The derivation of homogenized diffusion kurtosis models for diffusion MRI.

We use homogenization theory to establish a new macroscopic model for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of biological tissue with impermeable membranes. In this model, new higher-order diffusion tensors emerge and offer more information about the structure of the biological tissues. We explicitly solve the macroscopic model to obtain an ordinary differential equation for the diffusion MRI signal that has similar structure as diffusional kurtosis imaging models. We finally present some validating numerical results on synthetic examples showing the accuracy of the model with respect to signals obtained by solving the Bloch-Torrey equation.

Approximate joint diagonalization and geometric mean of symmetric positive definite matrices.

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.

BHATTACHARYYA MEDIAN OF SYMMETRIC POSITIVE-DEFINITE MATRICES AND APPLICATION TO THE DENOISING OF DIFFUSION-TENSOR FIELDS.

In this paper, we present algorithms for the computation of the median of a set of symmetric positive-definite matrices using different distances/divergences. The novelty of this paper lies in the median computation using the Bhattacharya distance on diffusion tensors. The numerical computation of the median is achieved using the gradient descent algorithm and the fixed point algorithm. We present an application namely, one of denoising tensor-valued data using median filters constructed using several distance/divergences and compare their performance.

Chaotic granular mixing.

Several models for convective mixing of coarse, freely flowing in granular tumblers have been proposed over the past decade. Powders of practical interest, by contrast, are frequently fine and cohesive, and cannot be analyzed with these models. Moreover, even in the freely flowing regime, mixing transverse to the dominant, convective, direction is typically slow and inefficient. In this paper, we examine two chaotic mixing mechanisms, the first of which can be intentionally applied to increase transverse mixing rates severalfold, with new prospects for further improvements in three-dimensional mixing through judicious process design. The second mechanism occurs spontaneously in fine grains, resulting in mixing rates overwhelmingly exceeding what would be possible in freely flowing grains. Finally, we show that the same chaotic mixing mechanisms seen in simple drum mixers are also found to be at work in more complex blender configurations widely used in batch industrial operations. (c) 1999 American Institute of Physics.