MPISuperRes-PnP: A Super-Resolution Zero-Shot Plug-and-Play Reconstruction Algorithm for Magnetic Particle Imaging

Magnetic Particle Imaging (MPI) is an emerging medical imaging modality. MPI is based on the non-linear response of magnetic nanoparticles to an applied magnetic field and avoids ionizing radiation. The measured signal is the voltage induced in receive coils by the particles' response. Reconstructing the particle concentration from the signal constitutes the imaging task. Even using state-of-the-art measurement-based reconstruction, the associated spatial grid is very coarse, hence super-resolution (SR) techniques are important. In this work, we propose an approach for SR in MPI inspired by energy minimization. Different methods have been proposed for SR in MPI, ranging from upscaling of the associated system matrix to interpolation of the reconstruction. Here we incorporate SR into the reconstruction task via an energy minimization formulation. Following the plug-and-play approach to energy minimization we derive a splitting scheme and a SR method for MPI where the arising Gaussian denoising task is treated with a pre-trained learned Gaussian denoiser in a zero-shot fashion. This way, we incorporate benefits of deep learning without training and avoid the need of training data. Further, we provide a quantitative and qualitative evaluation of the proposed method. Hyper-parameter are selected via an extended parameter search. The found parameters are applied for reconstruction on real data. We show the applicability of our method on synthetic and on real data (MPIData: EquilibriumModelWithAnisotropy and 2D-OpenMPI Data). The proposed method employs a deep-learning denoiser without training -- thus it does not require presently scarcely available MPI training data. The denoiser behaves conservatively, i.e., no hallucination artifacts were observed. The SR approach is generic such that it can be applied in future MPI contexts involving different regularizers or different imaging tasks.

New Synchronous Computation Dynamics for Hopfield Networks

The dynamics of the original Hopfield network is asynchronous (sequential) (updates the state of only one neuron per time step). In this paper, we propose a new tool and a new dynamics to reduce the processing time by updating one or more neurons simultaneously per instant while ensuring process convergence and aiming for the maximum energy decrease at each step, thus guaranteeing the shortest total processing time. From the point of view of synchronous dynamics, calculating the next network state at which energy decreases the most from the current state while ensuring convergence is itself a combinatorial optimization problem. We develop and use a new tool to solve it. We call this new tool Discrete Differential Filter (DDF) and, based upon it, we develop a new synchronous dynamics which we call SD-DDF (Synchronous Dynamics based upon Discrete Differential Filter). In this paper, we review the original asynchronous dynamics for Hopfield networks and present a new tool and a new synchronous dynamics with its theoretical justification and four computational experiments to assess the speed up in processing time empirically.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning

We illustrate in several examples that even neural networks of infinite width (specifically, Barron functions) may encounter substantial obstacles when used as a model class for problems in the calculus of variations. An instance of practical relevance concerns the bending, stretching and folding of a thin elastic shell with anchored or clamped boundary conditions where elastic energy could be reduced by folding along a circular line, but the neural networks can only describe straight folds along entire lines. Conversely, we show that there is no gap between the energy that Barron functions and Lipschitz functions can achieve for a large class of integral first-order functionals.