  In the shape analysis approach to computer vision problems, one treats shapes
as points in an infinite-dimensional Riemannian manifold, thereby facilitating
algorithms for statistical calculations such as geodesic distance between
shapes and averaging of a collection of shapes. The performance of these
algorithms depends heavily on the choice of the Riemannian metric. In the
setting of plane curve shapes, attention has largely been focused on a
two-parameter family of first order Sobolev metrics, referred to as elastic
metrics. They are particularly useful due to the existence of simplifying
coordinate transformations for particular parameter values, such as the
well-known square-root velocity transform. In this paper, we extend the
transformations appearing in the existing literature to a family of isometries,
which take any elastic metric to the flat $L^2$ metric. We also extend the
transforms to treat piecewise linear curves and demonstrate the existence of
optimal matchings over the diffeomorphism group in this setting. We conclude
the paper with multiple examples of shape geodesics for open and closed curves.
We also show the benefits of our approach in a simple classification
experiment.
