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, i.e., 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 in order 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.

Distribution on Warp Maps for Alignment of Open and Closed Curves

“…Landmark-constrained alignment, under a geometric, non-stochastic framework, was only recently studied (Strait et al, 2017;Bauer et al, 2017). In those methods, it is not possible to capture or model uncertainty in the optimal alignment.…”

Nonlinear flag manifolds as coadjoint orbits

“…Some nonlinear flag manifolds have already appeared in the literature too. Landmark-constrained planar curves, for instance, have been used in a statistical elastic shape analysis framework in [ 26 ]. Landmark-constrained surfaces in the context of shape analysis are being discussed in [ 17 , Chapter 6].…”

Analysis of shape data: From landmarks to elastic curves

“…A detailed description of this dataset (section 1.4.1), along with a thorough statistical analysis under the landmark-based representation, are given in Dryden and Mardia (2016). More recently, this data was also analyzed under different representations in Cheng, Dryden, and Huang (2016), Strait, Kurtek, Bartha, and MacEachern (2017) and Cho, Asiaee, and Kurtek (2019). The entire dataset is available in R as part of the shapes package.…”