In this work, we explore the ability of NN (Neural Networks) to serve as a tool for finding eigen-pairs of ordinary differential equations. The question we aime to address is whether, given a self-adjoint operator, we can learn what are the eigenfunctions, and their matching eigenvalues. The topic of solving the eigen-problem is widely discussed in Image Processing, as many image processing algorithms can be thought of as such operators. We suggest an alternative to numeric methods of finding eigenpairs, which may potentially be more robust and have the ability to solve more complex problems. In this work, we focus on simple problems for which the analytical solution is known. This way, we are able to make initial steps in discovering the capabilities and shortcomings of DNN (Deep Neural Networks) in the given setting.

Inverse problems in partial differential equations are fundamental in science and mathematics with wide applications in medical imaging, signal processing, computer vision, remote sensing, electromagnetism and more. Classical methods such as finite differences, finite volume and finite elements are numerical discretization-based methods where the domain is divided into a uniform grid or polygon mesh. The differential equation is then reduced to a system of algebraic equations. These methods may have some limitations: the solution is numeric and may suffer from high condition number, highly dependent on the discretization and even the second derivative is sensitive to noise.

Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a non invasive method for brain neuronal fibers delineation. Here we show a modification for DT-MRI that allows delineation of neuronal fibers which are infiltrated by edema. We use the Muliple Tensor Variational (MTV) framework which replaces the diffusion model of DT-MRI with a multiple component model and fits it to the signal attenuation with a variational regularization mechanism. In order to reduce free water contamination we estimate the free water compartment volume fraction in each voxel, remove it, and then calculate the anisotropy of the remaining compartment. The variational framework was applied on data collected with conventional clinical parameters, containing only six diffusion directions. By using the variational framework we were able to overcome the highly ill posed fitting. The results show that we were able to find fibers that were not found by DT-MRI.

Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a non invasive methodfor brain neuronal fibers delineation. Here we show a modification forDT-MRI that allows delineation of neuronal fibers which are infiltrated by edema. We use the Muliple Tensor Variational (MTV) framework which replaces the diffusion model of DT-MRI with a multiple componentmodel and fits it to the signal attenuation with a variational regularizationmechanism. In order to reduce free water contamination weestimate the free water compartment volume fraction in each voxel, remove it, and then calculate the anisotropy of the remaining compartment.