Spaces emerging from Grenander's Pattern Theory have a
special interest, because of their generality and flexibility.
These spaces derive from the structure induced by groups of
diffeomorphisms through the deformation induced by their actions
on shapes. In this context, a shape is not represented as such but
as a deformation of another (fixed) shape, called template. The
deformable template paradigm is rooted in the work of
D'Arcy-Thompson in his celebrated treatise (On Growth and Form),
and developed in Grenander's theory. Even if Pattern Theory is
more general, recent models of deformable templates in shape
analysis focus on deformations represented by diffeomorphisms
acting on landmarks, curves, surfaces or other structures that can
represent shapes. More precisely, if $T_0$ is the template, one
represents shapes via the map $\pi(\varphi) = \varphi \cdot T_0$,
which denotes the action of a diffeomorphism, $\varphi$ on $T_0$
($\varphi$ being a diffeomorphism in the ambient space, in
contrast to changes of parametrization, which are diffeomorphisms
of the parametrization space). With this model, the diffeomorphism
can be interpreted as an extrinsic parameter for the
representation.
INSERT IMAGE "fromTechnology.jpg" HERE WITH
CAPTION: A cartoon depiction of a shape
space with template-centered coordinate systems
(from Miller et al., Technology, 2014).
Letting $\mathrm{Diff}$ denote the space of diffeomorphisms, and
$\mathcal Q$ be the shape space, one can use the transformation
$\pi : \mathrm{Diff} \to \mathcal Q$ to "project" a
mathematical structure defined on diffeomorphisms to the shape
space. Using this paradigm, one can, from a single modeling effort
(on $\mathrm{Diff}$) design many shape spaces, like spaces of
landmarks, curves surfaces, images, density functions or measures,
etc.
The space of diffeomorphisms, which forms an algebraic group,
is a well studied mathematical object. The relationship between
right-invariant Riemannian metric on this space and classical
equations in fluid mechanics has been described in V.I. Arnold's
seminal work, followed by a large literature, by J.E. Marsden, T.
Ratiu, D.D. Holm and others. It is remarkable that the same
construction induces interesting shape spaces leading to concrete
applications in domains like medical image analysis.
Because the transformation $\varphi \rightarrow \pi(\varphi)$ is
many-to-one in general, the projection mechanism from
$\mathrm{Diff}$ to $\mathcal Q$ involves an optimization step over
the diffeomorphism group: given a target shape $T$, one looks for
an optimal diffeomorphism $\varphi$ such that $\pi(\varphi) =
\varphi\cdot T_0 = T$. Shapes are then compared by comparing these
optimal diffeomorphisms, or some parametrization that
characterizes them. Optimality is based on the Riemannian metric
on $\mathrm{Diff}$, and more precisely on the distance between
$\varphi$ and the identity mapping $\mathrm{id}$ for this metric.
The resulting $\pi$ then has the properties of what is called a Riemannian
submersion. Because the constraint \(\pi(\varphi)=T\) is hard
to achieve numerically in general, one preferably replaces this
constraint by a penalty term in the minimization, so that
the diffeomorphism representing a shape is sought via the
minimization of
\[
\varphi \mapsto \mathrm{dist}(\mathrm{id}, \varphi) + \lambda
E(\varphi\cdot T_0, T)
\]
where $E$ is an error function. This formulation leads to the
LDDMM (large deformation diffeomorphic metric mapping) algorithm,
first introduced for landmarks and images, then for curves and
surfaces. In this approach, the optimal $\varphi$ is
computed as the flow of an ODE (ordinary differential equation),
so that $\varphi(x) = \psi(1,x)$ with
\[
\frac{d\psi}{dt}(t,x) = v(t, \psi(t,x))
\]
where $v$ is a time-dependent vector field in the ambient space.
The problem can then be reformulated as an optimal control
problem where $v$ is the control, minimizing
\[
(v , \psi) \mapsto \int_0^1 \|v(t, \cdot)\|^2_V dt + E(\psi(1,
\cdot)\cdot T_0, T)
\]
subject to $\frac{d\psi}{dt}(t,x) = v(t, \psi(t,x))$, where
$\|\cdot\|_V$ is a norm over a Hilbert space $V$ of smooth vector
fields (e.g., reproducing kernel Hilbert space). Introducing the
time variables results in a continuous deformation from the
template to the target.
INSERT VIDEO: rotatingShape.avi WITH CAPTION: A surface
progressively deforming to a target (wireframe).
Shape descriptors characterizing the deformation are overlaid on
the deforming surface.
Using the optimal diffeomorphism estimated by the LDDMM algorithm
as a shape descriptor, one is able to analyze shape datasets, and
define statistically significant shape variations in the
considered structure. This has been applied at multiple times in
computational anatomy studies, in which one can exhibit patterns
of brain atrophy that are associated to neurodegenerative diseases
like Alzheimer's and Huntington's.
More details can be found in the following papers, and in other work of M. Miller, S. Joshi, A. Trouv�, J. Glaun�s, D.D. Holm, F-X Vialard, S. Durleman etc.
Matching
deformable objects, A Trouv�, L Younes, Traitement du Signal
20 (3), 295-302, 2003
The
metric spaces, Euler equations, and normal geodesic image
motions of computational anatomy, M.I. Miller, A. Trouv�, L.
Younes, Image Processing, 2003. ICIP 2003. Proceedings. 2003
International Conference on, 2003
Computing large deformation metric mappings via geodesic flows
of diffeomorphisms, M.F. Beg, M.I. Miller, A. Trouv�, L.
Younes, International journal of computer vision 61 (2), 139-157,
2005
The Euler-Lagrange equation for interpolating sequence of
landmark datasets, MF Beg, M Miller, A Trouv�, L Younes,
Medical Image Computing and Computer-Assisted Intervention-MICCAI
2003, 918-925, 2003
On the metrics and Euler-Lagrange equations of computational
anatomy, MI Miller, A Trouv�, L Younes, Annual review of
biomedical engineering 4 (1), 375-405, 2002
Diffeomorphic matching of distributions: A new approach for
unlabelled point-sets and sub-manifolds matching, J.
Glaunes, A. Trouv�, L. Younes, Computer Vision and Pattern
Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE
Computer Society Conference on, 2004
Large
deformation diffeomorphic metric mapping of vector fields,
Y. Cao, M.I. Miller, R.L. Winslow, L. Younes, Medical Imaging,
IEEE Transactions on 24 (9), 1216-1230, 2005
Modeling
planar shape variation via Hamiltonian flows of curves, J.
Glaun�s, A. Trouv�, L. Younes, Statistics and analysis of shapes,
335-361, 2006
Diffeomorphic
matching of diffusion tensor images, Y. Cao, M.I. Miller, S.
Mori, R.L. Winslow, L. Younes, Computer Vision and Pattern
Recognition Workshop, 2006. CVPRW'06. Conference on, 2006
Large
deformation diffeomorphic metric curve mapping, J. Glaun�s,
A. Qiu, M.I. Miller, L. Younes, International journal of computer
vision 80 (3), 317-336, 2008
A
kernel class allowing for fast computations in shape spaces
induced by diffeomorphisms, A. Jain, L. Younes, Journal of
Computational and Applied Mathematics, 2012
Diffeomorphometry
and geodesic positioning systems for human anatomy, MI
Miller, L Younes, A Trouv�, Technology 2 (01), 36-43