For analyzing shapes of planar, closed curves, we propose
differential geometric representations of curves using their
direction functions and curvature functions. Shapes are
represented as elements of infinite-dimensional spaces and their
pairwise differences are quantified using the lengths of geodesics
connecting them on these spaces. We use a Fourier basis to
represent tangents to the shape spaces and then use a
gradient-based shooting method to solve for the tangent that
connects any two shapes via a geodesic. Using the Surrey fish
database, we demonstrate some applications of this approach: (i)
interpolation and extrapolations of shape changes, (ii) clustering
of objects according to their shapes, (iii) statistics on shape
spaces, and (iv) Bayesian extraction of shapes in low-quality
images.