Riemann-Finsler geometry and its applications to diffusion magnetic resonance imaging

Riemannian geometry has become a popular mathematical framework for the analysis of diffusion tensor images (DTI) in diffusion weighted magnetic resonance imaging (DWMRI). If one declines from the a priori constraint to model local anisotropic diffusion in terms of a 6-degrees-of-freedom rank-2 DTI tensor, then Riemann-Finsler geometry appears to be the natural extension. As such it provides an interesting alternative to the Riemannian rationale in the context of the various high angular resolution diffusion imaging (HARDI) schemes proposed in the literature. The main advantages of the proposed Riemann-Finsler paradigm are its manifest incorporation of the DTI model as a limiting case via a "correspondence principle" (operationalized in terms of a vanishing Cartan tensor), and its direct connection to the physics of DWMRI expressed by the (appropriately generalized) Stejskal-Tanner equation and Bloch-Torrey equations furnished with a diffusion term

On the Riemannian rationale for diffusion tensor imaging

One of the approaches in the analysis of brain diffusion MRI data is to consider white matter as a Riemannian manifold, with a metric given by the inverse of the diffusion tensor. Such a metric is used for white matter tractography and connectivity analysis. Although this choice of metric is heuristically justified it has not been derived from first principles. We propose a modification of the metric tensor motivated by the underlying mathematics of diffusion

Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers

The apparent range of spin movement in diffusion MRI data

In this work we investigate the potential of diffusion MRI to measure the maximum range of motion due to diffusion within spatially homogeneous voxels. We show that it is possible to characterize this range even in clinical scanners, and show in data of the human brain how this leads to interesting new ways to extract information from diffusion MRI

Reconstruction of convex polynomial diffusion MRI models using semi-definite programming

In this work we describe and perform the reconstruction of general polynomial diffusion MRI models, with the added constraint that the polynomials are convex. This is done by requiring the Hessian of the model to be sum-of-squares. The resulting optimization problem is shown to be solvable in a reasonable amount of time for the scale of data typical in clinical diffusion MRI acquisitions.<br/

Numerical schemes for linear and non-linear enhancement of DW-MRI

We consider left-invariant di??usion processes on DTI data by embedding the data into the space R3 o S2 of 3D positions and orientations. We then define and solve the diffusion equation in a moving frame of reference defined using left-invariant derivatives. The diffusion process is made adaptive to the data in order to do Perona-Malik-like edge preserving smoothing, which is necessary to handle fiber structures near regions of large isotropic diffusion such as the ventricles of the brain. The corresponding partial differential systems are solved using finite difference stencils. We include experiments both on synthetic data and on DTI-images of the brain

Reconstruction of convex polynomial diffusion MRI models using semi-definite programming

In this work we describe and perform the reconstruction of general polynomial diffusion MRI models, with the added constraint that the polynomials are convex. This is done by requiring the Hessian of the model to be sum-of-squares. The resulting optimization problem is shown to be solvable in a reasonable amount of time for the scale of data typical in clinical diffusion MRI acquisitions

Finslerian diffusion and the Bloch–Torrey equation

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings.An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownian motion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal