Deep Learning Solution of the Eigenvalue Problem for Differential Operators.

Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization and yield nonanalytic or nonsmooth approximations. We introduce a novel neural network-based solver for the eigenvalue problem of differential self-adjoint operators, where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose several training procedures for solving increasingly challenging tasks toward the general eigenvalue problem. The proposed solver is capable of finding the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator, which is of particular interest in image processing, computer vision, and shape analysis among many other applications. In addition, we solve the Legendre differential equation. Our proposed method simultaneously solves several eigenpairs and can be easily used on free-form domains. We exemplify it on L-shape and circular cut domains. A significant contribution of this work is an analysis of the numerical error of this method. In particular an upper bound for the (unknown) solution error is given in terms of the (measured) truncation error of the partial differential equation and the network structure.

Microstructural information from angular double-pulsed-field-gradient NMR: From model systems to nerves.

PURPOSE: To evaluate the ability of angular double-pulsed-field gradient (d-PFG) MR to provide microstructural information in complex phantoms and fixed nerves. METHODS: We modeled the signal in angular d-PFG MR experiments performed on phantoms of increasing complexity where the ground truth is known a priori. After analyzing the microstructural features of such phantoms the same methodology was used to study microstructural features in fixed nerves. RESULTS: We found that our modeling is able to determine with high accuracy and with very little prior knowledge the sizes and relative fractions of the restricted components as well as the fraction of the free diffusing water molecules. The same approach was used to study nerve microstructure. We found the apparent averaged axonal diameter (AAD) to be 2.3 ± 0.2 µm. However, here the results depended, to some extent, on the parameters used to collect the data and were affected by the diffusion time. CONCLUSION: Modeling of the angular d-PFG MR signal provides a means to obtain accurate microstructural information in complex phantoms where the ground truth is known. This approach also seems to be suitable for obtaining microstructural features in fixed nerves. Magn Reson Med 74:25-32, 2015. © 2014 Wiley Periodicals, Inc.

Modeling of the diffusion MR signal in calibrated model systems and nerves.

Diffusion NMR is a powerful tool for gleaning microstructural information on opaque systems. In this work, the signal decay in single-pulsed-field gradient diffusion NMR experiments performed on a series of phantoms of increasing complexity, where the ground truth is known a priori, was modeled and used to identify microstructural features of these complex phantoms. We were able to demonstrate that, without assuming the number of components or compartments, the modeling can identify the number of restricted components, detect their sizes with an accuracy of a fraction of a micrometer, determine their relative populations, and identify and characterize free diffusion when present in addition to the components exhibiting restricted diffusion. After the accuracy of the modeling had been demonstrated, this same approach was used to study fixed nerves under different experimental conditions. It seems that this approach is able to characterize both the averaged axon diameter and the relative population of the different diffusing components in the neuronal tissues examined.

The CONNECT project: Combining macro- and micro-structure.

In recent years, diffusion MRI has become an extremely important tool for studying the morphology of living brain tissue, as it provides unique insights into both its macrostructure and microstructure. Recent applications of diffusion MRI aimed to characterize the structural connectome using tractography to infer connectivity between brain regions. In parallel to the development of tractography, additional diffusion MRI based frameworks (CHARMED, AxCaliber, ActiveAx) were developed enabling the extraction of a multitude of micro-structural parameters (axon diameter distribution, mean axonal diameter and axonal density). This unique insight into both tissue microstructure and connectivity has enormous potential value in understanding the structure and organization of the brain as well as providing unique insights to abnormalities that underpin disease states. The CONNECT (Consortium Of Neuroimagers for the Non-invasive Exploration of brain Connectivity and Tracts) project aimed to combine tractography and micro-structural measures of the living human brain in order to obtain a better estimate of the connectome, while also striving to extend validation of these measurements. This paper summarizes the project and describes the perspective of using micro-structural measures to study the connectome.

Measuring small compartments with relatively weak gradients by angular double-pulsed-field-gradient NMR.

NMR diffusion-diffraction patterns observed in compartments in which restricted diffusion occurs are a useful tool for direct extraction of compartment sizes. Such diffusion-diffraction patterns may be observed when the signal intensity E(q,∆) is plotted against the wave-vector q (when q=(2π)(-1)γδG). However, the smaller the compartment sizes are, the higher are the q-values needed to observe such diffractions. Moreover, these q-values should be achieved using short gradient pulses requiring extremely strong gradient systems. The angular double-pulsed-field gradient (d-PFG) NMR methodology has been proposed as a tool to extract compartment sizes using relatively low q-values. In this study, we have used single-PFG (s-PFG) NMR and angular d-PFG NMR to characterize the size of microcapillaries of about 2±1µm in diameter. We found that these microcapillaries are characterized by relatively strong background gradients that completely masked the effects of the microscopic anisotropy (µA) of the sample, resulting in a completely unexpected E(φ) profile in the angular d-PFG NMR experiments. We also show that bipolar angular d-PFG NMR experiments can largely suppress the effect of these background gradients resulting in the expected E(φ) profile from which the compartment dimensions could be obtained with relatively weak gradient pulses. These results demonstrate that the above methodology provides a quick, reliable, non-invasive means for estimating small pore sizes with relatively weak gradients in the presence of large magnetic susceptibility.

Mapping apparent eccentricity and residual ensemble anisotropy in the gray matter using angular double-pulsed-field-gradient MRI.

Conventional diffusion MRI methods are mostly capable of portraying microarchitectural elements such as fiber orientation in white matter from detection of diffusion anisotropy, which arises from the coherent organization of anisotropic compartments. Double-pulsed-field-gradient MR methods provide a means for obtaining microstructural information such as compartment shape and microscopic anisotropies even in scenarios where macroscopic organization is absent. Here, we apply angular double-pulsed-gradient-spin-echo MRI in the rat brain both ex vivo and in vivo for the first time. Robust angular dependencies are detected in the brain at long mixing time (t(m) ). In many pixels, the oscillations seem to originate from residual directors in randomly oriented media, i.e., from residual ensemble anisotropy, as corroborated by quantitative simulations. We then developed an analysis scheme that enables one to map of structural indices such as apparent eccentricity (aE) and residual phase (φ) that enables characterization of the rat brain in general, and especially the rat gray matter. We conclude that double-pulsed-gradient-spin-echo MRI may in principle become important in characterizing gray matter morphological features and pathologies in both basic and applied neurosciences.

Deblurring of color images corrupted by impulsive noise.

We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g., salt-and-pepper noise). Using the variational framework, we consider the L1 fidelity term and several possible regularizers. In particular, we use generalizations of the Mumford-Shah (MS) functional to color images and gamma-convergence approximations to unify deblurring and denoising. Experimental comparisons show that the MS stabilizer yields better results with respect to Beltrami and total variation regularizers. Color edge detection is a beneficial by-product of our methods.

Semi-blind image restoration via Mumford-Shah regularization.

Image restoration and segmentation are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of image restoration and segmentation processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates semi-blind image deconvolution (parametric blur-kernel), and Mumford-Shah segmentation. The functional is formulated using the T-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified treatment of the semi-blind restoration and segmentation problems, the important special case of known blur is also considered and promising results are obtained.