Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart - Modeling in Computational Biology and Biomedicine - page 165

Fig. 5.3 Riemannian statistics on the sphere: the Frechet mean is the point minimizing the squared
Riemannian distance. It corresponds to the point for which the development of the geodesics to the
data points on the tangent space is optimally centered (the mean in that tangent space is zero). The
covariance matrix is then defined in that tangent space (Figure adapted from [ 22 ])

if it is centered, i.e. if it minimizes (among all possible atlases) the sum of squared
distances to the shapes [ 32 ]. In this sense, this is the Frechet mean of the shapes.
Then, each shape is represented by the momentum of the deformation that allows
to regenerate the shape by deforming the atlas. These momentum all belong to the
tangent space (more precisely to the cotangent space) of the diffeomorphisms at the
identity, which is a vector space in which we can perform a PCA [ 15 ]: shooting in
the space of deformations along the first eigenvectors of the covariance matrix of
the momentum gives deformation modes that represent the main shape variability
when applied to the atlas.

5.2.2 From Points to Surfaces: The Formalism of Currents

Landmarks can be encoded by the probability of their location in space (typically
a Gaussian around their expected value). When the noise is going to zero, then
this probability density function (pdf) becomes singular. However, we can continue
to deal with these types of singularities by considering distributions (generalized
functions) instead of functions, which include Diracs. Mathematically, a Dirac
is not a function, but an object which can be characterized by the result of its
integration against any function of a sufficiently smooth functional space: ∀ f ∈

W, δ x (

. In a finite dimensional Euclidean vector space, any
linear form φ (function from the vector space to

y ) .f ( y ) .dy = f ( x )

) can be characterized using the
scalar product: Thanks to the Rietz representation theorem, there exists a vector x
such that φ

R

. The equivalent in infinite dimensional functional spaces
is the Dirac distribution, which is actually an element of the linear functionals over
the space W. In that framework, a set of N points x k can be represented by the 'pdf'
p

( w )= x | w

/N i ( x )=1 δ x i ( x ) , and its evaluation on a function f ( x ) results in the mean of

the values of f at the points x k .