Wasserstein GANs with Gradient Penalty (WGAN-GP) are a very popular method for training generative models to produce high quality synthetic data. While WGAN-GP were initially developed to calculate the Wasserstein 1 distance between generated and real data, recent works (e.g. [23]) have provided empirical evidence that this does not occur, and have argued that WGAN-GP perform well not in spite of this issue, but because of it. In this paper we show for the first time that WGAN-GP compute the minimum of a different optimal transport problem, the so-called congested transport [7]. Congested transport determines the cost of moving one distribution to another under a transport model that penalizes congestion. For WGAN-GP, we find that the congestion penalty has a spatially varying component determined by the sampling strategy used in [12] which acts like a local speed limit, making congestion cost less in some regions than others. This aspect of the congested transport problem is new, in that the congestion penalty turns out to be unbounded and depends on the distributions to be transported, and so we provide the necessary mathematical proofs for this setting. One facet of our discovery is a formula connecting the gradient of solutions to the optimization problem in WGAN-GP to the time averaged momentum of the optimal mass flow. This is in contrast to the gradient of Kantorovich potentials for the Wasserstein 1 distance, which is just the normalized direction of flow. Based on this and other considerations, we speculate on how our results explain the observed performance of WGAN-GP. Beyond applications to GANs, our theorems also point to the possibility of approximately solving large scale congested transport problems using neural network techniques.

We present a quantitative validation study to assess the accuracy of low-frequency conductivity imaging methods, based on a testing current measured using Current Density Imaging (CDI). We tested the proposed procedure to study the influence of tissue anisotropy on the accuracy of conductivity reconstruction methods, using a finite element model of anisotropic brain tissue. Simulations were carried out for three different levels of tissue anisotropy to compare the results obtained by our recently developed anisotropic conductivity method with those obtained by our well-established conductivity method that assumes isotropic conductivity. The validation results clearly show that the conductivity imaging method which takes into account tissue anisotropy yields significantly superior accuracy.

Radio-frequency current density imaging (RF-CDI) is a technique that noninvasively measures current density distributions at the Larmor frequency utilizing magnetic resonance imaging. Previously implemented RF-CDI methods reconstruct the applied current density component J(z) along the static magnetic field of the imager [(B)\vec](0) (the z direction) based on the assumption that the z-directional change of the magnetic field component H(z) can be ignored compared to J(z). However, this condition may be easily violated in biomedical applications. We propose a new reconstruction method for RF-CDI, which does not rely on the aforementioned assumption. Instead, the sample is rotated by 180 (°) in the horizontal plane to collect magnetic resonance data from two opposite positions. Using simulations and experiments, we have verified that this approach can fully recover one component of current density. Furthermore, this approach can be extended to measure three dimensional current density vectors by one additional sample orientation in the horizontal plane. We have therefore demonstrated for the first time the feasibility of imaging the magnitude and phase of all components of a radio-frequency current density vector field.

Radio-frequency current density imaging (RF-CDI) is an imaging technique that noninvasively measures current density distribution at the Larmor frequency utilizing magnetic resonance imaging (MRI). Previously implemented RF-CDI techniques were only able to image a single slice transverse to the static magnetic field B(0) . This paper describes the first realization of a multislice RF-CDI sequence on a 1.5 T clinical imager. Multislice RF current density images have been reconstructed for two phantoms. The influence of MRI random noise on the sensitivity of the multislice RF-CDI measurement has also been studied by theoretical analysis, simulation and phantom experiments.

Current density impedance imaging (CDII) is a new impedance imaging technique that can noninvasively measure the conductivity distribution inside a medium. It utilizes current density vector measurements which can be made using a magnetic resonance imager (MRI) (Scott , 1991). CDII is based on a simple mathematical expression for inverted Delta sigma / sigma = inverted Delta ln sigma, the gradient of the logarithm of the conductivity sigma, at each point in a region where two current density vectors J1 and J2 have been measured and J1 x J2 not equal 0. From the calculated inverted Delta ln sigma and a priori knowledge of the conductivity at the boundary, the logarithm of the conductivity ln sigma is integrated by two different methods to produce an image of the conductivity sigma in the region of interest. The CDII technique was tested on three different conductivity phantoms. Much emphasis has been placed on the experimental validation of CDII results against direct bench measurements by commercial LCR meters before and after CDII was performed.