# Trajectory Comparison

Depending on the trajectory estimation methods, trajectories can be compared with corresponding strategies:

- • Parallel transport When the trajectories are just simple geodesics as explained above, comparison of trajectories can be achieved by the well-known
*parallel transport*operation to translate their corresponding tangent space representatives to a common reference. But for piecewise geodesic trajectories, the problem remains open (see Fig. 2.18b). The reader is referred to [93, 108, 111] for more details on the parallel transport operation in LDDMM and SVF frameworks and their relationship. - • Trajectory registration For general interpolated trajectories, in [86] a trajectory registration strategy has been proposed to register spatiotemporally the trajectory of one object with the observed data of the other object, resulting in an atemporal spatial transformation ф and a time wrap ф, which can be used to represent the difference between trajectories.

# Spatiotemporal Atlas Construction

Similar to the statistical atlas of anatomic shapes, a spatiotemporal atlas of the evolutionary trajectories of anatomic shapes can also be constructed. In [86, 112] a subject-specific framework has been proposed to construct such an atlas. The basic assumptions are:

- • All the individuals in the population share a common mean evolutionary trajectory
*M(t) = Xt*(M_{0}) with*Xt*a time-dependent spatial transformation. - • The trajectory of object
*S*is a deformation of_{n}*M(t)*by a spatial morphological deformation*ф*and a time wrap '_{п}_{n}, given by*I*ф_{n}(t) =_{п}(М('_{п}(0)).

Then the spatiotemporal atlas can be constructed as an optimization procedure to find the optimal {M_{0}, *Xt* > {ФЛ, {'* _{n}*}} to fit the observed dataset. The reader is referred to [86, 112] for more algorithmic details and applications.

## Applications and Future Works

As a computational framework on shape manifolds, diffeomorphism-based CA has been widely used for general image registration [91, 107, 111], morphology-based disease diagnosis [108,113], SSA construction [90,105,114-116], and longitudinal data analysis [93, 102, 112, 117] even beyond the medical image processing field [118, 119].

Future work may be carried out on the following aspects:

- • Extending the diffeomorphic registration framework of CA to various image modalities and multimodality image registration
- • Extending the applications of SSA to achieve shape segmentation, registration, and classification

• Building longitudinal data analysis frameworks beyond the limitations of the framework of [112], which is essentially not a general and generative spatiotem- poral model that can cover the variabilities of the evolutionary shape trajectories