We consider morphological and linear scale spaces on the space R 3 S 2 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 strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them.The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on R 3 S 2 and can be solved by convolution with the corresponding Green's functions. The morphological scale spaces are Bellman equations of cost processes on R 3 S 2 and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green's function. For theoretical underpinning of our scale spaces on R 3 S 2 we introduce Lagrangians and Hamiltonians on R 3 S 2 indexed by a parameter η ∈ [1, ∞). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on R 3 S 2 .By means of the logarithm on SE(3) we provide tangible estimates for both the linear-and the morphological Green's functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI.We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.
One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can also be employed in the sharpening framework. Tractography experiments on synthetic and real brain diffusion data show improvement for high-curvature tracts and in the vicinity of isotropic diffusion regions relative to most results for inverse (sharpened) diffusion tensors, and especially on real data. In addition, adjugate tensors are shown to be more robust to noise.
In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R 3 S 2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3Drigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossingpreserving 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 R 3 S 2 and can be solved by R 3 S 2 -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 R 3 S 2 and they are solved by a morphological R 3 S 2 -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 R 3 S 2 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.
In this work we demonstrate how Finsler geometry-and specifically the related geodesic tractography-5can be levied to analyze structural connections between different brain regions. We present new 6 theoretical developments which support the definition of a novel Finsler metric and associated con-7 nectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. 8Using data from the Human Connectome Project, as well as population data from an autism spec-9 trum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity 10 measures, results in connectivity maps that are much closer to known tract anatomy compared to 11 previous geodesic connectivity methods. Our implementation can be used to compute geodesic 12 distance and connectivity maps for segmented areas, and is publicly available. 13
Diffusion-weighted magnetic resonance imaging (MRI) is sensitive to ensemble-averaged molecular displacements, which provide valuable information on e.g. structural anisotropy in brain tissue. However, a concrete interpretation of diffusion-weighted MRI data in terms of physiological or structural parameters turns out to be extremely challenging. One of the main reasons for this is the multi-scale nature of the diffusion-weighted signal, as it is sensitive to the microscopic motion of particles averaged over macroscopic volumes. In order to analyze the geometrical patterns that occur in (diffusion-weighted measurements of) biological tissue and many other structures, we may invoke tools from the field of stochastic geometry. Stochastic geometry describes statistical methods and models that apply to random geometrical patterns of which we may only know the distribution. Despite its many uses in geology, astronomy, telecommunications, etc., its application in diffusion-weighted MRI has so far remained limited. In this work we review some fundamental results in the field of diffusion-weighted MRI from a stochastic geometrical perspective, and discuss briefly for which other questions stochastic geometry may prove useful. The observations presented in this paper are partly inspired by the Workshop on Diffusion MRI and Stochastic Geometry held at Sandbjerg Estate (Denmark) in 2019, which aimed to foster communication and collaboration between the two fields of research.
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