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Nested Sphere Statistics of Skeletal Models

Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous three-space, and then discrete representations formed by a tuple of spoke vectors are derived. We describe means of fitting these skeletal models into manual or automatic segmentations of objects in a way stable enough to support statistical analysis, and we sketch means of modifying these fits to provide good correspondences of spoke vectors across a training population of objects. Understanding will be developed that these discrete skeletal models live in an abstract space made of a Cartesian product of a Euclidean space and a collection of spherical spaces. Based on this understanding and the way objects change under various rigid and nonrigid transformations, a method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods. The methods are illustrated by application to hippocampi.

Keywords

  • Object Boundary
  • Shape Operator
  • Shape Space
  • Distance Image
  • Training Population

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

We thank Jörn Schulz for studying the effects of global rotation, folding, and twisting on skeletal spoke vectors; Martin Styner for the hippocampus data; and Anna Snyder for help with the references.

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Correspondence to Stephen M. Pizer .

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Pizer, S.M. et al. (2013). Nested Sphere Statistics of Skeletal Models. In: Breuß, M., Bruckstein, A., Maragos, P. (eds) Innovations for Shape Analysis. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34141-0_5

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