  We propose a novel sparse dictionary learning method for planar shapes in the
sense of Kendall, namely configurations of landmarks in the plane considered up
to similitudes. Our shape dictionary method provides a good trade-off between
algorithmic simplicity and faithfulness with respect to the nonlinear geometric
structure of Kendall's shape space. Remarkably, it boils down to a classical
dictionary learning formulation modified using complex weights. Existing
dictionary learning methods extended to nonlinear spaces either map the
manifold to a reproducing kernel Hilbert space or to a tangent space. The first
approach is unnecessarily heavy in the case of Kendall's shape space and causes
the geometrical understanding of shapes to be lost, while the second one
induces distortions and theoretical complexity. Our approach does not suffer
from these drawbacks. Instead of embedding the shape space into a linear space,
we rely on the hyperplane of centered configurations, including pre-shapes from
which shapes are defined as rotation orbits. In this linear space, the
dictionary atoms are scaled and rotated using complex weights before summation.
Furthermore, our formulation is more general than Kendall's original one: it
applies to discretely-defined configurations of landmarks as well as
continuously-defined interpolating curves. We implemented our algorithm by
adapting the method of optimal directions combined to a Cholesky-optimized
order recursive matching pursuit. An interesting feature of our shape
dictionary is that it produces visually realistic atoms, while guaranteeing
reconstruction accuracy. Its efficiency can mostly be attributed to a clear
formulation of the framework with complex numbers. We illustrate the strong
potential of our approach for the characterization of datasets of shapes up to
similitudes and the analysis of patterns in deforming 2D shapes.
