Dictionary Learning for Two-Dimensional Kendall Shapes

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 map the manifold either 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 preshapes 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 two-dimensional shapes.