Since shapes outlines are often represented with two-dimensional curves, many shape-space models arise from the design of spaces of plane curves. Such spaces are generally associated to smoothness (differentiability) requirements for the curves, and come with a metric space structure since comparing shapes has important practical applications. Most of the spaces of plane curves that have been proposed in the literature actually possess richer structures, many of them providing infinite-dimensional Riemannian manifolds. In such spaces, the distance is associated to paths of least energy in shape space, yielding optimal shape evolutions, or geodesics. It is important to remark that shape spaces are not exactly spaces of curves, but are built upon them via identification of equivalent classes. For example, curves that are deduced for each other by translation, rotation and scaling correspond to the same shape and can be considered as equal. This results in spaces of shapes represented as quotients spaces of curves via the action of some linear transformations. Finally, one should define shapes regardless of their parametrization, which induces a new quotient, this time by the group of diffeomorphism of the unit interval, or of the unit circle for closed curves.
The following articles all discuss a special shape space on which
geodesics are relatively easy to compute, because the associated
space of curves turns out to be isometric to a Hilbert space. In
references 1 - 4, it is noticed that quotienting for scale
results in identifying the shape space to Hilbert sphere. This
argument is pushed further in reference 5, in which it is noticed that
quotienting for rotation with closed curves results in an
identification of the shape space with a Grassmann manifold.
The images below illustrate some geodesic paths in curve space
between selected shapes.
next two papers discuss a class of curve matching
problems that include the one introduced in reference 1-4
above, and study their properties, regarding, in particular,
the existence of minimizers.