Landmark-Constrained Elastic Shape Analysis of Planar Curves

ABSTRACT

Various approaches to statistical shape analysis exist in current literature. They mainly differ in the representations, metrics, and/or methods for alignment of shapes. One such approach is based on landmarks, that is, mathematically or structurally meaningful points, which ignores the remaining outline information. Elastic shape analysis, a more recent approach, attempts to fix this by using a special functional representation of the parametrically defined outline to perform shape registration, and subsequent statistical analyses. However, the lack of landmark identification can lead to unnatural alignment, particularly in biological and medical applications, where certain features are crucial to shape structure, comparison, and modeling. The main contribution of this work is the definition of a joint landmark-constrained elastic statistical shape analysis framework. We treat landmark points as constraints in the full shape analysis process. Thus, we inherit benefits of both methods: the landmarks help disambiguate shape alignment when the fully automatic elastic shape analysis framework produces unsatisfactory solutions. We provide standard statistical tools on the landmark-constrained shape space including mean and covariance calculation, classification, clustering, and tangent principal component analysis (PCA). We demonstrate the benefits of the proposed framework on complex shapes from the MPEG-7 dataset and two real data examples: mice T2 vertebrae and Hawaiian Drosophila fly wings. Supplementary materials for this article are available online.

KEYWORDS:

• Elastic metric
• Geodesics
• Karcher mean
• Statistics on shape spaces
• Tangent principal component analysis

Acknowledgments

The authors thank the associate editor and reviewers for providing valuable feedback.

Funding

This research has been supported by the NSF under grant number DMS-1209194. The views in this article are not necessarily those of the NSF.