poor approach. In this paper, we survey variational approaches for the reconstruction of phase-encoded magnetic resonance velocity images from sub-sampled k-space data. We are particularly interested in regularisers that correctly treat both smooth and geometric features of the image. These features are common to velocity imaging, where the flow field will be smooth but interfaces between the fluid and surrounding material will be sharp, but are challenging to represent sparsely. As an example we demonstrate the variational approaches on velocity imaging of water flowing through a packed bed of solid particles. We evaluate Wavelet regularisation against Total Variation and the relatively recent second order Total Generalised Variation regularisation. We combine these regularisation schemes with a contrast enhancement approach called Bregman iteration. We verify for a variety of sampling patterns that Morozovs discrepancy principle provides a good criterion for stopping the iterations. Therefore, given only the noise level, we present a robust guideline for setting up a variational reconstruction scheme for MR velocity imaging.

An adaptive inverse scale space method for compressed sensing

In this paper we introduce a novel adaptive approach for solving `-minimization problems as frequently arising in compressed sensing, which is based on the recently introduced inverse scale space method. The scheme allows to efficiently compute minimizers by solving a sequence of low-dimensional nonnegative least-squares problems. We provide a detailed convergence analysis in a general setup as well as refined results under special conditions. In addition we discuss experimental observations in several numerical examples.

Variational Depth From Focus Reconstruction

This paper deals with the problem of reconstructing a depth map from a sequence of differently focused images, also known as depth from focus (DFF) or shape from focus. We propose to state the DFF problem as a variational problem, including a smooth but nonconvex data fidelity term and a convex nonsmooth regularization, which makes the method robust to noise and leads to more realistic depth maps. In addition, we propose to solve the nonconvex minimization problem with a linearized alternating directions method of multipliers, allowing to minimize the energy very efficiently. A numerical comparison to classical methods on simulated as well as on real data is presented.

A nonlinear variational method for improved quantification of myocardial blood flow using dynamic H 2 15 O PET

H215O as a PET-tracer offers the opportunity to examine perfusion of blood into tissue non-invasively (cf. [1]). It features a short radioactive half-life (≈ 2 min.) and therefore adds a smaller radiation exposure to the patient in comparison to other tracers. The disadvantages arising from the short radioactive half-life are noisy, low-resolution reconstructions. Previous algorithms first reconstruct images from each dynamic H215O dataset independently, e.g. via the standard EM-algorithm (cf. [2]) or FBP. Hence, temporal correlation is neglected. The myocardial blood flow (MBF) and other important parameters, like tissue fraction, arterial and venous spillover effects are computed subsequently from these reconstructed images. Our new method interprets the direct computation of parameters as a nonlinear inverse problem. This implies the need for inversion of a nonlinear operator G(p) (with p denoting the parameters to compute), but allows to skip the process of generating noisy images. The process is schematically described in Figure 1. Therefore, our method takes into account the temporal correlation between the datasets, and not the correlation between noisy, low resolution images. The problem is transferred to a nonlinear parameter identification problem. Furthermore, regularization can be added to each parameter independently, assuring meaningful results.

Preconditioned ADMM with nonlinear operator constraint

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).

Fast imaging of laboratory core floods using 3D compressed sensing RARE MRI.

Three-dimensional (3D) imaging of the fluid distributions within the rock is essential to enable the unambiguous interpretation of core flooding data. Magnetic resonance imaging (MRI) has been widely used to image fluid saturation in rock cores; however, conventional acquisition strategies are typically too slow to capture the dynamic nature of the displacement processes that are of interest. Using Compressed Sensing (CS), it is possible to reconstruct a near-perfect image from significantly fewer measurements than was previously thought necessary, and this can result in a significant reduction in the image acquisition times. In the present study, a method using the Rapid Acquisition with Relaxation Enhancement (RARE) pulse sequence with CS to provide 3D images of the fluid saturation in rock core samples during laboratory core floods is demonstrated. An objective method using image quality metrics for the determination of the most suitable regularisation functional to be used in the CS reconstructions is reported. It is shown that for the present application, Total Variation outperforms the Haar and Daubechies3 wavelet families in terms of the agreement of their respective CS reconstructions with a fully-sampled reference image. Using the CS-RARE approach, 3D images of the fluid saturation in the rock core have been acquired in 16min. The CS-RARE technique has been applied to image the residual water saturation in the rock during a water-water displacement core flood. With a flow rate corresponding to an interstitial velocity of vi=1.89±0.03ftday(-1), 0.1 pore volumes were injected over the course of each image acquisition, a four-fold reduction when compared to a fully-sampled RARE acquisition. Finally, the 3D CS-RARE technique has been used to image the drainage of dodecane into the water-saturated rock in which the dynamics of the coalescence of discrete clusters of the non-wetting phase are clearly observed. The enhancement in the temporal resolution that has been achieved using the CS-RARE approach enables dynamic transport processes pertinent to laboratory core floods to be investigated in 3D on a time-scale and with a spatial resolution that, until now, has not been possible.

Quantitative mapping of chemical compositions with MRI using compressed sensing.

In this work, a magnetic resonance (MR) imaging method for accelerating the acquisition time of two dimensional concentration maps of different chemical species in mixtures by the use of compressed sensing (CS) is presented. Whilst 2D-concentration maps with a high spatial resolution are prohibitively time-consuming to acquire using full k-space sampling techniques, CS enables the reconstruction of quantitative concentration maps from sub-sampled k-space data. First, the method was tested by reconstructing simulated data. Then, the CS algorithm was used to reconstruct concentration maps of binary mixtures of 1,4-dioxane and cyclooctane in different samples with a field-of-view of 22mm and a spatial resolution of 344μm×344μm. Spiral based trajectories were used as sampling schemes. For the data acquisition, eight scans with slightly different trajectories were applied resulting in a total acquisition time of about 8min. In contrast, a conventional chemical shift imaging experiment at the same resolution would require about 17h. To get quantitative results, a careful weighting of the regularisation parameter (via the L-curve approach) or contrast-enhancing Bregman iterations are applied for the reconstruction of the concentration maps. Both approaches yield relative errors of the concentration map of less than 2mol-% without any calibration prior to the measurement. The accuracy of the reconstructed concentration maps deteriorates when the reconstruction model is biased by systematic errors such as large inhomogeneities in the static magnetic field. The presented method is a powerful tool for the fast acquisition of concentration maps that can provide valuable information for the investigation of many phenomena in chemical engineering applications.

A Solver for Dynamic PET Reconstructions based on Forward‐Backward‐Splitting

Dynamic Positron Emission Tomography allows monitoring physiological processes within the body that can be described by kinetic parameters. However, recovery of these parameters often requires the solution of complex and nonlinear operator equations. Advanced operator splitting techniques allow incorporating a‐priori knowledge, e.g. sparsity of minimizers with respect to an exponential basis, into the reconstruction process.

Nonlinear Spectral Image Fusion

In this paper we demonstrate that the framework of nonlinear spectral decompositions based on total variation (TV) regularization is very well suited for image fusion as well as more general image manipulation tasks. The well-localized and edge-preserving spectral TV decomposition allows to select frequencies of a certain image to transfer particular features, such as wrinkles in a face, from one image to another. We illustrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing, linear osmosis, wavelet fusion and Laplacian pyramid fusion. We conclude that the proposed spectral TV image decomposition framework is a valuable tool for semi- and fully-automatic image editing and fusion.

A Primal-Dual Approach for a Total Variation Wasserstein Flow

We consider a nonlinear fourth-order diffusion equation that arises in denoising of image densities. We propose an implicit time-stepping scheme that employs a primal-dual method for computing the subgradient of the total variation seminorm. The constraint on the dual variable is relaxed by adding a penalty term, depending on a parameter that determines the weight of the penalisation. The paper is furnished with some numerical examples showing the denoising properties of the model considered.