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(M0) with Xt a time-dependent spatial transformation.
  • • The trajectory of object Sn is a deformation of M(t) by a spatial morphological deformation фп and a time wrap 'n, given by In(t) = фп(М('п(0)).

Then the spatiotemporal atlas can be constructed as an optimization procedure to find the optimal {M0, 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

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