SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 141-175, January 2020.
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.

中文翻译：

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二维Kendall形状的字典学习

SIAM影像科学杂志，第13卷，第1期，第141-175页，2020年1月。
我们提出一种针对Kendall的平面形状稀疏字典学习方法，即考虑到相似度的平面中地标的配置。对于Kendall形状空间的非线性几何结构，我们的形状字典方法在算法简单性和忠实度之间提供了良好的折衷。值得注意的是，它归结为使用复杂权重进行修改的经典词典学习公式。现有的扩展到非线性空间的字典学习方法将流形映射到再现核Hilbert空间或切线空间。在Kendall形状空间的情况下，第一种方法不必要地繁重，并且会丢失对形状的几何理解，而第二种方法则会引起变形和理论上的复杂性。我们的方法没有这些缺点。我们没有将形状空间嵌入线性空间，而是依靠居中配置的超平面，其中包括将形状定义为旋转轨道的预形。在此线性空间中，字典原子在累加之前使用复数权重进行缩放和旋转。此外，我们的公式比Kendall最初的公式更笼统：适用于离散定义的地标配置以及连续定义的插值曲线。我们通过将最佳方向方法与Cholesky优化的顺序递归匹配追踪相结合来实现算法。我们的形状字典的一个有趣的功能是它可以产生视觉上逼真的原子，同时又可以保证重建的准确性。它的效率主要归因于框架的明确表述和复数。我们说明了我们的方法在表征形状到相似度的数据集以及对二维形状变形中的图案进行分析方面的强大潜力。