In many scientific applications, one needs to compare the shapes of curves from different sources in a quantitative manner. These curves might correspond to real objects in imaging applications, e.g. cell populations in microscopy, or they might be a representation of an important physical characteristic, such as the spectrum of material specimens. A researcher working with such applications would often need to state of how similar or dissimilar the shapes are, and might also want to use the shape dissimilarity numbers (or distances) to pursue large-scale statistical analyses of the shape samples.
A rigorous and quantitative tool to compare curves and to compute their dissimilarity is shape geodesics. For this, one chooses an appropriate shape representation for the curves and identifies the Riemannian manifolds where the shapes lie and then tries to compute the shape geodesics between the shapes. The computation of shape geodesics usually reduces to optimization on Riemannian manifolds, specifically manifolds of curves.
In this work, we propose an efficient algorithm to compute the geodesic
distances of shapes on a practical realization of such a shape manifold.
The shape representation that we consider is the square root velocity
representation of Srivastava et al (IEEE PAMI, 2011). For this choice,
the Riemannian manifold is the 
unit hypersphere of curves and the
computation of the geodesic distances can be reduced to minimizing the
distance between the derived curves. As we require this distance to
be invariant under various transformations of the curves, such as
rotations and reparameterizations, this setup induces an optimization
problem, in which the starting point of the curve, possible rotations and
reparameterizations are the variables. We derive the optimality
conditions of the corresponding energy formulation, and use the
optimality conditions to devise an effective optimization scheme, in
which one-dimensional global optimization and nonlinear least squares are
key ingredients. We test and demonstrate our method on several synthetic
and real examples, and achieve significant improvement in both speed and
accuracy of the geodesic distance computations.