Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart - Modeling in Computational Biology and Biomedicine - page 164

5.2.1.2 Deformations

An alternative modeling of shapes was proposed by D'Arcy Thompson in 1917
[ 37 ]. The idea is to assume that there is a template object, which represents the
reference shape. In medical image analysis, the template is most often built from
real observations and is called an atlas. 1 The atlas can be seen as an equivalent
of the mean shape. Then, the variability of the shape is analyzed through the
deformations of this reference object towards the actual observations: a shape
difference is encoded by the transformation that deforms one onto the other. As
the deformation of a smooth object should be a smooth object, we have to work
with diffeomorphisms (invertible, one-to-one mappings with smooth inverses).
This formalism was promoted to a generic shape analysis tool by Grenander
and Miller [ 8 , 20 ] based on advanced mathematical tools to compute on infinite
dimensional groups of diffeomorphisms [ 38 ]. One key feature of this lift of the
shape characteristics from the object space to the transformation space is that it
allows to apply the typical deformations to other objects than the ones analyzed in
the first place (provided that they also live in the coordinate system of the atlas).
For instance, the surface of the cortex in the brain has deep foldings called sulci
which are important anatomical features. Trouv´eetal.[ 28 ] analyzed the variability
of the trace of these sulci on the surface of the brain cortex to extract the main
3D deformation modes. These modes could then be used to deform accordingly the
surface of the cortex or the full 3D volume of the brain.
However, the problem is even more complex than previously as we want here
to perform statistics on large deformations, which are known to belong to a non-
linear and infinite dimensional manifold. The Riemannian setting is one of the most
powerful structures to generalize simple statistics to non-linear spaces: it provides
a definition of the distance between points of our manifold and a notion of shortest
path/straight lines using geodesics [ 22 , 23 ]. The main difficulty is that the mean
value cannot be defined through an integral or a sum as in Euclidean spaces. Instead,
one must look for points in the manifold that minimize the dispersion of the other
points around it, which is commonly measured by the variance (the mean squared
distance). This is what is called the Frechet or Karcher mean (Fig. 5.3 ).
Then, one can compute the covariance matrix (the directional dispersion around
the mean) by developing the manifold onto its tangent space at the mean point.
Basically, this amounts to representing each data point by the momentum (initial
speed vector) needed to shoot 2 a geodesic to it from the mean.
This generalization of statistics has to be slightly modified to fit the atlas
deformation model: the 'distance' between the atlas and a shape is given here by the
length of the shortest path in the space of deformations. The atlas is called unbiased

1 In this paper, atlas is always taken in the sense of template and not in the sense of the atlas of
differential geometry.
2 We use the term “geodesic shooting” to define the integration of the Euler-Lagrange equations,
which plays the role of the exponential map in Riemannian geometry.