Highly accelerated aortic 4D flow MR imaging with variable-density random undersampling

Purpose

To investigate an effective time-resolved variable-density random undersampling scheme combined with an efficient parallel image reconstruction method for highly accelerated aortic 4D flow MR imaging with high reconstruction accuracy.

Materials and Methods

Variable-density Poisson-disk sampling (vPDS) was applied in both the phase-slice encoding plane and the temporal domain to accelerate the time-resolved 3D Cartesian acquisition of flow imaging. In order to generate an improved initial solution for the iterative self-consistent parallel imaging method (SPIRiT), a sample-selective view sharing reconstruction for time-resolved random undersampling (STIRRUP) was introduced. The performance of different undersampling and image reconstruction schemes were evaluated by retrospectively applying those to fully sampled data sets obtained from three healthy subjects and a flow phantom.

Results

Undersampling pattern based on the combination of time-resolved vPDS, the temporal sharing scheme STIRRUP, and parallel imaging SPIRiT, were able to achieve 6-fold accelerated 4D flow MRI with high accuracy using a small number of coils (N = 5). The normalized root mean square error between aorta flow waveforms obtained with the acceleration method and the fully sampled data in three healthy subjects was 0.04 ± 0.02, and the difference in peak-systolic mean velocity was − 0.29 ± 2.56 cm/s.

Conclusion

Qualitative and quantitative evaluation of our preliminary results demonstrate that time-resolved variable-density random sampling is efficient for highly accelerating 4D flow imaging while maintaining image reconstruction accuracy.     

Harada–Tsutsui gauge recovery procedure: From Abelian gauge anomalies to the Stueckelberg mechanism

Revisiting a path-integral procedure developed by Harada and Tsutsui for recovering gauge invariance from anomalous effective actions, it is shown that there are two ways to achieve gauge symmetry: one already presented by the authors, which is shown to preserve the anomaly in the sense of standard current conservation law, and another one which is anomaly-free, preserving current conservation. It is also shown that the application of the Harada–Tsutsui technique to other models which are not anomalous but do not exhibit gauge invariance allows the identification of the gauge invariant formulation of the Proca model, also done by the referred authors, with the Stueckelberg model, leading to the interpretation of the gauge invariant map as a generalization of the Stueckelberg mechanism.     

On gravity localization in scalar braneworlds with a super-exponential warp factor

The optical medium analogy of a given spacetime was developed decades ago and has since then been widely applied to different gravitational contexts. Here we consider the case of a colliding gravitational wave spacetime, generalizing previous results concerning single gravitational pulses. Given the complexity of the nonlinear interaction of two gravitational waves in the framework of general relativity, typically leading to the formation of either horizons or singularities, the optical medium analogy proves helpful to simply capture some interesting effects of photon propagation.     

Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

This study examines various statistical distributions in connection with random Vandermonde matrices and their extension to \(d\) -dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be \(O(\log ^{1/2}{N^{d}})\) and \(\Omega ((\log N^{d} /(\log \log N^d))^{1/2})\) , respectively, where \(N\) is the dimension of the matrix, generalizing the results in Tucci and Whiting (IEEE Trans Inf Theory 57(6):3938–3954, 2011). We further study the behavior of the minimum singular value of these random matrices. In particular, we prove that the minimum singular value is at most \(N\exp (-C\sqrt{N}))\) with high probability where \(C\) is a constant independent of \(N\) . Furthermore, the value of the constant \(C\) is determined explicitly. The main result is obtained in two different ways. One approach uses techniques from stochastic processes and in particular a construction related to the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a lower bound for the maximum absolute value of a random complex polynomial on the unit circle, which may be of independent mathematical interest. Lastly, for each sequence of positive integers \(\{k_p\}_{p=1}^{\infty }\) we present a generalized version of the previously discussed matrices. The classical random Vandermonde matrix corresponds to the sequence \(k_{p}=p-1\) . We find a combinatorial formula for their moments and show that the limit eigenvalue distribution converges to a probability measure supported on \([0,\infty )\) . Finally, we show that for the sequence \(k_p=2^{p}\) the limit eigenvalue distribution is the famous Marchenko–Pastur distribution.     

The cluster problem revisited

In continuous branch-and-bound algorithms, a very large number of boxes near global minima may be visited prior to termination. This so-called cluster problem (J Glob Optim 5(3):253–265, 1994) is revisited and a new analysis is presented. Previous results are confirmed, which state that at least second-order convergence of the relaxations is required to overcome the exponential dependence on the termination tolerance. Additionally, it is found that there exists a threshold on the convergence order pre-factor which can eliminate the cluster problem completely for second-order relaxations. This result indicates that, even among relaxations with second-order convergence, behavior in branch-and-bound algorithms may be fundamentally different depending on the pre-factor. A conservative estimate of the pre-factor is given for $\alpha $ BB relaxations.     

A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k  -graph C^?$C^{?}$-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k  -graph C^?$C^{?}$-algebra, it is sufficient that the representation be injective on a distinguished abelian C^?$C^{?}$-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C^?$C^{?}$-algebra, each of which is the unique extension of a state on the distinguished abelian C^?$C^{?}$-subalgebra.     

On intransitive graph-restrictive permutation groups

Let Γ be a finite connected G-vertex-transitive graph and let v be a vertex of Γ. If the permutation group induced by the action of the vertex-stabiliser G _v on the neighbourhood Γ(v) is permutation isomorphic to L, then (Γ,G) is said to be locally L. A permutation group L is graph-restrictive if there exists a constant c(L) such that, for every locally L pair (Γ,G) and a vertex v of Γ, the inequality |G _v |≤c(L) holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.