A main goal in the field of statistical shape analysis is to define
computable and informative metrics on spaces of immersed manifolds, such as the
space of curves in a Euclidean space. The approach taken in the elastic shape
analysis framework is to define such a metric by starting with a
reparameterization-invariant Riemannian metric on the space of parameterized
shapes and inducing a metric on the quotient by the group of diffeomorphisms.
This quotient metric is computed, in practice, by finding a registration of two
shapes over the diffeomorphism group. For spaces of Euclidean curves, the
initial Riemannian metric is frequently chosen from a two-parameter family of
Sobolev metrics, called elastic metrics. Elastic metrics are especially
convenient because, for several parameter choices, they are known to be locally
isometric to Riemannian metrics for which one is able to solve the geodesic
boundary problem explictly -- well-known examples of these local isometries
include the complex square root transform of Younes, Michor, Mumford and Shah
and square root velocity (SRV) transform of Srivastava, Klassen, Joshi and
Jermyn. In this paper, we show that the SRV transform extends to elastic
metrics for all choices of parameters, for curves in any dimension, thereby
fully generalizing the work of many authors over the past two decades. We give
a unified treatment of the elastic metrics: we extend results of Trouv\'{e} and
Younes, Bruveris as well as Lahiri, Robinson and Klassen on the existence of
solutions to the registration problem, we develop algorithms for computing
distances and geodesics, and we apply these algorithms to metric learning
problems, where we learn optimal elastic metric parameters for statistical
shape analysis tasks