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

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## Abstract

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.

## Keywords

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

### Acknowledgements

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.

## References

- Bauer M, Bruveris M, Charon N, Moller-Andersen J (2018) A relaxed approach for curve matching with elastic metrics. arXiv:1803.10893v2
- Bharath K, Kurtek S, Rao A, Baladandayuthapani V (2018) Radiologic image-based statistical shape analysis of brain tumours. J R Stat Soc Ser C 67(5):1357–1378MathSciNetCrossRefGoogle Scholar
- Bookstein FL (1984) A statistical method for biological shape comparisons. J Theor Biol 107(3):475–520CrossRefGoogle Scholar
- Bookstein FL (1992) Morphometric tools for landmark data: geometry and biology. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- Bookstein FL (1996) Biometrics, biomathematics and the morphometric synthesis. Bull Math Biol 58(2):313–365CrossRefzbMATHGoogle Scholar
- Boothby W (1975) An introduction to differentiable manifolds and Riemannian geometry. Pure and applied mathematics. Elsevier Science, AmsterdamzbMATHGoogle Scholar
- Bruveris M (2016) Optimal reparametrizations in the square root velocity framework. SIAM J Math Anal 48(6):4335–4354MathSciNetCrossRefzbMATHGoogle Scholar
- Cheng W, Dryden IL, Huang X (2016) Bayesian registration of functions and curves. Bayesian Anal 11(2):447–475MathSciNetCrossRefzbMATHGoogle Scholar
- Cope JS, Corney D, Clark JY, Remagnino P, Wilkin P (2012) Plant species identification using digital morphometrics: a review. Expert Syst Appl 39(8):7562–7573CrossRefGoogle Scholar
- Dryden IL, Mardia KV (1993) Multivariate shape analysis. Sankhya Ser A 55(3):460–480MathSciNetzbMATHGoogle Scholar
- Dryden IL, Mardia KV (1998) Statistical analysis of shape. Wiley, New YorkzbMATHGoogle Scholar
- Dryden IL, Mardia KV (2016) Statistical shape analysis: with applications in R, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
- Fletcher PT, Venkatasubramanian S, Joshi S (2009) The geometric median on Riemannian manifolds with application to robust atlas estimation. Neuroimage 45(1):S143–S152CrossRefGoogle Scholar
- Hasegawa H, Holm L (2009) Advances and pitfalls of protein structural alignment. Curr Opin Struct Biol 19(3):341–348CrossRefGoogle Scholar
- Jolliffe I (2002) Principal component analysis. Springer, BerlinzbMATHGoogle Scholar
- Joshi SH, Klassen E, Srivastava A, Jermyn IH (2007) A novel representation for Riemannian analysis of elastic curves in \(\mathbb{R}^n\). In: IEEE conference on computer vision and pattern recognition, pp 1–7Google Scholar
- Kendall DG (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull Lond Math Soc 16:81–121MathSciNetCrossRefzbMATHGoogle Scholar
- Klassen E, Srivastava A (2006) Geodesics between 3D closed curves using path-straightening. In: European conference on computer vision, pp 95–106Google Scholar
- Klassen E, Srivastava A, Mio W, Joshi SH (2004) Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans Pattern Anal Mach Intell 26(3):372–383CrossRefGoogle Scholar
- Kolodny R, Petrey D, Honig B (2006) Protein structure comparison: implications for the nature of ‘fold space’, and structure and function prediction. Curr Opin Struct Biol 16(3):393–398CrossRefGoogle Scholar
- Kurtek S, Needham T (2018) Simplifying transforms for general elastic metrics on the space of plane curves. arXiv:1803.10894v1
- Kurtek S, Xie Q (2015) Elastic prior shape models of 3D objects for Bayesian image analysis. In: Current trends in Bayesian methodology with applications, pp 347–366Google Scholar
- Kurtek S, Srivastava A, Klassen E, Ding Z (2012) Statistical modeling of curves using shapes and related features. J Am Stat Assoc 107(499):1152–1165MathSciNetCrossRefzbMATHGoogle Scholar
- Kurtek S, Su J, Grimm C, Vaughan M, Sowell R, Srivastava A (2013) Statistical analysis of manual segmentations of structures in medical images. Comput Vis Image Underst 117:1036–1050CrossRefGoogle Scholar
- Laga H, Kurtek S, Srivastava A, Miklavcic SJ (2014) Landmark-free statistical analysis of the shape of plant leaves. J Theor Biol 363:41–52MathSciNetCrossRefzbMATHGoogle Scholar
- Lahiri S, Robinson D, Klassen E (2015) Precise matching of PL curves in \(\mathbb{R}^n\) in the Square Root Velocity framework. Geom Imaging Comput 2:133–186MathSciNetCrossRefzbMATHGoogle Scholar
- Lang S (2001) Fundamentals of differential geometry. Graduate texts in mathematics. Springer, BerlinGoogle Scholar
- Le H (2001) Locating Frechet means with application to shape spaces. Adv Appl Prob 33(2):324–338MathSciNetCrossRefzbMATHGoogle Scholar
- Liu W, Srivastava A, Zhang J (2010) Protein structure alignment using elastic shape analysis. In: ACM international conference on bioinformatics and computational biology, pp 62–70Google Scholar
- Liu W, Srivastava A, Zhang J (2011) A mathematical framework for protein structure comparison. PLOS Comput Biol 7(2):1–10MathSciNetCrossRefGoogle Scholar
- Mardia KV, Dryden IL (1989) The statistical analysis of shape data. Biometrika 76(2):271–281MathSciNetCrossRefzbMATHGoogle Scholar
- Mio W, Srivastava A, Joshi SH (2007) On shape of plane elastic curves. Int J Comput Vis 73(3):307–324CrossRefGoogle Scholar
- Morgan VL, Mishra A, Newton AT, Gore JC, Ding Z (2009) Integrating functional and diffusion magnetic resonance imaging for analysis of structure-function relationship in the human language network. PLoS One 4(8):e6660CrossRefGoogle Scholar
- O’Higgins P, Dryden IL (1992) Studies of craniofacial development and evolution. Archaeol Phys Anthropol Ocean 27:105–112CrossRefGoogle Scholar
- Pennec X (2006) Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25(1):127–154MathSciNetCrossRefGoogle Scholar
- Robinson DT (2012) Functional data analysis and partial shape matching in the square root velocity framework. Ph.D. thesis, Florida State UniversityGoogle Scholar
- Sangalli LM, Secchi P, Vantini S (2014) Aneurisk65: a dataset of three-dimensional cerebral vascular geometries. Electron J Stat 8(2):1879–1890MathSciNetCrossRefzbMATHGoogle Scholar
- Small CG (1996) The statistical theory of shape. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Spivak M (1979) A comprehensive introduction to differential geometry: volumes 1–5. Publish or Perish Inc., HoustonzbMATHGoogle Scholar
- Srivastava A, Klassen EP (2016) Functional and shape data analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Srivastava A, Joshi SH, Mio W, Liu X (2005) Statistical shape anlaysis: clustering, learning and testing. IEEE Trans Pattern Anal Mach Intell 27(4):590–602CrossRefGoogle Scholar
- Srivastava A, Klassen E, Joshi SH, Jermyn IH (2011) Shape analysis of elastic curves in Euclidean spaces. IEEE Trans Pattern Anal Mach Intell 33:1415–1428CrossRefGoogle Scholar
- Strait J, Kurtek S, Bartha E, MacEachern SM (2017) Landmark-constrained elastic shape analysis of planar curves. J Am Stat Assoc 112(518):521–533MathSciNetCrossRefGoogle Scholar
- Wu SG, Bao FS, Xu EY, Wang YX, Chang YF, Xiang QL (2007) A leaf recognition algorithm for plant classification using probabilistic neural network. In: IEEE international symposium on signal processing and information technology, pp 11–16Google Scholar
- Xie Q, Kurtek S, Srivastava A (2014) Analysis of AneuRisk65 data: elastic shape registration of curves. Electron J Stat 8(2):1920–1929MathSciNetCrossRefzbMATHGoogle Scholar
- Younes L (1998) Computable elastic distance between shapes. SIAM J Appl Math 58(2):565–586MathSciNetCrossRefzbMATHGoogle Scholar
- Zahn CT, Roskies RZ (1972) Fourier descriptors for plane closed curves. IEEE Trans Comput 21(3):269–281MathSciNetCrossRefzbMATHGoogle Scholar