## Electronic Journal of Statistics

### Analysis of AneuRisk65 data: Elastic shape registration of curves

#### Abstract

Shapes of carotid arteries are analyzed here as elastic curves under the framework introduced in [2]. Using a mathematical representation of parameterized curves, termed square-root velocity function (SRVF), in conjunction with an elastic Riemannian metric, the framework provides (1) parameterization-invariant shape metrics for comparing curves, (2) simultaneous registration of coordinate functions across curves, and (3) computation of statistical summaries and models of shapes of given curves. The method is applicable to curves in $\mathbb{R} ^{n}$ for $n\geq1$. Thus, we study the shapes and alignments of carotid arteries using their 3D coordinates and other geometric properties along the curves, such as radii and curvatures. The results show a significant improvement in curve alignments, leading to a compact phase-amplitude PCA representation and modeling of artery data.

#### Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 1920-1929.

Dates
First available in Project Euclid: 29 October 2014

https://projecteuclid.org/euclid.ejs/1414588180

Digital Object Identifier
doi:10.1214/14-EJS938D

Mathematical Reviews number (MathSciNet)
MR3273612

Zentralblatt MATH identifier
1305.62381

#### Citation

Xie, Qian; Kurtek, Sebastian; Srivastava, Anuj. Analysis of AneuRisk65 data: Elastic shape registration of curves. Electron. J. Statist. 8 (2014), no. 2, 1920--1929. doi:10.1214/14-EJS938D. https://projecteuclid.org/euclid.ejs/1414588180

#### References

• [1] Dryden, I. L. and Mardia, K. V. (1998)., Statistical Shape Analysis, John Wiley & Sons.
• [2] Srivastava, A., Klassen, E., Joshi, S. H. and Jermyn, I. H. (2011)., Shape analysis of elastic curves in Euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33(7) 1415–1428.
• [3] Tucker, J. D., Wu, W. and Srivastava, A. (2013)., Generative models for functional data using phase and amplitude separation, Computational Statistics & Data Analysis, 61 50–66.
• [4] Wu, W. and Srivastava, A. (2014)., Analysis of spike train data: Alignment and comparisons using extended Fisher-Rao metric, Electronic Journal of Statistics, 8 1776–1785, Special Section on Statistics of Time Warpings and Phase Variations.
• [5] Michor, W., Mumford, D., Shah, J. and Younes, L. (2008)., A metric on shape space with explicit geodesics, Matematica E Applicazioni, 19(1) 25–57.
• [6] Sangalli, L. M., Secchi, P., Vantini, S. and Veneziani, A. (2009)., A case study in exploratory functional data analysis: Geometrical features of the internal carotid artery, Journal of the American Statistical Association, 104(485) 37–48.
• [7] Sangalli, L. M., Secchi, P. and Vantini, S. (2014)., AneuRisk65: A dataset of three-dimensional cerebral vascular geometries, Electronic Journal of Statistics, 8 1879–1890, Special Section on Statistics of Time Warpings and Phase Variations.
• [8] Srivastava, A., Wu, W., Kurtek, S., Klassen E. and Marron, J. S. (2011)., Registration of functional data using Fisher-Rao metric, arXiv:1103.3817.