Sergei M. Prigarin

Edge Preserving Regularization and Tracking for Diffusion Tensor Imaging

Two major problems in MR Diffusion Tensor Imaging, regularization and tracking are addressed. Regularization is performed on a variance homogenizing transformation of the tensor field via a nonlinear filter chain to preserve discontinuities. The suitability of the smoothing procedure is validated by Monte Carlo simulations. For tracking, the tensor field is diagonalized and a local bilinear interpolation of the corresponding direction field is performed. The track curves, which are not restricted to the measured grid, are modeled by following stepwise the interpolated directions. The presented methods are illustrated by applications to measured data.

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 Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense.

Random fields of broken clouds and their associated direct solar radiation, scattered transmission and albedo

To model fields of broken clouds, we consider two methods based on Gaussian random fields and one method based on truncated random paraboloids. These models are determined by a few parameters only: the amount of cloud, the mean cloud base diameter and the mean cloud height. A field of broken clouds is a realization of such a random field, it is obtained by stochastic simulation and special spectral methods for Gaussian fields. The direct solar radiation S, the scattered transmission T and the albedo A of such a field of broken clouds are obtained by solving the stationary radiation transfer equation. Numerically we do this by variance reduction Monte Carlo methods. Varying the cloud parameters, the angle of incidence of the solar radiation we get interesting new functional relations between S, T and A for visible and infrared wavelengths.

Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion

Using Monte Carlo simulation techniques, we look at statistical properties of two numerical methods (the extended counting method and the variance counting method) developed to estimate the Hausdorff dimension of a time series and applied to the fractional Brownian motion.

Fitting of two-tensor models without ad hoc assumptions to detect crossing fibers using clinical DWI data

Analysis of crossing fibers is a challenging topic in recent diffusion-weighted imaging (DWI). Resolving crossing fibers is expected to bring major changes to present tractography results based on the standard tensor model. Model free approaches, like Q-ball or diffusion spectrum imaging, as well as multi-tensor models are used to unfold the different diffusion directions mixed in a voxel of DWI data. Due to its seeming simplicity, the two-tensor model (TTM) is applied frequently to provide two positive-definite tensors and the relative population fraction modeling two crossing fiber branches. However, problems with uniqueness and noise instability are apparent. To stabilize the fit, several of the 13 physical parameters are fixed ad hoc, before fitting the model to the data. Our analysis of the TTM aims at fitting procedures where ad hoc parameters are avoided. Revealing sources of instability, we show that the models inherent ambiguity can be reduced to one scalar parameter which only influences the fraction and the eigenvalues of the TTM, whereas the diffusion directions are not affected. Based on this, two fitting strategies are proposed: the parsimonious strategy detects the main diffusion directions without extra parameter fixation, to determine the eigenvalues and the population fraction an empirically motivated condition must be added. The expensive strategy determines all 13 physical parameters of the TTM by a fit to DWIs alone; no additional assumption is necessary. Ill-posedness of the model in case of noisy data is cured by denoising of the data and by L-curve regularization combined with global minimization performing a least-squares fit of the full model. By model simulations and real data applications, we demonstrate the feasibility of our fitting strategies and achieve convincing results. Using clinically affordable diffusion acquisition paradigms (encoding numbers: 21, 2*15, 2*21) and b values (b=500-1500 s/mm(2)), this methodology can place the TTM parameters involved in crossing fibers on a more empirical basis than fitting procedures with technical assumptions.

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

BackgroundNetworks 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 ℝn

Estimation of fractal dimension of random fields on the basis of variance analysis of increments

\mathbb {R}^{n}

Simulation of vector semibinary homogeneous random fields and modeling of broken clouds

.ResultsThe 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.ConclusionWe 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.

Monte Carlo simulation of radiation transfer in optically anisotropic clouds

This paper deals with estimation of fractal dimension of realizations of random fields. Numerical methods are based on analysis of variance of increments. It is proposed to study fractal properties with the use of a specific characteristic of randomfields called a “variational dimension.” For a class of Gaussian fields with homogeneous increments the variational dimension converges to the Hausdorff dimension. Several examples are presented to illustrate that the concept of variational dimension can be used to construct effective computational methods.

Simulation of binary random fields with Gaussian numerical models

A vector-valued homogeneous random field is said to be semibinary if its single-point marginal distribution is a sum of a singular distribution and a continuous one. In this paper, we present methods of numerical simulation of semibinary fields on the basis of the correlation structure and the marginal distribution. As an example we construct a combined model of cloud top height and optical thickness using satellite observations.