Finslerian 3-spinors and the generalized Duffin-Kemmer equation

The main facts of the geometry of Finslerian 3-spinors are formulated. The close connection between Finslerian 3-spinors and vectors of the 9-dimensional linear Finslerian space is established. The isometry group of this space is described. The procedure of dimensional reduction to 4-dimensional quantities is formulated. The generalized Duffin-Kemmer equation for a Finslerian 3-spinor wave function of a free particle in the momentum representation is obtained. From the viewpoint of a 4-dimensional observer, this 9-dimensional equation splits into the standard Dirac and Klein-Gordon equations.Comment: LaTeX2e, 11 pages, no figures, will be published in "Fundamental and Applied Mathematics

Permutonestohedra

There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini–Procesi models. In this paper, starting from any root system with finite Coxeter group W and any W -invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) that lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of W or also, depending on the building set, of Aut ( ). The permutonestohedra are natural generalizations of Kapranov’s permutoassociahedra

Discrete Ladders for Parallel Transport in Transformation Groups with an Affine Connection Structure

International audienceModeling the temporal evolution of the tissues of the body is an important goal of medical image analysis, for instance for understanding the structural changes of organs affected by a pathology, or for studying the physiological growth during the life span. For such purposes we need to analyze and compare the observed anatomical differences between follow-up sequences of anatomical images of different subjects. Non-rigid registration is one of the main instruments for modeling anatomical differences from images. The aim of non-rigid registration is to encode the observed structural changes as deformation fields of the image space, which represent the warping required to match observed differences. This way, anatomical changes can be modeled and quantified by analyzing the associated deformations. The comparison of temporal evolutions thus requires the transport (or "normalization") of longitudinal deformations in a common reference frame. Normalization of longitudinal deformations can be done in different ways, depending on the feature of interest. For instance, local volume changes encoded by the scalar Jacobian determinant of longitudinal deformations can be compared by scalar resampling in a common reference frame via inter-subject registration. However, if we consider vector-valued deformation trajectories instead of scalar quantities, the transport is not uniquely defined anymore. Among the different normalization methods for deformation trajectories, the parallel transport is a powerful and promising tool which can be used within the ''diffeomorphic registration'' setting. Mathematically, parallel transporting a vector along a curve consists in translating it across the tangent spaces to the curve by preserving its parallelism according to a given derivative operation called (affine) connection. This chapter focuses on explicitly discrete algorithms for parallel transporting diffeomorphic deformations. Schild's ladder is an efficient and simple method proposed in theoretical Physics for the parallel transport of vectors along geodesics paths by iterative construction of infinitesimal geodesics parallelograms on the manifold. The base vertices of the parallelogram are given by the initial tangent vector to be transported. By iteratively building geodesic diagonals along the path, Schild's Ladder computes the missing vertex which corresponds to the transported vector. In this chapter we first show that the Schild ladder can lead to an effective computational scheme for the parallel transport of diffeomorphic deformations parameterized by tangent velocity fields. Schild's ladder may be however inefficient for transporting longitudinal deformations from image time series of multiple time points, in which the computation of the geodesic diagonals is required several times. We propose therefore a new parallel transport method based on the Schild's ladder, the "pole ladder", in which the computation of geodesics diagonals is minimized. Differently from the Schild's ladder, the pole ladder is symmetric with respect to the baseline-to-reference frame geodesic. From the theoretical point of view, we show that the pole ladder is rigorously equivalent to the Schild's ladder when transporting along geodesics. From the practical point of view, we establish the computational advantages and demonstrate the effectiveness of this very simple method by comparing with standard methods of transport on simulated images with progressing brain atrophy. Finally, we illustrate its application to a clinical problem: the measurement of the longitudinal progression in Alzheimer's disease. Results suggest that an important gain in sensitivity could be expected in group-wise comparisons

Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms

International audienceIn computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to important numerical errors. In many cases, it is well known that the error can be drastically reduced in the presence of a symmetry. In this work, the leading idea is to approximate the space of deformations and images with a possibly non-metric symmetric space structure using an involution, with the aim to perform parallel transport. Through basic properties of symmetries, we investigate how the implementations of a midpoint and the involution compare with the ones of the Riemannian exponential and logarithm on diffeomorphisms and propose a modification of these maps using registration errors. This leads us to identify transvections, the composition of two symmetries, as a mean to measure how far from symmetric the underlying structure is. We test our method on a set of 138 cardiac shapes and demonstrate improved numerical consistency in the Pole Ladder scheme