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Parallel transport in shape analysis: a scalable

numerical scheme

Maxime Louis†12 , Alexandre Bˆone†12 , Benjamin Charlier23 , Stanley

Durrleman12 , and the Alzheimer’s Disease Neuroimaging Initiative

1Sorbonne Universit´es, UPMC Universit´e Paris 06, Inserm, CNRS, Institut du

Cerveau et de la Moelle (ICM) – Hˆopital Piti´e-Salpˆetri`ere, 75013 Paris, France,

2Inria Paris, Aramis project-team, 75013 Paris, France,

3Universit´e de Montpellier, France

Abstract. The analysis of manifold-valued data requires eﬃcient tools

from Riemannian geometry to cope with the computational complexity

at stake. This complexity arises from the always-increasing dimension of

the data, and the absence of closed-form expressions to basic operations

such as the Riemannian logarithm. In this paper, we adapt a generic

numerical scheme recently introduced for computing parallel transport

along geodesics in a Riemannian manifold to ﬁnite-dimensional manifolds

of diﬀeomorphisms. We provide a qualitative and quantitative analysis

of its behavior on high-dimensional manifolds, and investigate an appli-

cation with the prediction of brain structures progression.

1 Introduction

Riemannian geometry is increasingly meeting applications in statistical learning.

Indeed, working in ﬂat space amounts to neglecting the underlying geometry of

the laws which have produced the considered data. In other words, such a sim-

plifying assumption ignores the intrinsic constraints on the observations. When

prior knowledge is available, top-down methods can express invariance properties

as group actions or smooth constraints and model the data as points in quotient

spaces, as for Kendall shape space. In other situations, manifold learning can be

used to ﬁnd a low-dimensional hypersurface best describing a set of observations.

Once the geometry has been modeled, classical statistical approaches for

constrained inference or prediction must be adapted to deal with structured

data, as it is done in [4,5,11,13]. Being an isometry, the parallel transport arises

as a natural tool to compare features deﬁned at diﬀerent tangent spaces.

In a system of coordinates, the parallel transport is deﬁned as the solution

to an ordinary diﬀerential equation. The integration of this equation requires

to compute the Christoﬀel symbols, which are in general hard to compute –e.g.

in the case of the Levi-Civita connection– and whose number is cubic in the

dimension. The Schild’s ladder [5], later improved into the Pole ladder [7] when

transporting along geodesics, is a more geometrical approach which only requires

†Equal contributions.

the computation of Riemannian exponentials and logarithms. When the geodesic

equation is autonomous, the scaling and squaring procedure [6] allows to com-

pute exponentials very eﬃciently. In Lie groups, the Baker-Campbell Haussdorﬀ

formula allows fast computations of logarithms with a controlled precision. In

such settings, the Schild’s ladder is computationnally tractable. However, no

theoretical study has studied the numerical approximations or has provided a

convergence result. In addition, in the more general case of Riemannian mani-

folds, the needed logarithm operators are often computationally intractable.

The Large Deformation Diﬀeomorphic Metric Mapping (LDDMM) frame-

work [1] focuses on groups of diﬀeomorphisms, for shape analysis. Geodesic

trajectories can be computed by integrating the Hamiltonian equations, which

makes the exponential operator computationally tractable, when the logarithm

remains costly and hard to control in its accuracy. In [12] is suggested a numeri-

cal scheme which approximates the parallel transport along geodesics using only

the Riemannian exponential and the metric. The convergence is proved in [8].

In this paper, we translate this so-called fanning sheme to ﬁnite-dimensional

manifolds of diﬀeomorphisms built within the LDDMM framework [2]. We pro-

vide a qualitative and quantitative analysis of its behavior, and investigate a

high-dimensional application with the prediction of brain structures progression.

Section 2 gives the theoretical background and the detailed steps of the algo-

rithm, in the LDDMM context. Section 3 describes the considered application

and discusses the obtained results. Section 4 concludes.

2 Theoretical background and practical description

2.1 Notations and assumptions

Let Mbe a ﬁnite-dimensional Riemannian manifold with metric gand tangent

space norm k · kg. Let γ:t→[0,1] be a geodesic whose coordinates are known

at all time. Given t0, t ∈[0,1], the parallel transport of a vector w∈Tγ(s)M

from γ(t0) to γ(t) along γwill be noted Pt0,t(w)∈Tγ(t)M. We recall that

this mapping is uniquely deﬁned by the integration from u=t0to tof the

diﬀerential equation ∇˙γ(u)Pt0,u(w) = 0 with Pt0,t0(w) = wwhere ∇is the Levi-

Civita covariant derivative.

We denote Exp the exponential map, and for hsmall enough we deﬁne

Jw

γ(t)(h), the Jacobi Field emerging from γ(t) in the direction w∈Tγ(t)Mby:

Jw

γ(t)(h) = ∂

∂ε

ε=0

Expγ(t)h[ ˙γ(t) + εw]∈Tγ(t+h)M.(1)

2.2 The key identity

The following proposition relates the parallel transport to a Jacobi ﬁeld [12]:

Proposition. For all t > 0and w∈Tγ(0)M, we have:

P0,t(w) = Jw

γ(0)(t)

t+ Ot2.(2)

Proof. Let X(t) be the time-varying vector ﬁeld corresponding to the parallel

transport of w, i.e. such that ˙

Xi+Γi

klXl˙γk= 0 with X(0) = w. At t= 0, in

normal coordinates the diﬀerential equation simpliﬁes into ˙

Xi(0) = 0. Besides,

near t= 0 in the same local chart, the Taylor expansion of X(t) writes Xi(t) =

wi+ Ot2. Noticing that the ith normal coordinate of Expγ(0) (t[ ˙γ(t) + εw])

is t(vi

0+εwi), the ith coordinate of Jw

γ(0)(t) = ∂

∂ε |ε=0 Expγ(0) (t[ ˙γ(0) + εw]) is

therefore twi, and we thus obtain the desired result.

Subdividing [0,1] into Nintervals and iteratively computing the Jacobi ﬁelds

1

NJw

γ(k/N)(1

N) should therefore approach the parallel transport P0,1(w). With

an error in O1

N2at each step, a global error in O1

Ncan be expected. We

propose below an implementation of this scheme in the context of a manifold of

diﬀeomorphisms parametrized by control points and momenta. Its convergence

with a rate of O1

Nis proved in [8].

2.3 The chosen manifold of diﬀeomorphisms

The LDDMM-derived construction proposed in [2] provides an eﬀective way to

build a ﬁnite-dimensional manifold of diﬀeomorphims acting on the d-dimensional

ambient space Rd. Time-varying vector ﬁelds vt(.) are generated by the convolu-

tion of a Gaussian kernel k(x, y) = exp −kx−yk2/2σ2over ncp time-varying

control points c(t) = [ci(t)]i, weighted by ncp associated momenta α(t)=[αi(t)]i,

i.e. vt(.) = Pncp

i=1 k[. , ci(t)] αi(t). The set of such vector ﬁelds forms a Repro-

ducible Kernel Hilbert Space (RKHS).

Those vector ﬁelds are then integrated along ∂tφt(.) = vt[φ(.)] from φ0= Id

into a ﬂow of diﬀeomorphisms. In [10], the authors showed that the kernel-

induced distance between φ0and φ1–which can be seen as the deformation

kinetic energy– is minimal i.e. the obtained ﬂow is geodesic when the control

points and momenta satisfy the Hamiltonian equations :

˙c(t)=Kc(t)α(t),˙α(t) = −1

2gradc(t)α(t)TKc(t)α(t),(3)

where Kc(t)is the kernel matrix. A diﬀeomorphism is therefore fully parametrized

by its initial control points cand momenta α.

Those Hamiltonian equations can be integrated with a Runge-Kutta scheme

without computing the Christoﬀel symbols, thus avoiding the associated curse

of dimensionality. The obtained diﬀeomorphisms then act on shapes embedded

in Rd, such as images or meshes.

For any set of control points c= (ci)i∈{1,..,n}, we deﬁne the ﬁnite-dimensional

subspace Vc= span k(., ci)ξ|ξ∈Rd, i ∈ {1, .., n}of the vector ﬁelds’ RKHS.

We ﬁx an initial set c= (ci)i∈{1,..,n}of distinct control points and deﬁne the set

Gc={φ1|∂tφt=vt◦φt, v0∈Vc, φ0= Id}. Equipped with Kc(t)as –inverse–

metric, Gcis a Riemannian manifold such that Tφ1Gc=Vc(1), where for all tin

[0,1], c(t) is obtained from c(0) = cthrough the Hamiltonian equations (3) [9].

2.4 Summary of the algorithm

We are now ready to describe the algorithm on the Riemannian manifold Gc.

Algorithm. Divide [0,1] into Nintervals of length h=1

Nwhere N∈N. We

note ωkthe momenta of the transported diﬀeomorphism, ckthe control points

and αkthe momenta of the geodesic γat time k

N. Iteratively :

(i) Compute the main geodesic control points ck+1 and momenta αk+1, using

a Runge-Kutta 2 method.

(ii) Compute the control points c±h

k+1 of the perturbed geodesics γ±hwith initial

momenta and control points (αk±hωk,ck), using a Runge-Kutta 2 method.

(iii) Approximate the Jacobi ﬁeld Jk+1 by central ﬁnite diﬀerence :

Jk+1 =c+h

k+1 −c−h

k+1

2h.(4)

(iv) Compute the transported momenta ˜ωk+1 according to equation (2) :

Kck+1 ˜ωk+1 =Jk+1

h.(5)

(v) Correct this value with ωk+1 =βk+1 ˜ωk+1 +δk+1αk+1 , where βk+1 and δk+1

are normalization factors ensuring the conservation of kωkVc=ωT

kKckωk

and of hαk, ωkick=αT

kKckωk.

As step of the scheme is illustrated in Figure 1. The Jacobi ﬁeld is com-

puted with only four calls to the Hamiltonian equations. This operation scales

quadratically with the dimension of the manifold, which makes this algorithm

practical in high dimension, unlike Christoﬀel-symbol-based solutions. Step (iv)

–solving a linear system of size ncp– is the most expensive one, but remained

within reasonable computational time in the investigated examples.

Fig. 1: Step of the parallel transport of the vector w(blue arrow) along the

geodesic γ(solid black curve). Jw

γis computed by central ﬁnite diﬀerence with

the perturbed geodesics γεand γ−, integrated with a second-order Runge-Kutta

scheme (dotted black arrows). A fan of geodesics is formed.

In [8], the authors prove the convergence of this scheme, and show that

the error increases linearly with the size of the step used. The convergence is

guaranteed as long as the step (ii) is performed with a method of order at least

two. A ﬁrst order method in step (iii) is also theoretically suﬃcient to guarantee

convergence. Those variations will be studied in Section 3.3.

3 Application to the prediction of brain structures

3.1 Introducing the exp-parallelization concept

Fig. 2: Time-reparametrized exp-parallelization of a reference geodesic model.

The black dots are the observations, on which are ﬁtted a geodesic regression

(solid black curve, parametrized by the blue arrow) and a matching (leftmost

red arrow). The red arrow is then parallel-transported along the geodesic, and

exponentiated to deﬁne the exp-parallel curve (black dashes).

Fig. 3: Illustration of the exp-parallelization concept. Top row: the reference

geodesic at successive times. Bottow row: the exp-parallel curve. Blue arrows:

the geodesic momenta and velocity ﬁeld. Red arrows: the momenta describing

the initial registration with the target shape and its transport along the geodesic.

Exploiting the fanning scheme described in Section 2.4, we can parallel-

transport any set of momenta along any given reference geodesic. Figure 2

illustrates the procedure. The target shape is ﬁrst registered to the reference

geodesic : the diﬀeomorphism that best transforms the chosen reference shape

into the target one is estimated with a gradient descent algorithm on the initial

control points and momenta [2]. Such a procedure can be applied generically

to images or meshes. Once this geodesic is obtained, its initial set of momenta

is parallel-transported along the reference geodesic. Taking the Riemannian ex-

ponential of the transported vector at each point of the geodesic deﬁnes a new

trajectory, which we will call exp-parallel to the reference one.

As pointed out in [5], the parallel transport is quite intuitive in the context

of shape analysis, for it is an isometry which transposes the evolution of a shape

into the geometry of another shape, as illustrated by Figure 3.

3.2 Data and experimental protocol

Repeated Magnetic Resonance Imaging (MRI) measurements from 71 subjects

are extracted from the ADNI database and preprocessed through standard pipeli-

nes into aﬃnely co-registered surface meshes of hippocampi, caudates and putam-

ina. The geometries of those brain sub-cortical structures are altered along the

Alzheimer’s disease course, which all considered subjects ﬁnally convert to.

Two subjects are successively chosen as references, for they have fully de-

veloped the disease within the clinical measurement protocol. As illustrated on

Figure 2, a geodesic regression [3] is ﬁrst performed on each reference subject to

model the observed shape progression. The obtained trajectory on the chosen

manifold of diﬀeomorphisms is then exp-parallelized into a shifted curve, which

is hoped to model the progression of the target subject.

To account for the variability of the disease dynamics, for each subject two

scalar coeﬃcients encoding respectively for the disease onset age and the rate of

progression are extracted from longitudinal cognitive evaluations as in [11]. The

exp-parallel curve is time-reparametrized accordingly, and ﬁnally gives predic-

tions for the brain structures. In the proposed experiment, the registrations and

geodesic regressions typically feature around 3000 control points in R3, so that

the deformation can be seen as an element of a manifold of dimension 9000.

3.3 Estimating the error associated to a single parallel transport

To study the error in this high-dimensional setting, we compute the parallel

transport for a varying number of discretization steps N, thus obtaining in-

creasingly accurate estimations. We then compute the empirical relative errors,

taking the most accurate computation as reference.

Arbitrary reference and target subjects being chosen, Figure 4 gives the re-

sults for the proposed algorithm and three variations : without enforcing the

conservations at step (v) [WEC], using a Runge-Kutta of order 4 at step (ii)

[RK4], and using a single perturbed geodesic to compute Jat step (iii) [SPG].

0.005 0.01 0.015 0.02 0.025 0.03 0.035

0

0.5

Length of the step h=1

N

Empirical relative error (%)

Fig. 4: Empirical rela-

tive error of the paral-

lel transport in a high-

dimensional setting.

In black the proposed

algorithm, in green the

WEC variant, in red

the RK4 variant, and in

blue the SPG one.

We recover a linear behavior with the length of the step 1

Nin all cases. The SPG

variant converges much slower, and is excluded from the following considerations.

For the other algorithms, the empirical relative error remains below 5% with

15 steps or more, and below 1% with 25 steps or more. The slopes of the asymp-

totic linear behaviors, estimated with the last 10 experimental measurements,

range from 0.10 for the RK4 method to 0.13 for the WEC one. Finally, an iter-

ation takes respectively 4.26, 4.24 and 8.64 seconds for the proposed algorithm,

the WEC variant and the RK4 one. Therefore the initially detailed algorithm

in section 2.4 seems to achieve the best tradeoﬀ between accuracy and speed in

the considered experimental setting.

3.4 Prediction performance

Table 1 gathers the predictive performance of the proposed exp-parallelization

method. The performance metric is the Dice coeﬃcient, which ranges from 0 for

disjoint structures to 1 for a perfect match. A Mann-Witney test is performed

to quantify the signiﬁcance of the results in comparison to a naive methodology,

which keeps constant the baseline structures over time. Considering the very high

dimension of the manifold, failing to accurately capture the disease progression

Method

Predicted follow-up visit

M12 M24 M36 M48 M60 M72 M96

N=140 N=134 N=123 N=113 N=81 N=62 N=17

[exp] .882

.884

.852

.852

.825

.809∗

∗

.796

.764∗

∗

∗

.768

.734∗

∗

.756

.706∗

∗

∗

.730

.636∗

∗

[ref]

Table 1: Averaged Dice performance measures. In each cell, the ﬁrst line gives the

average performance of the exp-parallelization-based prediction [exp], and the

second line the reference one [ref]. Each column corresponds to an increasingly

remote predicted visit from baseline. Signiﬁcance levels [.05, .01, .001].

trend can quickly translates into unnatural predictions, much worse than the

naive approach.

The proposed paradigm signiﬁcantly outperforms the naive prediction three

years or later from the baseline, thus demonstrating the relevance of the exp-

parallelization concept for disease progression modeling, made computationally

tractable thanks to the operational qualities of the fanning scheme for high-

dimensional applications.

4 Conclusion

We detailed the fanning scheme for parallel transport on a high-dimensional

manifold of diﬀeomorphisms, in the shape analysis context. Our analysis unveiled

the operational qualities and computational eﬃciency of the scheme in high

dimensions, with a empirical relative error below 1% for 25 steps only. We then

took advantage of the parallel transport for accurately predicting the progression

of brain structures in a personalized way, from previously acquired knowledge.

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