A Fanning Scheme for the Parallel Transport along Geodesics on Riemannian Manifolds

Louis, Maxime; Charlier, Benjamin; Jusselin, Paul; Pal, Susovan; Durrleman, Stanley

doi:10.1137/17m1130617

Cited by 19 publications

(24 citation statements)

References 6 publications

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“…As parallel transport in curved shape spaces is rarely given in closed form, in general it has to be approximated numerically, e.g. employing Schild's ladder (Lorenzi et al, 2011) for fanning (Louis et al, 2018). For shapes in 2D, Kendall's shape space is isomorphic to the projective space, which is a symmetric space, so that the essential geometric quantities are well known (cf.…”

Section: Introductionmentioning

confidence: 99%

Geodesic Analysis in Kendall’s Shape Space with Epidemiological Applications

et al. 2020

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We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.

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Section: Introductionmentioning

confidence: 99%

Geodesic Analysis in Kendall’s Shape Space with Epidemiological Applications

et al. 2020

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“…The sequence of ρ [k] required by Algorithm 2 is chosen to be constantly equal to 1 in a preliminary "burn-in" phase of the calibration procedure, and then decreases with the iterations with an exponential decay. The fanning numerical scheme is used to compute the parallel transport along geodesics in a scalable manner [41,42,62]. A block Metropolis-Hastingwithin-Gibbs approach is used for the MCMC sampling step, where each variable α i , τ i and s i are successively sampled.…”

Section: Implementation Detailsmentioning

confidence: 99%

Learning the spatiotemporal variability in longitudinal shape data sets

2020

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In this paper, we propose a generative statistical model to learn the spatiotemporal variability in longitudinal shape data sets, which contain repeated observations of a set of objects or individuals over time. From all the short-term sequences of individual data, the method estimates a longterm normative scenario of shape changes and a tubular coordinate system around this trajectory. Each individual data sequence is therefore (i) mapped onto a specific portion of the trajectory accounting for differences in pace of progression across individuals, and (ii) shifted in the shape space to account for intrinsic shape differences across individuals that are independent of the progression of the observed process. The parameters of the model are estimated using a stochastic approximation of the expectation-maximization algorithm. The proposed approach is validated on a simulated data set, illustrated on the analysis of facial expression in video sequences, and applied to the modeling of the progressive atrophy of the hippocampus in Alzheimer's disease patients. These experiments show that one can use the method to reconstruct data at the precision of the noise, to highlight significant factors that may modulate the progression, and to simulate entirely synthetic longitudinal data sets reproducing the variability of the observed process.

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“…In the statistical analysis of temporal deformations, parallel transport along geodesics is commonly used to perform inter-subject normalization, that is the vector transport of velocity fields from each subject's space to a common atlas' space. An approximation based on Jacobi fields was proposed in [11,7]. A numerical implementation named Pole Ladder (PL) was proposed in [5] and relies only on the computation of exponential and logarithm maps.…”

Section: Fig 1: Pole Laddermentioning

confidence: 99%

Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms

Guigui

Jia

Sermesant

et al. 2019

Lecture Notes in Computer Science

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In 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.

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A Fanning Scheme for the Parallel Transport along Geodesics on Riemannian Manifolds

Cited by 19 publications

References 6 publications

Geodesic Analysis in Kendall’s Shape Space with Epidemiological Applications

Geodesic Analysis in Kendall’s Shape Space with Epidemiological Applications

Learning the spatiotemporal variability in longitudinal shape data sets

Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms

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