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Learning distributions of shape trajectories from longitudinal datasets:
a hierarchical model on a manifold of diffeomorphisms
Alexandre Bˆ
one Olivier Colliot Stanley Durrleman
The Alzheimer’s Disease Neuroimaging Initiative
Institut du Cerveau et de la Moelle ´
epini`
ere, Inserm, CNRS, Sorbonne Universit´
e, Paris, France
Inria, Aramis project-team, Paris, France
{alexandre.bone, olivier.colliot, stanley.durrleman}@icm-institute.org
Abstract
We propose a method to learn a distribution of shape tra-
jectories from longitudinal data, i.e. the collection of indi-
vidual objects repeatedly observed at multiple time-points.
The method allows to compute an average spatiotemporal
trajectory of shape changes at the group level, and the indi-
vidual variations of this trajectory both in terms of geometry
and time dynamics. First, we formulate a non-linear mixed-
effects statistical model as the combination of a generic sta-
tistical model for manifold-valued longitudinal data, a de-
formation model defining shape trajectories via the action
of a finite-dimensional set of diffeomorphisms with a man-
ifold structure, and an efficient numerical scheme to com-
pute parallel transport on this manifold. Second, we intro-
duce a MCMC-SAEM algorithm with a specific approach
to shape sampling, an adaptive scheme for proposal vari-
ances, and a log-likelihood tempering strategy to estimate
our model. Third, we validate our algorithm on 2D sim-
ulated data, and then estimate a scenario of alteration of
the shape of the hippocampus 3D brain structure during the
course of Alzheimer’s disease. The method shows for in-
stance that hippocampal atrophy progresses more quickly
in female subjects, and occurs earlier in APOE4 mutation
carriers. We finally illustrate the potential of our method for
classifying pathological trajectories versus normal ageing.
1. Introduction
1.1. Motivation
At the interface of geometry, statistics, and computer sci-
ence, statistical shape analysis meets a growing number of
applications in computer vision and medical image anal-
ysis. This research field has addressed two main statisti-
cal questions: atlas construction for cross-sectional shape
datasets, and shape regression for shape time series. The
former is the classical extension of a mean-variance analy-
sis, which aims to estimate a mean shape and a covariance
structure from observations of several individual instances
of the same object or organ. The latter extends the concept
of regression by estimating a spatiotemporal trajectory of
shape changes from a series of observations of the same in-
dividual object at different time-points. The emergence of
longitudinal shape datasets, which consist in the collection
of individual objects repeatedly observed at multiple time-
points, has raised the need for a combined approach. One
needs statistical methods to estimate normative spatiotem-
poral models from series of individual observations which
differ in shape and dynamics of shape changes across in-
dividuals. Such model should capture and disentangle the
inter-subject variability in shape at each time-point and the
temporal variability due to shifts in time or scalings in pace
of shape changes. Considering individual series as samples
along a trajectory of shape changes, this approach amounts
to estimate a spatiotemporal distribution of trajectories, and
has potential applications in various fields including silhou-
ette tracking in videos, analysis of growth patterns in biol-
ogy or modelling disease progression in medicine.
1.2. Related work
The central difficulty in shape analysis is that shape
spaces are either defined by invariance properties [21,40,
41] or by the conservation of topological properties [5,8,
13,20,34], and therefore have intrinsically a structure of
infinite dimensional Riemannian manifolds or Lie groups.
Statistical Shape Models [9] are linear but require consistent
points labelling across observations and have no topology
preservation guarantees. A now usual approach is to use
the action of a group of diffeomorphisms to define a metric
on a shape space [29,43]. This approach has been used to
compute a “Fr´
echet mean” together with a covariance ma-
trix in the tangent-space of the mean [1,16,17,33,44] from
a cross-sectional dataset, and regression from time series of
shape data [4,15,16,18,26,32]. In [12,14,31], these tools
have been used to estimate an average trajectory of shape
changes from a longitudinal dataset using the convenient
assumption that the parameters encoding inter-individual
variability are independent of time. The work in [27] in-
troduced the idea to use the parallel transport to translate
the spatiotemporal patterns seen in one individual into the
geometry of another one. The co-adjoint transport is used
in [38] for the same purpose. Both estimate a group average
trajectory from individual trajectories. The proposed mod-
els do not account for inter-individual variability in the time
dynamics, which is of key importance in the absence of tem-
poral markers of the progression to align the sequences. The
same remarks can be applied to [6], which introduces a nice
theoretical setting for spaces of trajectories, in the case of a
fixed number of temporal observations across subjects. The
need for temporal alignment in longitudinal data analysis
is highlighted for instance in [13] with a diffeomorphism-
based morphometry approach, or in [40,41] with quotient
manifolds. In [24,37], a generative mixed-effects model
for the statistical analysis of manifold-valued longitudinal
data is introduced for the analysis of feature vectors. This
model describes both the variability in the direction of the
individual trajectories by introducing the concept of “exp-
parallelization” which relies on parallel transport, and the
pace at which those trajectories are followed using “time-
warp” functions. Similar time-warps are used by the authors
of [23] to refine their linear modeling approach [22].
1.3. Contributions
In this paper, we propose to extend the approach of [37]
from low-dimensional feature vectors to shape data. Us-
ing an approach designed for manifold-valued data for
shape spaces defined by the action of a group of diffeomor-
phisms raises several theoretical and computational difficul-
ties. Are notably needed: a finite-dimensional set of dif-
feomorphisms with a Riemannian manifold structure; sta-
ble and efficient numerical schemes to compute Riemannian
exponential and parallel transport operators on this mani-
fold as no closed-form expressions are available; acceler-
ated algorithmic convergence to cope with an hundreds of
times larger dimensionality. To this end, we propose here:
•to formulate a generative non-linear mixed-effects
model for a finite-dimensional set of diffeomorphisms
defined by control points, to show that this set is sta-
ble under exp-parallelization, and to use an efficient
numerical scheme for parallel transport;
•to introduce an adapted MCMC-SAEM algorithm with
an adaptive block sampling of the latent variables, a
specific sampling strategy for shape parameters based
on random local displacements of the shape contours,
and a vanishing tempering of the target log-likelihood;
•to validate our method on 2D simulated data and a
large dataset of 3D brain structures in the context of
Alzheimer’s disease progression, and to illustrate the
potential of our method for classifying spatiotemporal
patterns, e.g. to discriminate pathological versus nor-
mal trajectories of ageing.
All in one, the proposed method estimates an average
spatiotemporal trajectory of shape changes from a longi-
tudinal dataset, together with distributions of space-shifts,
time-shifts and acceleration factors describing the variabil-
ity in shape, onset and pace of shape changes respectively.
2. Deformation model
2.1. The manifold of diffeomorphisms Dc0
We follow the approach taken in [12] built on the princi-
ples of the Large Deformation Diffeomorphic Metric Map-
ping (LDDMM) framework [30]. We note d2{2,3}the
dimension of the ambient space. We choose ka Gaussian
kernel of width 2R⇤
+and ca set of ncp 2N“con-
trol” points c=(c1,...,c
ncp )of the ambient space Rd. For
any set of “momentum” vectors m=(m1,...,m
ncn ), we
define the “velocity” vector field v:Rd!Rdas the con-
volution v(x)=Pncp
k=1 k(ck,x)mkfor any point xof the
ambient space Rd. From initial sets of ncp control points c0
and corresponding momenta m0, we obtain the trajectories
t!(ct,m
t)by integrating the Hamiltonian equations:
˙c=Kcm;˙m=1
2rcmTKcm (1)
where Kcis the ncp ⇥ncp “kernel” matrix [k(ci,c
j)]ij ,r(.)
the gradient operator, and (.)Tthe matrix transposition.
Those trajectories prescribe the trajectory t!vtin the
space of velocity fields. The integration along such a path
from the identity generates a flow of diffeomorphisms t!
tof the ambient space [5]. We can now define:
Dc0=1;@tt=vtt,0=Id,v
t=Conv(ct,m
t)
(˙ct,˙mt)=Ham(ct,m
t),m
02Rdncp (2)
where Conv(., .)and Ham(., .)are compact notations for
the convolution operator and the Hamiltonian equations (1)
respectively. Dc0has the structure of a manifold of finite
dimension, where the metric at the tangent space T1Dc0is
given by K1
c1. It is shown in [29] that the proposed paths
t!tare the paths of minimum deformation energy, and
are therefore the geodesics of Dc0. These geodesics are
fully determined by an initial set of momenta m0.
Then, any point x2Rdof the ambient space follows the
trajectory t!t(x). Such trajectories are used to deform
any point cloud or mesh embedded in the ambient space,
defining a diffeomorphic deformation of the shape. For-
mally, this operation defines a shape space Sc0,y0as the or-
bit of a reference shape y0under the action of Dc0. The
manifold of diffeomorphisms Dc0is used as a proxy to ma-
nipulate shapes: all computations are performed in Dc0or
more concretely on a finite set of control points and mo-
mentum vectors, and applied back to the template shape y0
to obtain a result in Sc0,y0.
2.2. Riemmanian exponentials on Dc0
For any set of control points c0, we define the exponen-
tial operator Expc0:m02Rdncp !12Dc0. Note that
Dc0=Expc0(m0)|m02Rdncp .
The following proposition ensures the stability of Dc0by
the exponential operator, i.e. that the control points obtained
by applying successive compatible exponential maps with
arbitrary momenta are reachable by an unique integration
of the Hamiltonian equations from c0:
Proposition. Let c0be a set of control points. 812Dc0,
8wmomenta, we have Exp1(c0)(w)2Dc0.
Proof. We note 0
1=Exp
1(c0)(w)2D1(c0)and c0
1=
0
11(c0). By construction, there exist two paths t!'t
in Dc0and s!'0
sin D1(c0)such that '0
1'1(c0)=c0
1.
Therefore there exist a diffeomorphic path u! usuch
that (c0)=c0
1. Concluding with [29], the path u! uof
minimum energy exists, and is written u!Expc0(u.m0
0)
for some m0
02Rdncp .⇤
As a physical interpretation might be given to the inte-
gration time twhen building a statistical model, we intro-
duce the notation Expc0,t0,t :m02Rdncp !t2Dc0
where tis obtained by integrating from t=t0. Note that
Expc0=Exp
c0,0,1.
On the considered manifold Dc0, computing exponen-
tials – i.e. geodesics – therefore consists in integrating or-
dinary differential equations. This operation is direct and
computationally tractable. The top line on Figure 1plots a
geodesic :t!tapplied to the top-left shape y0.
2.3. Parallel transport and exp-parallels on Dc0
In [37] is introduced the exp-parallelism, which is a
generalization of Euclidian parallelism to geodesically-
complete manifolds. It relies on the Riemmanian parallel
transport operator, which we propose to compute using the
fanning scheme [28]. This numerical scheme only requires
the exponential operator to approximate the parallel trans-
port along a geodesic, with proved convergence.
We note Pc0,m0,t0,t :Rdncp !Rdncp the parallel trans-
port operator, which transports any momenta walong the
geodesic :t!t=Exp
c0,t0,t(m0)from t0to t. For
any c0,m0,wand t0, we can now define the curve:
t!⌘c0,m0,t0,t(w)=Exp
(t)(c0)⇥Pc0,m0,t0,t(w)⇤.(3)
This curve, that we will call exp-parallel to , is well-
defined on the manifold Dc0, according to the proposition of
Section 2.2. Figure 1illustrates the whole procedure. From
the top-left shape, the computational scheme is as follow:
integrate the Hamiltonian equations to obtain the control
Figure 1: Samples from a geodesic (top) and an exp-
parallelized curve ⌘(bottom) on Sc0,y0. Parameters encod-
ing the geodesic are the blue momenta attached to con-
trol points and plotted together with the associated velocity
fields. Momenta in red are parallel transported along the
geodesic and define a deformation mapping each frame of
the geodesic to a frame of ⌘. Exp-parallelization allows to
transport a shape trajectory from one geometry to another.
points c(t)(red crosses) and momenta m(t)(bold blue ar-
rows); compute the associated velocity fields vt(light blue
arrows); compute the flow :t!t(shape progression);
transport the momenta walong (red arrows); compute the
exp-parallel curve ⌘by repeating the three first steps along
the transported momenta.
3. Statistical model
For each individual 1iNare available nilongitu-
dinal shape measurements y=(yi,j )1jniand associated
times (ti,j )1jni.
3.1. The generative statistical model
Let c0a set of control points and m0associated mo-
menta. We call the geodesic t!Expc0,t0,t(m0)of Dc0.
Let y0be a template mesh shape embedded in the ambi-
ent space. For a subject i, the observed longitudinal shape
measurements yi,0,...,y
i,niare modeled as sample points at
times i(ti,j )of an exp-parallel curve t!⌘c0,m0,t0,t(wi)
to this geodesic , plus additional noise ✏i,j :
yi,j =⌘c0,m0,t0, i(ti,j )(wi)y0+✏i,j .(4)
The time warp function iand the space-shift momenta wi
respectively encode for the individual time and space vari-
ability. The time-warp is defined as an affine reparametriza-
tion of the reference time t: i(t)=↵i(tt0⌧i)+t0
where the individual time-shift ⌧i2Rallows an inter-
individual variability in the stage of evolution, and the in-
dividual acceleration factor ↵i2R⇤
+a variability in the
pace of evolution. For convenience, we write ↵i= exp(⇠i).
In the spirit of Independent Component Analysis [19], the
space-shift momenta wiis modeled as the linear combi-
nation of nssources, gathered in the ncp ⇥nsmatrix A:
wi=Am?
0si. Before computing this superposition, each
column cl(A)of A has been projected on the hyperplane
m?
0for the metric Kc0, ensuring the orthogonality between
m0and wi. As argued in [37], this orthogonality is fun-
damental for the identifiability of the model. Without this
constraint, the projection of the space shifts (wi)ion m0
could be confounded with the acceleration factors (↵i)i.
3.2. Mixed-effects formulation
We use either the current [42] or varifold [7] noise model
for the residuals ✏i,j , allowing our method to work with in-
put meshes without any point correspondence. In this set-
ting, we note ✏i,j
iid
⇠N(0,2
✏). The other previously in-
troduced variables are modeled as random effects z, with:
y0⇠N(y0,2
y),c0⇠N(c0,2
c),m0⇠N(m0,2
m),A⇠
N(A, 2
A),⌧i
iid
⇠N(0,2
⌧),⇠i
iid
⇠N(0,2
⇠),si
iid
⇠N(0,1).
We define ✓=(y0, c0, m0, A, t0,2
⌧,2
⇠,2
✏)the fixed ef-
fects i.e. the parameters of the model. The remaining vari-
ance parameters 2
y,2
c,2
mand 2
Acan be chosen arbitrar-
ily small. Standard conjugate distributions are chosen as
Bayesian priors on the model parameters: y0⇠N(y0,&2
O),
c0⇠N(c0,&2
c),m0⇠N(m0,&2
m),A⇠N(A, &2
A),
t0⇠N(t0,&2
t),2
⌧⇠IG(m⌧,2
⌧,0),2
⇠⇠IG(m⇠,2
⇠,0),
and 2
✏⇠IG(m✏,2
✏,0)with inverse-gamma distributions
on variance parameters. Those priors ensure the existence
of the maximum a posteriori (MAP) estimator. In practice,
they regularize and guide the estimation procedure.
The proposed model belongs to the curved exponential
family (see supplementary material, which gives the com-
plete log-likelihood). In this setting, the algorithm intro-
duced in the following section has a proved convergence.
We have defined a distribution of trajectories that could
be noted t!y(t)=f✓,t(z)where zis a random variable
following a normal distribution. We call t!f✓,t(E[z]) the
average trajectory, which may not be equal to the expected
trajectory t!E[f✓,t(z)] in the general non-linear case.
4. Estimation
4.1. The MCMC-SAEM algorithm
The Expectation Maximization (EM) algorithm [11] al-
lows to estimate the parameters of a mixed-effects model
with latent variables, here the random effects z. It alter-
nates between an expectation (E) step and a maximiza-
tion (M) one. The E step is intractable in our case, due
to the non-linearity of the model. In [10] is introduced
and proved a stochastic approximation of the EM algo-
rithm, where the E step is replaced by a simulation (S)
step followed by an approximation (A) one. The S step
requires to sample q(z|y, ✓k), which is also intractable
in our case. In the case of curved exponential mod-
els, the authors in [2] show that the convergence holds
if the S step is replaced by a single transition of an er-
godic Monte-Carlo Markov Chain (MCMC) whose sta-
tionary distribution is q(z|y, ✓k). This global algorithm is
called the Monte-Carlo Markov Chain Stochastic Approxi-
mation Expectation-Maximization (MCMC-SAEM), and is
exploited in this paper to compute the MAP estimate of the
model parameters ✓map = max✓Rq(y, z|✓)dz.
4.2. The adaptative block sampler
We use a block formulation of the Metropolis-Hasting
within Gibbs (MHwG) sampler in the S-MCMC step. The
latent variables zare decomposed into nbnatural blocks:
z=y0,c
0,m
0,[cl(A)]l,[⌧i,⇠i,s
i]i . Those blocks have
highly heterogeneous sizes, e.g. a single scalar for ⌧iversus
possibly thousands for y0, for which we introduce a specific
proposal distribution in Section 4.3.
For all the other blocks, we use a symmetric random
walk MHwG sampler with normal proposal distributions
of the form N(0,2
bId)to perturb the current block state
zk
b. In order to achieve reasonable acceptance rates ar i.e.
around ar?= 30% [36], the proposal standard deviations
bare dynamically adapted every nadapt iterations by mea-
suring the mean acceptance rates ar over the last ndetect it-
erations, and applying, for any b:
b b+1
k
ar ar?
(1 ar?)arar?+ar?ar<ar?
(5)
with >0.5. Inspired by [3], this dynamic adaptation is
performed with a geometrically decreasing step-size k,
ensuring the vanishing property of the adaptation scheme
and the convergence of the whole algorithm [2,3]. It proved
very efficient in practice with nadapt =ndetect = 10 and
=0.51, for any kind of data.
4.3. Efficient sampling of smooth template shapes
The first block z1=y0i.e. the coordinates of the points
of the template mesh, is of very high dimension: naively
sampling over each scalar value of its numerical descrip-
tion would result both in unnatural distorted shapes and a
daunting computational burden.
We propose to take advantage of the geometrical na-
ture of y0and leverage the framework introduced in Sec-
tion 2by perturbing the current block state zk
1with a small
displacement field v, obtained by the convolution of ran-
dom momenta on a pre-selected set of control points. This
proposal distribution can be seen as a normal distribution
N(0,2
1DTD)where 2
1is the variance associated with
the random momenta, and Dthe convolution matrix. In
practice, dynamically adapting the proposal variance 2
1and
selecting regularly-spaced shape points as control points
proved efficient.
4.4. Tempering
The MCMC-SAEM is proved convergent toward a local
maximum of ✓!Rq(y, z|✓)dz. In practice, the dimen-
sionality of the energetic landscape q(y, z|✓)and the pres-
ence of multiple local maxima can make the estimation pro-
cedure sensitive to initialization. Inspired by the globally-
convergent simulated annealing algorithm, [25] proposes
to carry out the optimization procedure in a smoothed ver-
sion of the original landscape qT(y, z|✓). The temperature
parameter Tcontrols this smoothing, and should decrease
from large values to 1, for which qT=q.
We propose to introduce such a temperature parameter
only for the population variables zpop. The tempered version
of the complete log-likelihood is given as supplementary
material. In our experiments, the chosen temperature se-
quence Tkremains constant at first, and then geometrically
decreases to unity. Implementing this “tempering” feature
had a dramatic impact on the required number of iterations
before convergence, and greatly improved the robustness of
the whole procedure. Note that the theoretical convergence
properties of the MCMC-SAEM are not degraded, since the
tempered phase of the algorithm can be seen as an initializ-
ing heuristic, and may actually be improved.
Algorithm 1: Estimation of the longitudinal deformations
model with the MCMC-SAEM.
Code publicly available at: www.deformetrica.org.
input : Longitudinal dataset of shapes y=(yi,j )i,j . Initial
parameters ✓0and latent variables z0. Geometri-
cally decreasing sequence of step-sizes ⇢k.
output: Estimation of ✓map.Samples (zs)sapproximately
distributed following q(z|y, ✓map).
Initialization: set k=0and S0=S(z0).
repeat
Simulation:foreach block of latent variables zbdo
Draw a candidate zc
b⇠pb(.|zk
b).
Set zc=(zk+1
1,...,zk+1
b1,zc
b,zk
b+1,...,zk
nb).
Compute the geodesic :t!Expc0,t0,t(m0).
8i, compute wi=A?
m0si.
8i, compute w:t!Pc0,m0,t0,t(wi).
8i, j, compute Exp[ i(ti,j )](c0)w[ i(ti,j )] .
Compute the acceptation ratio !=minh1,q(zc|y,✓k)
q(zk|y,✓k)i.
if u⇠U(0,1) <!then zk+1
b zc
belse zk+1
b zk
b.
end
Stochastic approx.:Sk+1 Sk+⇢k⇥S(zk+1)Sk⇤.
Maximization:✓k+1 ✓?(Sk+1).
Adaptation:if remainder(k+1,n
adapt)=0then update
the proposal variances (2
b)bwith equation (5).
Increment: set k k+1.
until convergence;
4.5. Sufficient statistics and maximization step
Exhibiting the sufficient statistics S1=y0,S2=c0,S3=
m0,S4=A,S5=Pit0+⌧i,S6=Pi(t0+⌧i)2,S7=Pi⇠2
i
and S8=PiPjkyi,j ⌘c0,m0,t0, i(ti,j )(wi)y0k2, the
update of the model parameters ✓ ✓?in the M step of the
MCMC-SAEM can be derived in closed-form. The explicit
expressions are given as supplementary material.
5. Experiments
5.1. Validation with simulated data in R2
Convergence study. To validate the estimation procedure,
we first generate synthetic data directly from the model
without additional noise. Our choice of reference geodesic
is plotted on top line of the previously introduced Fig-
ure 1: the template y0is the top central shape, the chosen
five control points c0are the red crosses, and the momenta
m0the bold blue arrow. Those parameters are completed
with t0= 70,⌧=1,⇠=0.1. With ns=4independent
components, we simulate N= 100 individual trajectories
and sample hniii=5observations from each.
The algorithm is run ten times. Figure 2plots the evolu-
tion of the error on the parameters along the estimation pro-
cedure in log scale. Each color corresponds to a different
run: the algorithm converges to the same point each time,
as it is confirmed by the small variances on the residual er-
rors indicated in Table 1. Those residual errors come from
the finite number of observations of the generated dataset
0 10 20 30 40 50
Thousands of iterations
2
3
4
||y0
est - y 0
true||2
var
Varifold error on y0
0 10 20 30 40 50
Thousands of iterations
1
1.5
2
2.5
||v0
est - v 0
true||2
L2 error on v0
0 10 20 30 40 50
Thousands of iterations
10-1
100
101
|t0
est - t 0
true|
L1 error on t0
0 10 20 30 40 50
Thousands of iterations
10-3
10-2
10-1
100
101
|στ
est - στ
true|
L1 error on στ
0 10 20 30 40 50
Thousands of iterations
10-3
10-2
10-1
100
101
|σξ
est - σξ
true|
L1 error on σξ
0 10 20 30 40 50
Thousands of iterations
0.1
0.2
0.3
0.4
|σϵ
est - σϵ
true|
L1 error on σϵ
Figure 2: Error on the population parameters along the esti-
mation procedure, with logarithmic scales. The residual on
the template shape y0is computed with the varifold metric.
ky0k2
var. kv0k2|t0||⌧||⇠||✏|⌦kvik2↵i⌦|⇠i|↵i⌦|⌧i|↵i
1.43 ±5.6% 0.89 ±0.7% 0.19 ±2.7% 0.029 ±13.2% 0.017 ±7.6% 0.11 ±0.1% 2.47 ±1.7% 0.022 ±6.7% 0.19 ±0.8%
Table 1: Absolute residual errors on the estimated parameters and associated relative standard deviations across the 10 runs.
Are noted v0=Conv(c0,m
0)and vi=Conv(c0,w
i). The operator h.iiindicates an average over the index i. Residuals
are satisfyingly small, as it can be seen for |t0|for instance when compared with the time-span max|tij |=4. The low
standard deviations suggest that the stochastic estimation procedure is stable and reproduces very similar results at each run.
Figure 3: Estimated mean progression
(bottom line in bold), and three recon-
structed individual scenarii (top lines). In-
put data is plotted in red in the rele-
vant frames, demonstrating the reconstruc-
tion ability of the estimated model. Our
method is able to disentangle the variabil-
ity in shape, starting time of the arm move-
ment and speed.
and the Bayesian priors, but are satisfyingly small, as qual-
itatively confirmed by Figure 3. The estimated mean trajec-
tory, in bold, matches the true one, given by the top line of
Figure 1. Figure 3also illustrates the ability of our method
to reconstruct continuous individual trajectories.
Personalizing the model to unseen data. Once a model
has been learned i.e. the parameters ✓map have been esti-
mated, it can easily be personalized to the observations ynew
of a new subject by maximizing q(ynew,z
new |✓map)for the
low-dimensional latent variables znew. We implemented this
maximization procedure with the Powell’s method [35], and
evaluated it by registering the simulated trajectories to the
true model. Table 2gathers the results for the previously-
introduced dataset with hniii=5observations per subject,
and extended ones with hniii=7and 9. The parameters
are satisfyingly estimated in all configurations: the recon-
Experience |✏|⌦|si|↵i⌦|⇠i|↵i⌦|⌧i|↵i
hniii=5 0.110 3.34% 37.0% 5.45%
hniii=7 0.095 2.98% 16.2% 3.86%
hniii=9 0.087 2.38% 11.9% 3.28%
Table 2: Residual errors metrics for the longitudinal regis-
tration procedure, for three simulated datasets. The abso-
lute residual error on ✏is given, the other errors are given
in percentage of the simulation standard deviation.
struction error measured by |✏|remains as low as in the
previous experiment (see Table 1, Figure 3). The acceler-
ation factor is the most difficult parameter to estimate with
small observation windows of the individual trajectories; at
least two observations are needed to obtain a good estimate.
5.2. Hippocampal atrophy in Alzheimer’s disease
Longitudinal deformations model on MCIc subjects. We
extract the T1-weighted magnetic resonance imaging mea-
surements of N= 100 subjects from the ADNI database,
with hniii=7.6datapoints on average. Those sub-
jects present mild cognitive impairements, and are even-
tually diagnosed with Alzheimer’s disease (MCI convert-
ers, noted MCIc). In a pre-processing phase, the 3D
images are affinely aligned and the segmentations of the
right-hemisphere hippocampi are transformed into a surface
meshes. Each affine transformation is then applied to the
corresponding mesh, before rigid alignement of follow-up
meshes on the baseline one. The hippocampus is a subcorti-
cal brain structure which plays a central role in memory, and
experiences atrophy during the development of Alzheimer’s
disease. We initialize the geodesic population parameters
y0,c
0,m
0with a geodesic regression [15,16] performed on
a single subject. The reference time t0is initialized to the
mean of the observation times (ti,j)i,j and 2
⌧to the corre-
sponding variance. We choose to estimate ns=4indepen-
dent components and initialize the corresponding matrix A
to zero, so as the individual latent variables ⌧i,⇠i,s
i. After
10,000 iterations, the parameter estimates stabilized.
!. #$
!. !%
!. &%
!. '(
!. $(
!. ((
!. %(
!. )(
*!
++/-
Figure 4: Estimated mean progression of the right hippocampi. Successive ages: 69.3y, 71.8y (i.e. the template y0), 74.3y,
76.8y, 79.3y, 81.8y, and 84.3y. The color map gives the norm of the velocity field kv0kon the meshes.
!" #$
#" #%
&" #%
'" #(
$" #(
)" #)
(" #)
%" #$
*+,$
--./0
Figure 5: Third independent component. The plotted hippocampi correspond to si,3successively equal to: -3, -2, -1, 0 (i.e.
the template y0), 1, 2 and 3. Note that this component is orthogonal to the temporal trajectory displayed in Figure 4.
Figure 4plots the estimated mean progression, which ex-
hibits a complex spatiotemporal atrophy pattern during dis-
ease progression: a pinching effect at the “joint” between
the head and the body, combined with a specific atrophy of
the medial part of the tail. Figure 5plots an independent
component, which is orthogonal to the mean progression
by construction. This component seems to account for the
inter-subject variability in the relative size of the hippocam-
pus head compared to its tail.
We further examine the correlation between individual
parameters and several patients characteristics. Figure 6ex-
hibits the strong correlation between the estimated individ-
ual time-shifts ⌧iand the age of diagnostic tdiag
i, suggesting
that the hippocampal atrophy correlates well with the cog-
nitive symptoms. The few outliers above the regression line
55 60 65 70 75 80 85 90
( early onset) Onset age t0+t
i(late onset !)
60
70
80
90
Age at diagnosis tdiag
i
Figure 6: Comparison of the estimated individual time-
shifts ⌧iversus the age of diagnostic tdiag
i.R2=0.74.
might have resisted better to the atrophy of their hippocam-
pus with a higher brain plasticity, in line with the cognitive
reserve theory [39]. The few outliers below this line could
have developed a subform of the disease, with delayed atro-
phy of their hippocampi. Further investigation is required
to rule out potential convergence issues in the optimiza-
tion procedure. Figures 7,8,9propose group comparisons
based on the estimated individual parameters: the accelera-
tion factor ↵i, time-shift ⌧iand space-shift si,3in the direc-
tion of the third component (see Figure 5). The distributions
of those parameters are significantly different for the Mann-
Whitney statistical test when dividing the N= 100 MCIc
subjects according to gender, APOE4 mutation status, and
onset age t0+⌧irespectively.
0.5 1.0 1.5 2.0
( slow) Acceleration factor ai(fast !)
0
5
10
Male
Female
Figure 7: Distributions of acceleration factors ↵iaccord-
ing to the gender. Hippocampal atrophy is faster in female
subjects (p=0.045).
55 60 65 70 75 80 85 90
( early onset) Onset age t0+t
i(late onset !)
0
5
10
No APOE4 allele
One APOE4 allele
Two A P OE 4 a l le l es
Figure 8: Distributions of time-shifts ⌧iaccording to the
number of APOE4 alleles. Hippocampal atrophy occurs
earlier in carriers of 1 or 2 alleles (p=0.017 and 0.015).
3210123
Source parameter si,3
0.0
2.5
5.0
7.5
10.0 Early onset
Ave r a ge on s e t
Late onset
Figure 9: Distribution of the third source term si,3accord-
ing to the categories {⌧i3},{3<⌧i<3}, and
{3⌧i}. Hippocampal atrophy seems to occurs later in
subjects presenting a lower volume ratio of the hippocam-
pus tail over the hippocampus head (p=0.0049).
hniiiShape features Naive feature All features
1 71% ±4.5 [lr] 50% ±5.0 [nb] 58% ±5.0 [lr]
2 77% ±4.3 [lr] 58% ±4.9 [5nn] 68% ±4.7 [dt]
4 79% ±4.1 [svm] 67% ±4.7 [5nn] 80% ±4.0 [lr]
5 77% ±4.2 [nn] 77% ±4.2 [lr] 82% ±3.8 [nb]
6.86 83% ±3.7 [lr] 80% ±4.0 [lr] 86% ±3.4 [lr]
Table 3: Mean classification scores and associated standard
deviations, computed on 10,000 bootstrap samples from the
test dataset. Across all tested classifiers (sklearn default
hyperparameters), only the best performing one is reported
in each case: [lr] logistic regression, [nb] naive Bayes,
[5nn] 5 nearest neighbors, [dt] decision tree, [nn] neural net-
work, [svm] linear support vector machine.
Classifying pathological trajectories vs. normal ageing.
We processed another hundred of individuals from the
ADNI database (hniii=7.37), choosing this time control
subjects (CN). We form two balanced datasets, each con-
taining 50 MCIc and 50 CN. We learn two distinct longitu-
dinal deformations model on the training MCIc (N= 50,
hniii=8.14) and CN (N= 50,hniii=8.08) subjects.
We personalize both models to all the 200 subjects, and
use the scaled and normalized differences zMCIc
izCN
ias
feature vectors of dimension 6, on which a list of stan-
dard classifiers are trained and tested to predict the label
in {MCIc,CN}. For several number of observations per
test subject hniiiconfigurations, we compute confidence
intervals by bootstraping the test set. Table 3compares
the results with a naive approach, using as unique feature
the slope of individually-fitted linear regressions of the hip-
pocampal volume with age. Classifiers performed consis-
tently better with the features extracted from the longitu-
dinal deformations model, even with a single observation.
The classification performance increases with the number
of available observations per subject. Interestingly, from
hniii=4pooling the shape and volume features yields an
improved performance, suggesting complementarity.
6. Conclusion
We proposed a hierarchical model on a manifold of dif-
feomorphisms estimating the spatiotemporal distribution of
longitudinal shape data. The observed shape trajectories
are represented as individual variations of a group-average,
which can be seen as the mean progression of the popu-
lation. Both spatial and temporal variability are estimated
directly from the data, allowing the use of unaligned se-
quences. This feature is key for applications where no ob-
jective temporal markers are available, as it is the case for
Alzheimer’s disease progression for instance, whose onset
age and pace of progression vary among individuals. Our
model builds on the principles of a generic longitudinal
modeling for manifold-valued data [37]. We provided a
coherent theoretical framework for its application to shape
data, along with the needed algorithmic solutions for paral-
lel transport and sampling on our specific manifold. We es-
timated our model with the MCMC-SAEM algorithm both
with simulated and real data. The simulation experiments
confirmed the ability of the proposed algorithm to retrieve
the optimal parameters in realistic scenarii. The application
to medical imaging data, namely segmented hippocampi
brain structures of Alzheimer’s diseased patients, deliv-
ered results coherent with medical knowledge, and provides
more detailed insights into the complex atrophy pattern of
the hippocampus and its variability across patients. In fu-
ture work, the proposed method will be leveraged for auto-
matic diagnosis and prognosis purposes. Further investiga-
tions are also needed to evaluate the algorithm convergence
with respect to the number of individual samples.
Acknowledgments. This work has been partly funded by the Eu-
ropean Research Council with grant 678304, European Union’s
Horizon 2020 research and innovation program with grant 666992,
and the program Investissements d’avenir ANR-10-IAIHU-06.
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Appendix: supplementary material
We introduce the onset age individual random variable ti=t0+⌧i⇠N(t0,2
⌧)instead of the time shift ⌧i.
The obtained hierarchical model is equivalent to the one presented in Section 3, with unchanged parameters ✓=
(y0, c0, m0, A, t0,2
⌧,2
⇠,2
✏)and equivalent random effects z=(zpop ,z
1,...,z
N), where zpop =(y0,c
0,m
0,A)and
8i2J1,NK,z
i=(ti,⇠i,s
i). The complete log-likelihood writes:
log q(y, z, ✓)=
N
X
i=1
ni
X
j=1
log q(yi,j |z,✓) + log q(zpop|✓)+
N
X
i=1
log q(zi|✓) + log q(✓)(6)
where the densities q(yi,j |z,✓),q(zpop |✓),q(zi|✓)and q(✓)are given, up to an additive constant, by:
2 log q(yi,j |z, ✓)+cst
=⇤log 2
✏+kyi,j ⌘c0,m0,t0, i(ti,j )(wi)y0k2/2
✏(7)
2 log q(zpop|✓)+cst
=|y0|log 2
y+ky0y0k2/2
y+|c0|log 2
c+kc0c0k2/2
c(8)
+|m0|log 2
m+km0m0k2/2
m+|A|log 2
A+kAAk2/2
A
2 log q(zi|✓)+cst
= log 2
⌧+(tit0)2/2
⌧+ log 2
⇠+⇠2
i/2
⇠+ksik2(9)
2 log q(✓)+cst
=ky0y0k2/&2
y+kc0c0k2/&2
c+km0m0k2/&2
m+kAAk2/&2
A(10)
+(t0t0)2/&2
t+m⌧log 2
⌧+m⌧2
⌧,0/2
⌧+m⇠log 2
⇠+m⇠2
⇠,0/2
⇠
+m✏log 2
✏+m✏2
✏,0/2
✏
noting ⇤the dimension of the space where the residual kyi,j ⌘c0,m0,t0, i(ti,j )(wi)y0k2is computed, and |y0|,|c0|,|m0|
and |A|the total dimension of y0,c0,m0and Arespectively. We chose either the current [42] or the varifold [7] norm for the
residuals.
Noticing the identity ⌘c0,m0,t0, i(ti,j )=⌘c0,m0,0, i(ti,j )t0, the complete log-likelihood can be decomposed into
log q(y, z, ✓)=⌦S(y, z),(✓)↵Id (✓)i.e. the proposed mixed-effects model belongs the curved exponential family.
In this setting, the MCMC-SAEM algorithm presented in Section 4has a proved convergence.
Exhibiting the sufficient statistics S1=y0,S2=c0,S3=m0,S4=A,S5=Piti,S6=Pit2
i,S7=Pi⇠2
iand
S8=PiPjkyi,j ⌘c0,m0,t0, i(ti,j )(wi)y0k2(see Section 4.5), the update of the model parameters ✓ ✓?in the M step
of the MCMC-SAEM algorithm can be derived in closed form:
y0?=⇥&2
yS1+2
yy0⇤/⇥&2
y+2
y⇤t?
0=⇥&2
tS5+2
⌧
?t0⇤/⇥N&2
t+2
⌧
?⇤(11)
c0?=⇥&2
cS2+2
cc0⇤/⇥&2
c+2
c⇤2
⌧
?=⇥S62t?
0S5+Nt?
0
2+m⌧2
⌧,0⇤/⇥N+m⌧⇤(12)
m0?=⇥&2
mS3+2
mm0⇤/⇥&2
m+2
m⇤2
⇠
?=⇥S7+m⇠2
⇠,0⇤/⇥N+m⇠⇤(13)
A?=⇥&2
AS4+2
AA⇤/⇥&2
A+2
A⇤2
✏
?=⇥S8+m✏2
✏,0⇤/⇥⇤Nhniii+m✏⇤(14)
The intricate update of the parameters t0 t?
0and 2
⌧ 2
⌧
?can be solved by iterative replacement.
Similarly to Equation 6, the tempered complete log-likelihood writes:
log qT(y, z, ✓)=
N
X
i=1
ni
X
j=1
log qT(yi,j |z,✓) + log qT(zpop|✓)+
N
X
i=1
log q(zi|✓) + log q(✓)(15)
with: 2 log qT(yi,j |z, ✓)+cst
=⇤log(T2
✏)+kyi,j ⌘c0,m0,t0, i(ti,j )(wi)y0k2/(T2
✏)(16)
2 log qT(zpop|✓)+cst
=|y0|log(T2
y)+ky0y0k2/(T2
y)+|c0|log(T2
c)+kc0c0k2/(T2
c)(17)
+|m0|log(T2
m)+km0m0k2/(T2
m)+|A|log(T2
A)+kAAk2/(T2
A)
Tempering can therefore be understood as an artificial increase of the variances 2
✏,2
y,2
c,2
mand 2
Awhen computing
the associated acceptation ratios in the S-MCMC step of the algorithm. This intuition is well-explained in [25].
