Maher Moakher

The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.

The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data

In this paper we present a Riemannian framework for smoothing data that are constrained to live in

Approximate Joint Diagonalization and Geometric Mean of Symmetric Positive Definite Matrices

\mathcal{P}(n)

Riemannian curvature-driven flows for tensor-valued data

, the space of symmetric positive-definite matrices of order n. We start by giving the differential geometry of

Using the Bhattacharyya Mean for the Filtering and Clustering of Positive-Definite Matrices

\mathcal{P}(n)

Bhattacharyya median of symmetric positive-definite matrices and application to the denoising of diffusion-tensor fields

, with a special emphasis on

Numerical convergence and stability of mixed formulation with X-fem cut-off

\mathcal{P}(3)

Divergence Measures and Means of Symmetric Positive-Definite Matrices

, considered at a level of detail far greater than heretofore. We then use the harmonic map and minimal immersion theories to construct three flows that drive a noisy field of symmetric positive-definite data into a smooth one. The harmonic map flow is equivalent to the heat flow or isotropic linear diffusion which smooths data everywhere. A modification of the harmonic flow leads to a Perona-Malik like flow which is a selective smoother that preserves edges. The minimal immersion flow gives rise to a nonlinear system of coupled diffusion equations with anisotropic diffusivity. Some preliminary numerical results are presented for synthetic DT-MRI data.

Fiber Orientation Distribution Functions and Orientation Tensors for Different Material Symmetries

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.

Blind image deconvolution via Hankel based method for computing the GCD of polynomials

We present a novel approach for the derivation of PDE modeling curvature-driven flows for matrix-valued data. This approach is based on the Riemannian geometry of the manifold of symmetric positive-definite matrices P(n). The differential geometric attributes of P(n) -such as the bi-invariant metric, the covariant derivative and the Christoffel symbols- allow us to extend scalar-valued mean curvature and snakes methods to the tensor data setting. Since the data live on P(n), these methods have the natural property of preserving positive definiteness of the initial data. Experiments on three-dimensional real DT-MRI data show that the proposed methods are highly robust.