A Statistical Model for Smooth Shapes in Kendall Shape Space
This paper proposes a novel framework for learning a statistical shape model from image data, automatically without manual annotations. The framework proposes a generative model for image data of individuals within a group, relying on a model of group sha
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Abstract. This paper proposes a novel framework for learning a statistical shape model from image data, automatically without manual annotations. The framework proposes a generative model for image data of individuals within a group, relying on a model of group shape variability. The framework represents shape as an equivalence class of pointsets and models group shape variability in Kendall shape space. The proposed model captures a novel shape-covariance structure that incorporates shape smoothness, relying on Markov regularization. Moreover, the framework employs a novel model for data likelihood, which lends itself to an inference algorithm of low complexity. The framework infers the model via a novel expectation maximization algorithm that samples smooth shapes in the Riemannian space. Furthermore, the inference algorithm normalizes the data (via similarity transforms) by optimal alignment to (sampled) individual shapes. Results on simulated and clinical data show that the proposed framework learns better-fitting compact statistical models as compared to the state of the art. Keywords: Statistical shape model, Kendall shape space, geodesic distance, Markov model, shape sampling, shape alignment, image data normalization.
1 Introduction and Related Work The typical notion of object shape [4,13] is an equivalence class of object boundaries / silhouettes, where the equivalence relation is given by a similarity transformation. These geometric invariance properties make shape space non-Euclidean. Following Kendall [8], representing shapes via pointsets with known correspondences, we model (i) preshape space as a subset of Euclidean space, i.e., an intersection of the unit hypersphere (for scale invariance) with a hyperplane through the origin (for translation invariance) and (ii) shape space with an additional rotational-invariance structure. While typical pointset-based shape models ignore this structure, the proposed method adapts shape modeling and inference to this Riemannian structure. Unlike typical pointset-based shape models, the proposed model incorporates prior information that real-world objects have smooth boundaries. Such information is useful during (i) model learning: as regularization to counter the noise in image data (even errors in manual landmark placement) to produce more compact models; and (ii) model
Thanks to the Royal Academy of Engineering 1314RECI076 and IIT Bombay 14IRCCSG010.
c Springer International Publishing Switzerland 2015 N. Navab et al. (Eds.): MICCAI 2015, Part III, LNCS 9351, pp. 628–635, 2015. DOI: 10.1007/978-3-319-24574-4_75
A Statistical Model for Smooth Shapes in Kendall Shape Space
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application (e.g., segmentation with shape priors): to more effectively regularize the learned covariance, having several near-zero eigenvalues, to enforce smoother shapes. Many early methods for statistical shape analysis rely on manually defined landmarks on image data [3] or templates [11]. Later methods optimize point positions relying on model covariance or information-theor
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