Related Topics in Shape Analysis of Curves

Abstract

Previously we have developed theory and computational solutions for shape analysis associated with curves in Euclidean spaces. We end this textbook by presenting some related topics that fall outside the main framework. These miscellaneous topics do not fit an organized theme but are bunched together in this chapter for convenience. They include: (1) Investigate the use of shape in conjunction with other features, such as scale, pose, and position, to characterize curves. (2) Extend the group of shape-invariant transformations, from the similarity group to the affine group, in the case of planar closed curves. While most shape analysis works consider similarity transformations (rigid motions and global scales) as the main shape-preserving transformations, some applications, including imaging, may require us to nullify affine distortions of curves. (3) Develop techniques for analyzing trajectories on nonlinear Riemannian manifolds. While we have studied only the Euclidean curves so far, there is also a strong need for analyzing curves on other, perhaps nonlinear, domains. One may not use the word shape for characterizing the desired properties of these curves, but this analysis should be invariant at least to parameterizations of these trajectories.