The Annals of Applied Statistics

Persistent homology analysis of brain artery trees

Paul Bendich, J. S. Marron, Ezra Miller, Alex Pieloch, and Sean Skwerer

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Abstract

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.

Article information

Source
Ann. Appl. Stat. Volume 10, Number 1 (2016), 198-218.

Dates
Received: December 2014
Revised: September 2015
First available in Project Euclid: 25 March 2016

Permanent link to this document
http://projecteuclid.org/euclid.aoas/1458909913

Digital Object Identifier
doi:10.1214/15-AOAS886

Mathematical Reviews number (MathSciNet)
MR3480493

Zentralblatt MATH identifier
06586142

Keywords
Persistent homology statistics angiography tree-structured data topological data analysis

Citation

Bendich, Paul; Marron, J. S.; Miller, Ezra; Pieloch, Alex; Skwerer, Sean. Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10 (2016), no. 1, 198--218. doi:10.1214/15-AOAS886. http://projecteuclid.org/euclid.aoas/1458909913.


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References

  • Adcock, A., Carlsson, E. and Carlsson, G. (2013). The ring of algebraic functions on persistence bar codes. 2013. Available at arXiv:1304.0530.
  • Aydin, B., Pataki, G., Wang, H., Bullitt, E. and Marron, J. S. (2009). A principal component analysis for trees. Ann. Appl. Stat. 3 1597–1615.
  • Aylward, S. and Bullitt, E. (2002). Initialization, noise, singularities, and scale in height ridge traversal for tubular object centerline extraction. IEEE Trans. Med. Imag. 21 61–75.
  • Barden, D., Le, H. and Owen, M. (2013). Central limit theorems for Fréchet means in the space of phylogenetic trees. Electron. J. Probab. 18 no. 25, 25.
  • Barden, D., Le, H. and Owen, M. (2014). Limiting behaviour of Fréchet means in the space of phylogenetic trees. Preprint. Available at arXiv:1409.7602v1.
  • Bendich, P., Chin, S., Clarke, J., deSena, J., Harer, J., Munch, E., Newman, A., Porter, D., Rouse, D., Strawn, N. and Watkins, A. (2014). Topological and statistical behavior classifiers for tracking applications. Preprint. Available at arXiv:1406.0214.
  • Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 733–767.
  • Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16 77–102.
  • Bullitt, E., Zeng, D., Mortamet, B., Ghosh, A., Aylward, S. R. Lin, W., Marks, B. L. and Smith, K. (2010). The effects of healthy aging on intracerebral blood vessels visualized by magnetic resonance angiography. Neurobiol. Aging 31 290–300.
  • Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46 255–308.
  • Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L. J. and Oudot, S. (2009). Proximity of persistence modules and their diagrams. In Proc. of the 25th Ann. ACM Symp. on Comput. Geom. 237–246. ACM, New York.
  • Cohen-Steiner, D., Edelsbrunner, H. and Harer, J. (2007). Stability of persistence diagrams. Discrete Comput. Geom. 37 103–120.
  • Cohen-Steiner, D., Edelsbrunner, H., Harer, J. and Mileyko, Y. (2010). Lipschitz functions have $L_{p}$-stable persistence. Found. Comput. Math. 10 127–139.
  • Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction. Amer. Math. Soc., Providence, RI.
  • Feragen, A., Lauze, F., Lo, P., deBruijne, M. and Nielson, M. (2011). Geometries on spaces of treelike shapes. In Computer Vision—ACCV 2010. Lecture Notes in Computer Science 6493 160–173. Springer, Berlin.
  • Feragen, A., Lo, P., deBruijne, M., Nielson, M. and Lauze, F. (2013). Toward a theory of statistical tree-shape analysis. IEEE Trans. Pattern Anal. Mach. Intell. 35 2008–2021.
  • Gamble, J. and Heo, G. (2010). Exploring uses of persistent homology for statistical analysis of landmark-based shape data. J. Multivariate Anal. 101 2184–2199.
  • Harris, T. E. (1952). First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 471–486.
  • Holmes, S. P. (1999). Phylogenies: An overview. IMA Vol. Math. Appl. 112 81–118.
  • Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V. and Skwerer, S. (2013). Sticky central limit theorems on open books. Ann. Appl. Probab. 23 2238–2258.
  • Inselberg, A. (1997). Multidimensional detective. In Proc. of the IEEE Symp. on Information Visualization 100–107. IEEE Xplore Online Library.
  • Joliffe, I. (2005). Principal Component Analysis. Wiley Online Library.
  • Mileyko, Y., Mukherjee, S. and Harer, J. (2011). Probability measures on the space of persistence diagrams. Inverse Probl. 27 124007, 22.
  • Molina-Abril, H. and Frangi, A. F. (2014). Topo-geometric filtration scheme for geometric active contours and level sets: Application to cerebrovascular segmentation. In Medical Image Computing and Computer-Assisted Intervention. Lec. Notes in Computer Science 8673 755–762.
  • Munch, E., Turner, K., Bendich, P., Mukherjee, S., Mattingly, J. and Harer, J. (2015). Probabilistic Fréchet means for time varying persistence diagrams. Electron. J. Stat. 9 1173–1204.
  • Nye, T. M. W. (2011). Principal components analysis in the space of phylogenetic trees. Ann. Statist. 39 2716–2739.
  • Pieloch, A., Marron, J. S., Skwerer, S. and Bendich, P. (2016). Supplement to “Persistent homology analysis of brain artery trees.” DOI:10.1214/15-AOAS886SUPP.
  • Ramsay, J. O. (2006). Functional Data Analysis. Wiley Online Library.
  • Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.
  • Reininghaus, J., Huber, S., Bauer, U. and Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition 4741–4748.
  • Shen, D., Shen, H., Bhamidi, S., Muñoz Maldonado, Y., Kim, Y. and Marron, J. S. (2014). Functional data analysis of tree data objects. J. Comput. Graph. Statist. 23 418–438.
  • Skwerer, S., Bullitt, E., Huckemann, S., Miller, E., Oguz, I., Owen, M., Patrangenaru, V., Provan, S. and Marron, J. S. (2014). Tree-oriented analysis of brain artery structure. J. Math. Imaging Vision 50 126–143.
  • Wang, H. and Marron, J. S. (2007). Object oriented data analysis: Sets of trees. Ann. Statist. 35 1849–1873.
  • Wang, Y., Marron, J. S., Aydin, B., Ladha, A., Bullitt, E. and Wang, H. (2012). A nonparametric regression model with tree-structured response. J. Amer. Statist. Assoc. 107 1272–1285.
  • Wei, S., Lee, C., Wichers, L., Li, G. and Marron, J. S. (2015). Direction-projection-permutation for high dimensional hypothesis tests. J. Comput. Graph. Statist. To appear.
  • Wright, S. N., Kochunov, P., Mut, F., Bergamino, M., Brown, K. M., Mazziotta, J. C., Toga, A. W., Cebral, J. R. and Ascoli, G. A. (2013). Digital reconstruction and morphometric analysis of human brain arterial vasculature from magnetic resonance angiography. Neuroimage 15 170–181.

Supplemental materials

  • Supplement to “Persistent homology analysis of brain artery trees”. This archive contains brain tree data, persistence diagrams and statistical analysis pipeline for the paper.