Stochastic Algorithms for White Matter Fiber Tracking and the Inference of Brain Connectivity from MR Diffusion Tensor Data

We consider several stochastic algorithms for fiber tracking and compute the connectivity matrix from data obtained by magnetic resonance diffusion tensor imaging of the living human brain

Spatial Smoothing for Diffusion Tensor Imaging with low Signal to Noise Ratios

Though low signal to noise ratio (SNR) experiments in DTI give key information about tracking and anisotropy, e.g. by measurements with very small voxel sizes, due to the complicated impact of thermal noise such experiments are up to now seldom analysed. In this paper Monte Carlo simulations are presented which investigate the random fields of noise for different DTI variables in low SNR situations. Based on this study a strategy for spatial smoothing, which demands essentially uniform noise, is derived. To construct a convenient filter the weights of the nonlinear Aurich chain are adapted to DTI. This edge preserving three dimensional filter is then validated in different variants via a quasi realistic model and is applied to very new data with isotropic voxels of the size 1x1x1 mm3 which correspond to a spatial mean SNR of approximately 3

DTI denoising for data with low signal to noise ratios

Low signal to noise ratio (SNR) experiments in diffusion tensor imaging (DTI) give key information about tracking and anisotropy, e. g., by measurements with small voxel sizes or with high b values. However, due to the complicated and dominating impact of thermal noise such data are still seldom analysed. In this paper Monte Carlo simulations are presented which investigate the distributions of noise for different DTI variables in low SNR situations. Based on this study a strategy for the application of spatial smoothing is derived. Optimal prerequisites for spatial filters are unbiased, bell shaped distributions with uniform variance, but, only few variables have a statistics close to that. To construct a convenient filter a chain of nonlinear Gaussian filters is adapted to peculiarities of DTI and a bias correction is introduced. This edge preserving three dimensional filter is then validated via a quasi realistic model. Further, it is shown that for small sample sizes the filter is as effective as a maximum likelihood estimator and produces reliable results down to a local SNR of approximately 1. The filter is finally applied to very recent data with isotropic voxels of the size 1×1×1mm^3 which corresponds to a spatially mean SNR of 2.5. This application demonstrates the statistical robustness of the filter method. Though the Rician noise model is only approximately realized in the data, the gain of information by spatial smoothing is considerable

Numerical solution of boundary value problems for stochastic differential equations on the basis of the Gibbs sampler

To solve boundary value problems for linear systems of stochastic differential equations we propose and justify a numerical method based on the Gibbs sampler. In contrast to the technique which yields for linear systems an "exact" numerical solution, the proposed method is simpler to generalize for stochastic partial differential equations and nonlinear systems. Such generalizations are discussed as well

A Simple Stochastic Model for Generating Broken Cloud Optical Depth and Top Height Fields

A simple and fast algorithm for generating two correlated stochastic twodimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud top height retrieved from Moderate Resolution Imaging Spectrometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. For keeping the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper is served as a theoretical justification to the publicly available software that has been recently released by the authors and can be freely downloaded from http://i3rc.gsfc.nasa.gov/Public codes clouds.htm. Though 2D rather than full 3D, stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proved to be very useful to study 3D radiative transfer features of broken cumulus clouds for better understanding of shortwave radiation and interpretation of the remote sensing retrievals

Exact numerical algorithms for linear stochastic wave equation and stochastic Klein-Gordon equation

On the basis of integral representations we propose numerical methods to solve the stochastic wave equation and the stochastic Klein-Gordon equation. The algorithms are exact in a probabilistic sense

About the Influence of Randomness of Hydraulic Conductivity on Solute Transport in Saturated Soil: Numerical Experiments

Up-to-date methods of numerical modelling of random fields were applied to investigate some features of solute transport in saturated porous media with stochastic hydraulic conductivity. The paper describes numerical experiments which were performed and presents the first results

Exact and Fast Numerical Algorithms for the Stochastic Wave Equation

On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with the Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense

A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain

BACKGROUND: Networks or graphs play an important role in the biological sciences. Protein interaction networks and metabolic networks support the understanding of basic cellular mechanisms. In the human brain, networks of functional or structural connectivity model the information-flow between cortex regions. In this context, measures of network properties are needed. We propose a new measure, Ndim, estimating the complexity of arbitrary networks. This measure is based on a fractal dimension, which is similar to recently introduced box-covering dimensions. However, box-covering dimensions are only applicable to fractal networks. The construction of these network-dimensions relies on concepts proposed to measure fractality or complexity of irregular sets in [Formula: see text]. RESULTS: The network measure Ndim grows with the proliferation of increasing network connectivity and is essentially determined by the cardinality of a maximum k-clique, where k is the characteristic path length of the network. Numerical applications to lattice-graphs and to fractal and non-fractal graph models, together with formal proofs show, that Ndim estimates a dimension of complexity for arbitrary graphs. Box-covering dimensions for fractal graphs rely on a linear log-log plot of minimum numbers of covering subgraph boxes versus the box sizes. We demonstrate the affinity between Ndim and the fractal box-covering dimensions but also that Ndim extends the concept of a fractal dimension to networks with non-linear log-log plots. Comparisons of Ndim with topological measures of complexity (cost and efficiency) show that Ndim has larger informative power. Three different methods to apply Ndim to weighted networks are finally presented and exemplified by comparisons of functional brain connectivity of healthy and depressed subjects. CONCLUSION: We introduce a new measure of complexity for networks. We show that Ndim has the properties of a dimension and overcomes several limitations of presently used topological and fractal complexity-measures. It allows the comparison of the complexity of networks of different type, e.g., between fractal graphs characterized by hub repulsion and small world graphs with strong hub attraction. The large informative power and a convenient computational CPU-time for moderately sized networks may make Ndim a valuable tool for the analysis of biological networks

Numerical study of rainbows and glories in water-drop clouds

In this paper we study scattering phase functions in water-drop clouds for various distributions of droplet size and various conditions of glory, rainbow and corona formation, and discuss the hypothesis proposed by A. N. Nevzorov that a considerable amount of water in cold clouds can exist in a specific phase state with the refractive index ≈ 1.8 (so called A-water). Polarization and angular distributions are studied by the Monte Carlo method for radiation reflected by cloud layers with drops of water or hypothetical A-water taking into account multiple scattering. Computational results make it possible to develop procedures for analysis of microphysical structure of clouds and confirmation or disproof of the existence of A-water