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Deformetrica is an open-source software for the statistical analysis of images and meshes. It relies on a specific instance of the large deformation diffeomorphic metric mapping (LDDMM) framework, based on control points: local momenta vectors offer a low-dimensional and interpretable parametrization of global diffeomorphims of the 2/3D ambient space, which in turn can warp any single or collection of shapes embedded in this physical space. Deformetrica has very few requirements about the data of interest: in the particular case of meshes, the absence of point correspondence can be handled thanks to the current or var-ifold representations. In addition to standard computational anatomy functionalities such as shape registration or atlas estimation, a bayesian version of atlas model as well as temporal methods (geodesic regression and parallel transport) are readily available. Installation instructions, tutorials and examples can be found at http://www.deformetrica.org.
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Deformetrica 4: an open-source software for
statistical shape analysis
Alexandre Bˆone12345, Maxime Louis12 34 5, Benoˆıt Martin1 23 45 , and
Stanley Durrleman12345
1Institut du Cerveau et de la Moelle ´epini`ere, ICM, F-75013, Paris, France
2Inserm, U 1127, F-75013, Paris, France
3CNRS, UMR 7225, F-75013, Paris, France
4Sorbonne Universit´e, F-75013, Paris, France
5Inria, Aramis project-team, F-75013, Paris, France
Abstract. Deformetrica is an open-source software for the statistical
analysis of images and meshes. It relies on a specific instance of the
large deformation diffeomorphic metric mapping (LDDMM) framework,
based on control points: local momenta vectors offer a low-dimensional
and interpretable parametrization of global diffeomorphims of the 2/3D
ambient space, which in turn can warp any single or collection of shapes
embedded in this physical space. Deformetrica has very few requirements
about the data of interest: in the particular case of meshes, the absence
of point correspondence can be handled thanks to the current or var-
ifold representations. In addition to standard computational anatomy
functionalities such as shape registration or atlas estimation, a bayesian
version of atlas model as well as temporal methods (geodesic regression
and parallel transport) are readily available. Installation instructions,
tutorials and examples can be found at http://www.deformetrica.org.
Keywords: statistical shape analysis ·computational anatomy ·large
deformation diffeomorphic metric mapping ·open-source software.
1 Introduction
D’Arcy Thomson first proposed the idea to compare two distinct shapes through
the ambient-space deformations that transform one into the other [17]. Many
years later, this insight still proves relevant, and one of its state-of-the-art avatar
is the large deformation diffeomorphic metric mapping (LDDMM) [7,15], which
offers a modern and principled framework for the construction of such trans-
formations. Deformetrica relies on a specific instance of this framework, based
on control points [7]. Section 2 details this theoretical backbone of our soft-
ware, along with the current and varifold representations, which allow to handle
mesh without point correspondence. Section 3 reports the competitive execution
times of those core operations. Section 4 describes how this computation core is
leveraged to offer ready-to-use higher level models to study shape dataset.
Equal contributions.
2 Theoretical background
2.1 Control-points-based LDDMM: constructing diffeomorphisms
Deformetrica offers a low-dimensional and interpretable parametrization of dif-
feomorphisms of the ambient space Rd,d∈ {2,3}. Let (qk)k=1,...,p a set of p
“control” points in Rdand (µk)k=1,...,p be a set of p“momentum” vectors of Rd.
Those paired sets define a vector “velocity” field vof the ambient space through
a convolution filter:
v:xRdv(x) =
p
X
k=1
K(x, qk)·µk(1)
where Kis typically a gaussian kernel K(x, y) = exp −kxyk22of kernel
width σ > 0. The kernel width σwill control the typical width of the generated
deformation patterns. The set of vector fields vof the form (1) is a reproducible
kernel Hilbert space (RKHS) V, with norm:
kvk2
V=X
k,l=1,...,n
K(qk, ql)·µ>
kµl.(2)
Evolution equations are prescribed for the control point and momentum sets,
called the “Hamiltonian” equations:
(˙q(t) = K(q(t), q(t)) ·µ(t)
˙µ(t) = 1
2qK(q(t), q(t)) ·µ(t)>µ(t)(3)
These equations are integrated using an Euler or a Runge-Kutta of order 2
scheme. Is therefore obtained a time-varying velocity field v(x, t) that can be
computed at any time tusing equation (1) with the corresponding control points
q(t) and momenta µ(t).
Let xRdbe any point of the ambient space. We define the transformed
point Φ(x) as the value at time 1 of the function l: [0,1] 7→ Rdwith initial
condition l(0) = xand which obeys the ordinary differential equation:
l0(t) = v(l(t), t).(4)
The obtained mapping Φ:Rd7→ Rdis a diffeomorphism of the ambient space
Rd. Mathematical details are available in [19].
Overall, the obtained diffeomorphism Φis fully parametrized by initial sets of
control points qand momenta µ: we will note Φ=Φq,µ . This simple parametriza-
tion of a large family of diffeomorphisms paves the way to the optimization of
the initial control points qand momenta µto estimate a desired transformation
of the ambient space.
On a more theoretical note, for a fixed number of control points pthe obtained
set of diffeomorphisms has the structure of a finite-dimensional manifold, its
geodesics are defined by the Hamiltonian equations (3), its tangent space at any
point is the set of velocity fields obtained by the convolution of any momenta on
the corresponding control points, and its cometric is given by the kernel matrix
[K(qk, ql)]k,l=1,...,p.
2.2 Diffeomorphic action on shapes: deforming meshes or images
Once a diffeormorphism of the ambient space is constructed, the way it deforms
a shape must be specified. We distinguish the cases of mesh data and image
data. A diffeormophism acts on a mesh by direct and independent application
onto its vertices. On an image I:Rd7→ R, a diffeomorphism acts according to:
Φq,µ(I) = IΦ1
q,µ.
This computation is done the following way:
1. A initial regular grid of points (sk)k=1,...,r corresponding to the voxel posi-
tions of the original image Iis determined.
2. The positions Φ1(sk) are computed. This is achieved using equation (4) for
k∈ {1, . . . , r}, integrated from 1 to 0, with initial position l(sk) = skand
using the opposite of the momenta µ(t) describing the diffeomorphism. This
operation is exactly as expensive as the computation of the deformation of
a mesh with rvertices.
3. The intensities at the positions Φ1(sk) are computed by bi/tri-linear in-
terpolation from the original image intensities, and assigned as being the
intensity of the deformed image on the grid at position sk. Zero padding is
applied outside the original image. This operation is massively parellelizable.
In the rest of the paper, we will note Φq,µ ? S the result of the action of a
diffeomorphism Φq,µ on a shape S.
2.3 Shape attachments: evaluting deformation residuals
To evaluate if the deformed shape is close to its target, a metric is needed. For
images, the Euclidian `2distance is trivially available. For meshes, the same `2
metric can be used if there is a point-to-point correspondence. In the general case
of meshes without point correspondence, the “current” or “varifold” distances
are available, and described in the rest of this section.
Whether the connectivity of the mesh is made of segments or triangles, it is
possible to compute the centers (ck)k=1,...,r and the normals (nk)k=1,...,r of the
edges. Equipped with those, one can compute either the current distance [18]:
d(nα
k, cα
k)p=1,...,rα,(nβ
l, cβ
l)l=1,...,rβ2
=X
kX
l
KW(cα
k, cβ
l)·(nα
k)>nβ
l
or rather the varifold distance [5], which ignores the orientation of the normals:
d(nα
k, cα
k)k=1,...,rα,(nβ
l, cβ
l)l=1,...,rβ2
=X
kX
l
KW(cα
k, cβ
l)·(nα
k)>nβ
l2
knα
kkknβ
lk
where KWis a Gaussian kernel with width σW.
Deformetrica offers the possibility to compute simultaneous deformations of
several shapes all embedded in the same ambient space Rd. If Oα= (Sα
1, . . . , Sα
ns)
and Oβ= (Sβ
1, . . . , Sβ
ns) are two objects constituted of nshomologous shapes,
Deformetrica computes the squared distance via:
d(Oα, Oβ)2=
ns
X
k=1
d(Sα
k, Sβ
k)2
σ2
k
(5)
which is a weighted average of the squared distances of the corresponding objects.
The parameters σkcan be used to tune the relative importances of each part of
the composite “multi-object” of study.
2.4 A glimpse at optimization
Each Deformetrica model leverages those deformation and attachment mechanics
to define a specific cost function, that will then be optimized either by steepest
gradient descent or with the limited-memory Broyden-Fletcher-Goldfarb-Shanno
(L-BFGS) method [12]. Deformetrica 4 exploits the automatic differentiation
functionalities offered by the PyTorch project [16] to compute the required gra-
dients, as suggested in [11].
3 Performances
The deformation mechanics heavily rely on convolution operations, as well as
computing current or varifold attachments. Computing a convolution has a
quadratic numerical complexity with the number of considered points, and is
therefore a very critical operator in Deformetrica. A second constraint arise
with automatic differentiation memory requirements, which are also quadratic
with the input data sizes in the case of a naive implementation. Deformetrica
features two ways to perform convolution, both either on CPU or GPU:
using a naive PyTorch-based code [16], typically faster for small data sizes
but unreasonably memory-greedy with larger data;
using the dedicated PyKeops library [4] which offers a PyTorch-compatible
python wrapper for memory-efficient kernel operations with their derivatives.
This library is typically required to deal with real-size data.
An additional performance switch is offered by the PyTorch library: all linear
algebra operations can be ported directly on GPU with a single flag. Obviously,
this come at the cost of an increased GPU memory usage.
Figure 1 reports typical execution times against the data size, respectively for
the attachment and deformation atomic operations. The reported times include
the (automatic) computation of the gradient. This benchmark has been made
Fig. 1. Top: needed time to compute either the current or varifold attachment and
the associated gradient, versus the number of vertices in each mesh. Bottom: needed
time to compute either a landmark or image deformation and the associated gradient,
versus the number of vertices and voxels respectively. The reported times are averages
over 100 evaluations.
on an Ubuntu 14.04 machine, equipped with an Intel Xeon E5-1630 v3 CPU
and Nvidia Quadro M4000 GPU with Nvidia driver version 384.130. Note that
both the PyTorch and PyKeops libraries are quite recent, and can be expected
to improve their performances in the near future.
In all cases, the “torch”-based convolutions are faster for small data sizes,
but are overtaken by the “keops”-based ones at some point. The CPU-only
operations can prove efficient to compute the deformation of small shapes, but
quickly become order of magnitudes slower than their GPU equivalents for larger
data. The “full-gpu” option does not lower the execution times for attachments,
when it consistently does so for deformations. Note that the torch-based curves
are interrupted earlier than their keops-based counterparts, because the memory
requirements due to automatic differentiation becomes unreasonable for too large
data sizes.
We can finally underline the satisfyingly fast image deformation performances,
allowing to register two full-resolution (181 ×217 ×181) T1-weighted magnetic
resonance images (MRIs) in 1 minute and 42 seconds (after 50 iterations of the
L-BFGS estimator), with a GPU memory footprint around 2 gigabytes. Choos-
ing the slower but much less memory-intensive “keops-gpu” mode instead of
“keops-full-gpu”, the same registration takes 3 minutes and 22 seconds with a
GPU memory footprint of 60 megabytes. In absence of gpu, the “keops-cpu”
option allows to still estimate the registration, but requires around 10 hours.
4 Deformetrica applications
4.1 Atlas and registration
Cost function We consider here a cross-sectional collection of shapes (Si)i=1,...,n.
The atlas model offers to compute a mean Tof the shapes and a collection of
diffeomorphisms (Φi)i=1,...,n such that for all i∈ {1, . . . , n}, we have Φi?T 'Si.
This is achieved by minimization of the cost function:
C(T, q, µi=1,...,n) = X
i
d(Φq,µi? T, Si)22
+R(q, (µi)i=1,...,n ),(6)
with R(q, (µi)i=1,...,n ) = X
i
µ>
iK(q, q)µi(7)
noting K(q, q) the p-by-p“kernel” matrix [K(qk, ql)]k,l=1,...,p . The first term in
equation (6) controls the data attachment i.e. how well the collection of objects
is fitted by the deformation of the template, while the second term acts as a
regularizer by penalizing the kinetic energy of the deformations. The relative
importance of those two terms is specified by the user through the parameter
σ. The resulting atlas obtained from images of digits is displayed by Figure 4.1.
Smoothing the gradient. When working with meshes with boundaries, the
gradient of the cost function (6) with respect to the mesh vertices positions T
can be very large near the boundary, inducing the estimated template Tto have
a non-natural shape. A workaround consists in convolving the analytic gradient
with a Gaussian kernel. It provides a different descent direction which results in
a smoother estimated template.
Registration The registration problem is a particular instantiation of the atlas
cost function with a single target Sand a fixed template T:
C(q, µ) = d(Φq,µ ? T , S)22
+R(q, µ) (8)
Fig. 2. Illustration of an estimated
“deterministic” atlas model on the five
images represented at the bottom row.
The top row represents five repetitions
of the estimated template shape, when
the following rows represent the pro-
gressive deformations of this template
that eventually match well the input
dataset shown on the last row. The
somehow unnatural rightmost defor-
mation indicates that the σparam-
eter might advantageously be chosen
slightly greater, since less energetic de-
formations would be estimated.
It has numerous applications in medical imaging. For instance, registering MRIs
from two different patients allows to perform relevant voxel-wise intensity com-
parisons, after removal of their natural anatomical differences. Alternatively, it
can be leveraged to transfer some standard brain segmentation towards a new
particular subject.
4.2 Bayesian Atlas
The atlas cost function (6) can be seen as an approximation of the negative com-
plete log-likelihood of a generative, hierarchical, mixed-effects statistical model,
that we call the Bayesian atlas one [10].
Statistical model From a common template Tand control points q, the indi-
vidual shapes Siare considered as random deformations of Tplus noise:
Si=Φq,µi? T +i,with µi
iid
∼ N (0, Σµ) and i
iid
∼ N (0, σ).(9)
To fit the framework of mixed-effects models, we distinguish the model fixed
effects θ= (T, q, Σµ, σ) and the model random effects z= (µi)i. Inverse-Wishart
bayesian priors are chosen for the variance parameters: Σµ∼ IW(Γµ, mµ) and
σ∼ IW (γ, m). The introduced additional hyper-parameters are by default
automatically set following the heuristics given in [10].
Log-likelihood Noting S= (Si) the collection of all the observations, the
complete log-likelihood is given by:
2 log p(S, θ, z) = X
id(Φq,µi? T, Si)22
+µ
iΣ1
µµi(10)
+mµlog (det Σµ) + TrΣ
µΓ1
µ+mlog σ2
+γ2
2
.
The maximum a posteriori (MAP) estimate of the model parameters can be
approximated as follow:
θmap = argmaxθZp(S, θ, z)dzargmaxθ ,z p(S, θ, z).(11)
This classical ”max-max” or ”mode” approximation becomes an equality in the
limit case where p(z) is a Dirac distribution, i.e. Σµ= 0.
Note that computing this approximate MAP amounts to finding the mini-
mum of the negative log-likelihood (10), which echoes the previously introduced
atlas cost function (6). The introduced modeling provides a statistical interpre-
tation to the regularization term, which arises from assumed underlying ran-
dom structures on the momenta µiand the residuals i. Those assumptions are
weaker, more intrinsic than arbitrarily prescribing the regularization term (7):
the estimated atlas can therefore be expected to be more data-driven, or in other
words more representative of the input data.
Estimation The Bayesian atlas is estimated in Deformetrica with gradient-
based methods following the iterative procedure described in [10], which al-
ternates gradient steps over the current estimates of T, q , (µi) and closed-form
updates of the variance parameters Σµ, σ.
A second class of estimation methods, based on a stochastic approximation
of the classical expectation-maximization algorithm (see [1,6]) will be released in
Deformetrica 4.1. This so-called SAEM estimator will compute the exact θmap,
integrating out the full distribution of the momenta random effects.
4.3 Geodesic regression
Geodesic regression generalizes linear regression to manifold-valued data [8, 9].
We consider here a time-series dataset (Si)i=1,...,n observed at times (ti)i=1,...,n.
Practical examples could be repeated MRIs of the same individual, or repeated
observations of the growth of a plant. The cost function for geodesic regression is:
C(T, q, µ) = X
i
d(Φq,tiµ? T, Si)22
+R(q, µ).(12)
where R(q, µ) is given by equation (7). The first term in (12) controls the at-
tachment of the data while the second penalizes the “kinetic” energy of the
deformation. The data-attachment versus regularity tradeoff is addressed by the
Fig. 3. Estimated geodesic regression. Top row: the estimated trajectory. Bottom row:
observations from which the top trajectory is learned.
user-specified parameter σ. Note that the trajectory t7→ Φq,tµ ? T is the action
of a geodesic on the q-manifold of diffeomorphisms onto the template shape T.
Optimization of this cost yields an estimated template shape Tas well as
sets of control points and associated initial momenta, so that the induced time-
continuous flow of diffeomorphisms applied to the template shape t7→ Φq,tµ ? T
is as close as possible to the input observations. Figure 3 shows an example of
geodesic regression on 3D meshes of human faces (data courtesy of Paolo Piras,
Sapienza Universit´a di Roma, Italy).
4.4 Parallel transport in shape analysis
Deformetrica implements the parallel transport method for shape analysis de-
scribed in [13]. Given two sets of control points and momenta qα, qβand µα, µβ,
the parallel transport is a differential geometry notion which allows to consider
the translation of the deformation described by qβ, µβalong the deformation
defined by qα, µα. The computation of this transport can be done following a
procedure whose convergence is proven in [14].
An interesting example occurs when qα, µαdescribes a known progression,
for example a geodesic regression learned from repeated observation of a refer-
ence subject and when qβ, µβdescribes a registration between an observation of
the reference subject and a new subject. In that case, the flow of the parallel-
transported deformation can be used to obtain a prediction of the future state
of the subject [3]. It is in some sense a transfer learning operation.
Fig. 4. Parallel transport of the human face trajectory shown on Figure 3 onto a
different face.
Figure 4 shows an example of parallel translation of the geodesic progression
obtained on Figure 3 onto a face with a different form.
5 Conclusion
Deformetrica implements common computational anatomy methods both on
meshes and images. Future releases of the software will include probabilistic prin-
cipal geodesic analysis [20] as well as the longitudinal atlas statistical model [2].
One of the main limitation of the software for a wider range of applications
lies in the purely geometrical modeling of the shapes. Mainly, a deformation
model cannot change the topology of the deformed image, thus restricting the
range of applications. Using the metamorphosis framework or including func-
tional shapes could increase the impact of the software.
Acknowledgments. This work has been partly funded by the European Research
Council (ERC) under grant agreement No 678304, European Union’s Horizon 2020
research and innovation program under grant agreement No 666992, and the program
Investissements d’avenir ANR-10-IAIHU-06.
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The hominin mandible SK 15 was discovered in April 1949 in Swartkrans Member 2, dated to~1.4 Ma. Albeit distorted on the right side, the left and right corpus of SK 15 are relatively low and thick, even compared to most Early to Middle Pleistocene Homo specimens. It preserves the left molar row and the right M 2 and M 3 that show a distalward increase in mesiodistal diameter. SK 15 was originally attributed to Telanthropus capensis but is now generally attributed to Homo erectus/Homo ergaster, even if it was previously suggested to possibly belong to Australopithecus. Similarities between SK 15 and Homo naledi mandible and tooth morphology were also claimed. To clarify the taxonomy of SK 15, we used X-ray microtomography to investigate aspects of bone and tooth structural organization. Geometric morpho-metric analyses of the dental arcade shape, mandible symphysis outline, and the M 2 and M 3 enamel-dentine junction shape were conducted. For mandibular symphysis shape, SK 15 exhibits an australopith signal, whereas for both the dental arcade and enamel-dentine junction analyses, the specimen is statistically classified as Paranthropus. Altogether, the results show that SK 15 unambiguously falls outside the variation of H. erectus/H. ergaster and that it is most compatible with the morphology of Paranthropus, albeit showing smaller dimensions and an absence of some dental morphological features (e.g., developed protostylid, distally tapering M 3 , short molar roots) typically found in specimens of Paranthropus aethiopicus, Paranthropus boisei, and Paranthropus robustus. In particular, SK 15 differs markedly in size and morphology from mandibular remains of P. robustus from Swartkrans Member 2. We thus tentatively attribute SK 15 to Paranthropus capensis, a more gracile species of Paranthropus than the other three currently recognized species of this genus and discuss the implications for the existence of another species of Paranthropus in southern Africa during the Early Pleistocene.
... LAA SSA may also be applied implicitly on both the LAA only (Goparaju et al., 2022;Ahmad et al., 2024;Goparaju et al., 2018) and for the conjoint left atria with LAA (Corrado et al., 2020). Implicit approaches typically rely on the optimisation of deformations in a Riemannian space to warp shapes into others (Bône et al., 2018;Hartman et al., 2023). Established frameworks, such as Deformetrica (Bône et al., 2018), have been used (Goparaju et al., 2022;Goparaju et al., 2018), and recent works have also experimented with dedicated frameworks (Hartman et al., 2023) applied specifically to the LAA (Ahmad et al., 2024). ...
... Implicit approaches typically rely on the optimisation of deformations in a Riemannian space to warp shapes into others (Bône et al., 2018;Hartman et al., 2023). Established frameworks, such as Deformetrica (Bône et al., 2018), have been used (Goparaju et al., 2022;Goparaju et al., 2018), and recent works have also experimented with dedicated frameworks (Hartman et al., 2023) applied specifically to the LAA (Ahmad et al., 2024). However, to our knowledge, such methods do not allow the high complexity of the LAA surfaces to be considered. ...
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Parametrized Plane Curves.- Medial Axis.- Moment-Based Representation.- Local Properties of Surfaces.- Isocontours and Isosurfaces.- Evolving Curves and Surfaces.- Deformable templates.- Ordinary Differential Equations and Groups of Diffeomorphisms.- Building Admissible Spaces.- Deformable Objects and Matching Functionals.- Diffeomorphic Matching.- Distances and Group Actions.- Metamorphosis.