Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)

In this work we study the formulation of convection/diffusion equations on the 3D motion group SE(3) in terms of the irreducible representations of SO(3). Therefore, the left-invariant vector-fields on SE(3) are expressed as linear operators, that are differential forms in the translation coordinate and algebraic in the rotation. In the context of 3D image processing this approach avoids the explicit discretization of SO(3) or $S_2$, respectively. This is particular important for SO(3), where a direct discretization is infeasible due to the enormous memory consumption. We show two applications of the framework: one in the context of diffusion-weighted magnetic resonance imaging and one in the context of object detection

Analytical method to measure three-dimensional strain patterns in the left ventricle from single slice displacement data

Background: Displacement encoded Cardiovascular MR (CMR) can provide high spatial resolution measurements of three-dimensional (3D) Lagrangian displacement. Spatial gradients of the Lagrangian displacement field are used to measure regional myocardial strain. In general, adjacent parallel slices are needed in order to calculate the spatial gradient in the through-slice direction. This necessitates the acquisition of additional data and prolongs the scan time. The goal of this study is to define an analytic solution that supports the reconstruction of the out-of-plane components of the Lagrangian strain tensor in addition to the in-plane components from a single-slice displacement CMR dataset with high spatio-temporal resolution. The technique assumes incompressibility of the myocardium as a physical constraint. Results: The feasibility of the method is demonstrated in a healthy human subject and the results are compared to those of other studies. The proposed method was validated with simulated data and strain estimates from experimentally measured DENSE data, which were compared to the strain calculation from a conventional two-slice acquisition. Conclusion: This analytical method reduces the need to acquire data from adjacent slices when calculating regional Lagrangian strains and can effectively reduce the long scan time by a factor of two

Automatic eduction and statistical analysis of coherent structures in the wall region of a confine plane

This paper describes a vortex detection algorithm used to expose and statistically characterize the coherent flow patterns observable in the velocity vector fields measured by Particle Image Velocimetry (PIV) in the impingement region of air curtains. The philosophy and the architecture of this algorithm are presented. Its strengths and weaknesses are discussed. The results of a parametrical analysis performed to assess the variability of the response of our algorithm to the 3 user-specified parameters in our eduction scheme are reviewed. The technique is illustrated in the case of a plane turbulent impinging twin-jet with an opening ratio of 10. The corresponding jet Reynolds number, based on the initial mean flow velocity U0 and the jet width e, is 14000. The results of a statistical analysis of the size, shape, spatial distribution and energetic content of the coherent eddy structures detected in the impingement region of this test flow are provided. Although many questions remain open, new insights into the way these structures might form, organize and evolve are given. Relevant results provide an original picture of the plane turbulent impinging jet

A LES-Langevin model for turbulence

We propose a new model of turbulence for use in large-eddy simulations (LES). The turbulent force, represented here by the turbulent Lamb vector, is divided in two contributions. The contribution including only subfilter fields is deterministically modeled through a classical eddy-viscosity. The other contribution including both filtered and subfilter scales is dynamically computed as solution of a generalized (stochastic) Langevin equation. This equation is derived using Rapid Distortion Theory (RDT) applied to the subfilter scales. The general friction operator therefore includes both advection and stretching by the resolved scale. The stochastic noise is derived as the sum of a contribution from the energy cascade and a contribution from the pressure. The LES model is thus made of an equation for the resolved scale, including the turbulent force, and a generalized Langevin equation integrated on a twice-finer grid. The model is validated by comparison to DNS and is tested against classical LES models for isotropic homogeneous turbulence, based on eddy viscosity. We show that even in this situation, where no walls are present, our inclusion of backscatter through the Langevin equation results in a better description of the flow.Comment: 18 pages, 14 figures, to appear in Eur. Phys. J.