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.

A population quantity of interest in statistical shape analysis is the location of landmarks, which are points that aid in reconstructing and representing shapes of objects. We provide an automated, model-based approach to inferring landmarks given a sample of shape data. The model is formulated based on a linear reconstruction of the shape, passing through the specified points, and a Bayesian inferential approach is described for estimating unknown landmark locations. The question of how many landmarks to select is addressed in two different ways: (1) by defining a criterionbased approach, and (2) joint estimation of the number of landmarks along with their locations. Efficient methods for posterior sampling are also discussed. We motivate our approach using several simulated examples, as well as data obtained from applications in computer vision and biology; additionally, we explore placements and associated uncertainty in landmarks for various substructures extracted from magnetic resonance image slices.

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