# Statistical Analysis on Shape Manifold

Statistical analysis on anatomy shapes is one of the key goals of CA. Two of the

most commonly used statistical tools are the:

Statistical shape atlas, which aims to build statistics of organ shapes across diseases, populations, species, or ages. Its key task is to estimate representative organ anatomies and the intersubject shape variability.

Longitudinal shape data analysis is used to model the organ development across time, i.e., statistical analysis of the dynamic trajectories of organ shapes.

Tangent Space Statistical Shape Atlas

The abovementioned diffeomorphic registration frameworks provide two essential components for the statistical analysis on the shape space: the distance metric between anatomic shapes and the linearized representation of shape difference on a vector space.

We take one of the most commonly used statistical tools, the PCA atlas of construction of anatomic structures, as an example to explain how statistical analysis can be carried out. Usually, it is called “tangent space PCA,” since the underlying vector space is essentially the tangent space at the mean shape of the PCA model [86, 104].

Given a collection of anatomic shapes *S = {I _{0}, I,..., I_{N}-},* the tangent space PCA atlas consists of a

*mean*model

*I*and a covariance matrix of the deviations of the shapes from

*I.*The PCA model can be constructed thus:

1. The mean model *I* is computed from S.

- 2. The linearized shape deviation
*x*of^{n}*I*represented by the initial vector field_{n},*u*or its dual^{n}n*u*for LDDMM or the stationary vector field^{b}n^{n}*v*of each shape^{n}forSVF,*I*is computed by registering_{n}, n = 0,1,..., N — 1*I*to*I*_{n}. - 3. The covariance matrix of
*{x*is computed.^{n}}

In CA, the most commonly used (weighted) mean shape *I* of *S* is the Frechet mean or its local version, the Karcher mean with weighting factors *{w„}* defined as

where *dist.I*, *I*_{n}/ is a distance metric between *I* and *I _{n}. I* in (2.171) can be found using an iterative fixed point algorithm [90, 105, 106]. Given the mean model

*I, {x*and the covariance matrix can be easily computed to construct the generative PCA model from

^{n}g*S*as illustrated in Fig. 2.18a. For more details about the mean model such as its geometrical meaning, invariancy with group actions in LDDMM and SVF frameworks, and log-domain statistics, the reader is referred to [84, 107]. A methodological discussion on statistical atlas construction can be found in [86].

Fig. 2.18 Statistical analysis of diffeomorphism-based computational anatomy (CA) (a) Tangent space PCA model with LDDMM: *I* is the Frechet/Karcher mean of the training data; each image *I _{i}* is represented in the vector space

*TjM*as

*u*), and the statistical analysis is carried out on that vector space. (b) Longitudinal data analysis in LDDMM: Three objects Si,

_{n}(I_{i}*S*are observed at discrete time points represented as •, o, ?. Evolutionary trajectories are estimated by geodesics for Si,

_{2}, S_{3}*S*and a piecewise geodesic for S

_{2}_{3}. To compare trajectories of S

_{i}and S

_{2}, the tangent space representations of their trajectories would undergo parallel transporting u

_{n}(Ij

^{!2}),

*u*to a common reference

_{n}(I^{!2})*I*as

_{n}*PT(u*along geodesics

_{n}(I^{!2})), PT(u_{n}(I^{!2}))*g*and then comparison in

^{I}°^{!Ii}, g^{In!I2}*T*How this parallel transport and comparison of the piecewise geodesic trajectory of

_{In}M.*S*is to be achieved remains open because the parallel transport is path-dependent [98] and there are different ways to transport

_{3}*u*

_{n}(L^{2}j^{!3}) to

*T*

_{In}M