Abstract
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using death vectors and persistence landscapes to vectorize object data and perform statistical analysis. We apply this method to some common leaf images that were previously shown to be challenging to compare using a 3D shape techniques. Surprisingly, the most persistent features are shown to be “topological noise” and the statistical analysis depends on the less persistent features which we refer to as the “geometric signal”. We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces. We introduce a new Fréchet-Morse function technique for probability distribution on a compact object space, extending the Fréchet means lo a larger number of location parameters, including Fréchet antimeans. An example of 3D data analysis to distinguish two flowers using the new location parameters associated with a Veronese-Whitney (VW) embedding of random projective shapes of 3D configurations extracted from a set of pairs of their digital camera images is also given here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauer, U (2017). Ripser: a lean C++ code for the computation of Vietoris–Rips persistence barcodes. Software available at https://github.com/Ripser/ripser.
Bendich, P., Marron, J.S., Miller, E., Pieloch, A. and Skwerer, S. (2016). Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10, 198–218.
Bhattacharya, R.N., Ellingson, L., Liu, X., Patrangenaru, V. and Crane, M. (2012). Extrinsic analysis on manifolds is computa tionally faster than intrinsic analysis, with applications to quality control by machine vision. Applied Stochastic Models in Business and Industry. 28, 222–235.
Bhattacharya, R.N., Buibas, M., Dryden, I.L., Ellingson, L.A., Groisser, D., Hendriks, H., Huckemann, S., Le, H., Liu, X., Marron, J.S., Osborne, D.E., Patrângenaru, V., Schwartzman, A., Thompson, H.W. and Andrew, T.A.W. (2013). Extrinsic data analysis on sample spaces with a manifold stratification. Advances in Mathematics, Invited Contributions at the Seventh Congress of Romanian Mathematicians, Brasov, 2011, Publishing House of the Romanian Academy (Editors: Lucian Beznea, Vasile brîzanescu, Marius Iosifescu, Gabriela Marinoschi, Radu Purice and Dan Timotin), pp. 241–252.
Billera, L., Holmes, S. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27, 733–767.
Bobrowski, O. and Adler, R.J. (2014). Distance functions, critical points, and the topology of random Čech complexes. Homology, Homotopy and Applications 16, 2, 311–344.
Bubenik, P., Carlsson, G., Kim, P.T. and Luo, Z.-M. (2010). Statistical topology via Morse theory persistence and nonparametric estimation. Algebraic methods in statistics and probability II, 75—92, Contemp. Math., 516, Amer. Math. Soc.
Bubenik, P. (2015). Statistical Topological Data Analysis using Persistence Landscapes. J. of Machine Learning Research. 16, 77–102.
Bubenik, P. and Dlotko, P. (2017). A persistence landscapes toolbox for topological statistics. J. Symb. Comput. 78, 91–114.
Bubenik, P., Patel, D. and Whittle, B. (In preparation). Using Topological Data Analysis to estimate local geometry.
Chazal, F., Fasy, B.T., Lecci, F., Rinaldo, A., Singh, A. and Wasserman, L. (2014). On the bootstrap for persistence diagrams and landscapes. Modeling and Analysis of Information Systems 20, 96–105.
Chazal, F., Fasy, B.T., Lecci, F., Rinaldo, A. and Wasserman, L. (2015). Stochastic convergence of persistence landscapes and silhouettes. J. Comput. Geom. 6, 140–161.
Chen, Y.-C., Genovese, C.R. and Wasserman, L. (2017). Statistical inference using the Morse-Smale complex. Electron. J. Stat. 11, 1390–1433.
Cohen-Steiner, D., Edelsbrunner, H. and Harer, J. (2007). Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120.
Crane, M. and Patrangenaru, V. (2011). Random change on a lie group and mean glaucomatous projective shape change detection from stereo pair image. J. Multivar. Anal. 102, 225–237.
Deng, Y., Patrangenaru, V. and Balan, V. (2017). Anti-regression on Manifolds with an Applications to 3D Projective Shape Analysis. BSG Proceedings, 25, (In press).
Diaconis, P., Holmes, S. and Shahshahani, M. (2013). Sampling from a manifold. Advances in modern statistical theory and applications: a Festschrift in honor of Morris L. Eaton, 102–125, Inst. Math. Stat. (IMS) Collect., 10, Inst. Math Statist., Beachwood, OH.
Donaldson, S.K. and Kronheimer, P.B. (1990). The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications The Clarendon Press. Oxford University Press, New York.
Dryden, I.L. and Mardia, K.V. (2016). Statistical shape analysis with applications in r, 2nd edn. Wiley, Chichester.
Ellingson, L., Ruymgaart, F.H. and Patrangenaru, V. (2013). Nonparametric estimation of means on hilbert manifolds and extrinsic analysis of mean shapes of contours. Journal of Multivariate Analysis 122, 317–333.
Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S. and Singh, A. (2014). Statistical Inference For Persistent Homology: Confidence Sets For Persistence Diagrams. Annals of Statistics. Ann. Statist. 42, 6, 2301–2339.
Fasy, B.T., Jisu Kim, J., Lecci, F. and Maria, C. (2015). Introduction to the R package TDA.
Fisher, N.I. (1983). Statistical analysis of circular data. Cambridge University Press, Cambridge.
Fréchet, M. (1948). Les élements aléatoires de nature quelconque dans un espace distancié (In French). Ann. Inst. H. Poincaré, 10, 215–310.
Freedman, M.H. and Quinn, F. (1990). Topology of 4-manifolds. Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ.
Guo, R. and Patrangenaru, V. (2017). Testing for the Equality of two Distributions on High Dimensional Object Spaces. arXiv:1703.07856.
Harman, R. and Lacko, V. (2010). On decompositional algorithms for uniform sampling from n-spheres and n-balls. J. Multivariate Anal. 101, 2297–2304.
Hartley, R. and Zisserman, A. (2004). Multiple View Geometry in Computer Vision, Second Edition. Cambridge University Press.
Kovacev-Nikolic, V, Bubenik, P., Nikolic, D. and Heo, G. (2016). Using persistent homology and dynamical distances to analyze protein binding. Stat. Appl. Genet. Mol. Biol. 15, 1, 19–38.
Ma, Y., Soatto, S., Kosecka, J. and Sastry, S.S. (2006). An invitation to 3-D vision. Springer, New York.
Mardia, K.V. and Jupp, P.E. (2000). Directional statistics. Wiley, Chichester.
Mardia, K.V. and Patrangenaru, V. (2005). Directions and projective shapes. Annals of Statistics 33, 1666–1699.
Patrangenaru, V. (1996). Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by Cartan triples. Pacific J. Math. 173, 2, 511–532.
Patrangenaru, V., Liu, X. and Sugathadasa, S. (2010). Nonparametric 3D projective shape estimation from pairs of 2D images - i, in memory of W.P. Dayawansa. Journal of Multivariate Analysis 101, 11–31.
Patrangenaru, V. and Ellingson, L.E. (2015). Nonparametric Statistics on Manifolds and their Applications to Object Data Analysis. CRC.
Patrangenaru, V., Guo, R. and Yao, K.D. (2016a). Nonparametric Inference for Location Parameters via Fréchet Functions. 2nd International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management (SMRLO), Beer Sheva, Israel. (Edited by Frenkel, I and Lisnianski, A), 254–262.
Patrangenaru, V., Yao, K.D. and Guo R. (2016b). Extrinsic Means and Antimeans. In: Cao, R., Gonzlez Manteiga, W., Romo, J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol. 175. 161–178.
Patrangenaru, V., Paige, R., Yao, K.D., Qiu, M. and Lester, D. (2016c). Projective shape analysis of contours and finite 3D configurations from digital camera image. Statistical Papers 57, 1017–1040.
Perelman, G. (2003). Ricci Flow with Surgery on Three-Manifolds, arXiv:math.DG/0303109.
Scott, P. (1983). The geometries of 3-manifolds. Bull. London Math. Soc. 15, 5, 401–487.
Spanier, E.H. (1981). Algebraic topology. Corrected reprint. Springer, New York-Berlin.
Spivak, M. (1979). A Comprehensive Introduction to Differential Geometry. Vol I-II. Second Edition. Publish or Perish Inc. Wilmington, Del.
Thurston, W.P. (1982). Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Vol. 6.
Acknowledgments
VP would like to thank Karthik Bharath for the invitation for a contribution to the special volume on Statistics on Manifolds for Sankya, and for suggestions that helped us improve the original manuscript. PB would like to acknowledge that this project was supported by the UFII SEED Funds.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Patrangenaru, V., Bubenik, P., Paige, R.L. et al. Challenges in Topological Object Data Analysis. Sankhya A 81, 244–271 (2019). https://doi.org/10.1007/s13171-018-0137-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-018-0137-7
Keywords
- VW-means of index r
- Topological data analysis
- Relative homology
- Object data analysis
- Persistence landscapes