# Longitudinal Shape Data Analysis

Longitudinal shape data analysis in CA analyzes the spatiotemporal variability of anatomic shapes to reveal dynamic development patterns of organs across diseases, disease course, ages, genders, etc. Its key task is to provide a model of how one individual’s trajectory changes relative to those of other subjects.

Given a longitudinal data set for a collection of *N* objects *S = {S _{0}, Si,..., S_{N-1}} *observed at different time slots. The observed data set for

*S*includes

_{n}*T*observations at time slots {t0,

_{n}*t*

_{n},...,t_{n}

^{n}} as

*I*,

_{n}= {In, In*..,1%*}. Longitudinal data analysis should answer the following questions:

- • How is the trajectory of each object evolution estimated and how is its shape predicted at any time
*t*from limited observations? - • How are trajectories of different objects compared?
- • How can a statistical atlas of evolutionary trajectories be constructed?

# Trajectory Estimation

Trajectory estimation involves finding a continuous (smooth) curve in the spatiotemporal shape space that best fits the observed discrete time data of a single object. In diffeomorphism-based CA, existing solutions include:

Ut° ti ^{T}n

*ПП, If,..., If* } at *T _{n}*

time slots, the evolutionary trajectory can be computed as a piecewise geodesic (Riemannian geodesic or group geodesic for LDDMM and SVF, respectively) that connects successive observed data, which can be computed by diffeomorphic registration between successive observations by LDDMM and SVF. If there are only two observations at *t _{0}* and t

_{n}, the trajectory is just the geodesic connecting them. Also, such a trajectory holds a correspondent (piecewise) tangent space representation, which might be explored for the trajectory comparison and statistical analysis. The disadvantage is that such a greedy method may over-fit the observed data and lose the global smoothness of the trajectory [93,108,109]. • Trajectory regression or interpolation Another solution is to fit a smooth time- dependent curve simultaneously to all the observed data by kernel regression or the first- or second-order interpolations as in [86, 110]. Such methods result in smooth trajectories, but at a cost of losing the tangent space trajectory representation.