Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters e.g., relaxation times T1, T2, or proton density ρ. Recently in [Ma et al., Nature, 2013], Magnetic Resonance Fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two step procedure: (i) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a pre-computed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into a perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is non-convex and that the accuracy of the MRF-type algorithms is limited by the "discretization size" of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two step method, our model is dictionary-free and is rather governed by a single non-linear equation, which is studied analytically. This non-linear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples for which also improvement over MRF and its variants is documented.Here m, yielding m = ρm, is the macroscopic magnetization of (Hydrogen) proton of some unitary density in the tissue under an external magnetic field B, and the relaxation rates T 1 and T 2 are associated model parameters. Further, m 0 represents an initial state. System (1.1) is instrumental in our quantification process established below and will be further described in Section 2.1.Although qMRI techniques are still in their infancy, several interesting ideas and methods have already been conceived. Early approaches [25] are based on a set of spin echo or inversion recovery images that are reconstructed from k-space data with respect to various repetition times (T R) and echo times (T E). In that context, acquisitions are designed for each parameter individually. The overall technique is often referred to as parametric mapping method and consists of two steps: (i) reconstruct a sequence of images as in qualitative MRI, and (ii) for each pixel of those images fit its intensity to an ansatz curve characterized by the magnetic parameter associated to the tissue imaged at that pixel. Based on this idea, many improvements have been suggested in the literature; see for instance [17]. The associated approaches aim to simplify the physical model and handle tissue parameters separately, as these are considered to be time consuming for the patient.Another line of research, initiated by Ma et al. in [27] and named Magnetic Resonance Fingerprinting (MRF), has recently gained considerable attention. First, in an offline phase, it builds a database (dictio...
Measuring the performance of Freight Villages (FVs) has important implications for logistics companies and other related companies as well as governments. In this paper we apply Data Envelopment Analysis (DEA) to measure the performance of European FVs in a purely data-driven way incorporating the nature of FVs as complex operations that use multiple inputs and produce several outputs. We employ several DEA models and perform a complete sensitivity analysis of the appropriateness of the chosen input and output variables, and an assessment of the robustness of the efficiency score. It turns out that about half of the 20 FVs analyzed are inefficient, with utilization of the intermodal area and warehouse capacity and level of goods handed the being the most important areas of improvement. While we find no significant differences in efficiency between FVs of different sizes and in different countries, it turns out that the FVs Eurocentre Toulouse, Interporto Quadrante Europa and GVZ Nürnberg constitute more than 90% of the benchmark share.
Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.
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