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

Abstract

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...