Bulletin of Mathematical Biology

, Volume 81, Issue 7, pp 2052–2073 | Cite as

Elastic Statistical Shape Analysis of Biological Structures with Case Studies: A Tutorial

  • Min Ho Cho
  • Amir Asiaee
  • Sebastian KurtekEmail author
Education Article
Part of the following topical collections:
  1. Topological Data Analysis


We describe a recent framework for statistical shape analysis of curves and show its applicability to various biological datasets. The presented methods are based on a functional representation of shape called the square-root velocity function and a closely related elastic metric. The main benefit of this approach is its invariance to reparameterization (in addition to the standard shape-preserving transformations of translation, rotation and scale), and ability to compute optimal registrations (point correspondences) across objects. Building upon the defined distance between shapes, we additionally describe tools for computing sample statistics including the mean and covariance. Based on the covariance structure, one can also explore variability in shape samples via principal component analysis. Finally, the estimated mean and covariance can be used to define Wrapped Gaussian models on the shape space, which are easy to sample from. We present multiple case studies on various biological datasets including (1) leaf outlines, (2) internal carotid arteries, (3) Diffusion Tensor Magnetic Resonance Imaging fiber tracts, (4) Glioblastoma Multiforme tumors, and (5) vertebrae in mice. We additionally provide a MATLAB package that can be used to produce the results given in this manuscript.


Shape Elastic metric Square-root velocity function Karcher mean Principal component analysis Wrapped Gaussian model 



We thank the two reviewers for their feedback, which helped significantly improve this manuscript. We also thank Arvind Rao for sharing the GBM tumor data, and acknowledge Joonsang Lee, Juan Martinez, Shivali Narang and Ganesh Rao for their roles in processing the MRIs used to produce the tumor outlines. We thank Zhaohua Ding for providing the DT-MRI fiber dataset. Finally, we acknowledge the Mathematical Biosciences Institute (MBI) for organizing the 2012 Workshop on Statistics of Time Warpings and Phase Variations, during which the internal carotid artery dataset was discussed and analyzed. Sebastian Kurtek’s work was supported in part by grants NSF DMS-1613054, NSF CCF-1740761, NSF CCF-1839252, NSF CCF-1839356 and NIH R37-CA214955.


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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of StatisticsThe Ohio State UniversityColumbusUSA
  2. 2.Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA

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