Persistent Homology Analysis of Brain Artery Trees.

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.

[1]  T. E. Harris.  First passage and recurrence distributions , 1952 .

[2]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[3]  Alfred Inselberg,et al.  Multidimensional detective , 1997, Proceedings of VIZ '97: Visualization Conference, Information Visualization Symposium and Parallel Rendering Symposium.

[4]  Susan Holmes,et al.  Phylogenies: An Overview , 1997 .

[5]  Louis J. Billera,et al.  Geometry of the Space of Phylogenetic Trees , 2001, Adv. Appl. Math..

[6]  Stephen R. Aylward,et al.  Initialization, noise, singularities, and scale in height ridge traversal for tubular object centerline extraction , 2002, IEEE Transactions on Medical Imaging.

[7]  T. Auton.  Applied Functional Data Analysis: Methods and Case Studies , 2004 .

[8]  Stephen R. Aylward,et al.  Analyzing attributes of vessel populations , 2005, Medical Image Anal..

[9]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[10]  J. Marron,et al.  Object oriented data analysis: Sets of trees , 2007, 0711.3147.

[11]  Gabor Pataki,et al.  A Principal Component Analysis for Trees , 2008, 0810.0944.

[12]  Leonidas J. Guibas,et al.  Proximity of persistence modules and their diagrams , 2009, SCG '09.

[13]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[14]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[15]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[16]  Jennifer Gamble,et al.  Exploring uses of persistent homology for statistical analysis of landmark-based shape data , 2010, J. Multivar. Anal..

[17]  Marleen de Bruijne,et al.  Geometries on Spaces of Treelike Shapes , 2010, ACCV.

[18]  David Cohen-Steiner,et al.  Lipschitz Functions Have Lp-Stable Persistence , 2010, Found. Comput. Math..

[19]  Tom M. W. Nye,et al.  Principal components analysis in the space of phylogenetic trees , 2011, 1202.5132.

[20]  S. Mukherjee,et al.  Probability measures on the space of persistence diagrams , 2011 .

[21]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[22]  J. Marron,et al.  A Nonparametric Regression Model With Tree-Structured Response , 2012 .

[23]  J. S. Marron,et al.  Direction-Projection-Permutation for High-Dimensional Hypothesis Tests , 2013, 1304.0796.

[24]  J. S. Marron,et al.  Sticky central limit theorems on open books , 2012, 1202.4267.

[25]  Aaron B. Adcock,et al.  The Ring of Algebraic Functions on Persistence Bar Codes , 2013, 1304.0530.

[26]  Ipek Oguz,et al.  Tree-Oriented Analysis of Brain Artery Structure , 2013, Journal of Mathematical Imaging and Vision.

[27]  Marleen de Bruijne,et al.  Toward a Theory of Statistical Tree-Shape Analysis , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Huiling Le,et al.  Central limit theorems for Fréchet means in the space of phylogenetic trees , 2013 .

[29]  Arthur W. Toga,et al.  Digital reconstruction and morphometric analysis of human brain arterial vasculature from magnetic resonance angiography , 2013, NeuroImage.

[30]  James Stephen Marron,et al.  Object-Oriented Data Analysis of Cell Images , 2014 .

[31]  Shankar Bhamidi,et al.  Functional Data Analysis of Tree Data Objects , 2014, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[32]  H. Le,et al.  Limiting behaviour of Fréchet means in the space of phylogenetic trees , 2014, 1409.7602.

[33]  J Steve Marron,et al.  Overview of object oriented data analysis , 2014, Biometrical journal. Biometrische Zeitschrift.

[34]  Alejandro F. Frangi,et al.  Topo-Geometric Filtration Scheme for Geometric Active Contours and Level Sets: Application to Cerebrovascular Segmentation , 2014, MICCAI.

[35]  G. Petri,et al.  Homological scaffolds of brain functional networks , 2014, Journal of The Royal Society Interface.

[36]  Ulrich Bauer,et al.  A stable multi-scale kernel for topological machine learning , 2014, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[37]  Peter Bubenik,et al.  Statistical topological data analysis using persistence landscapes , 2012, J. Mach. Learn. Res..

[38]  Steve Oudot,et al.  Eurographics Symposium on Geometry Processing 2015 Stable Topological Signatures for Points on 3d Shapes , 2022 .

[39]  Probabilistic Fréchet means for time varying persistence diagrams , 2013, 1307.6530.

[40]  Adam Watkins,et al.  Topological and statistical behavior classifiers for tracking applications , 2014, IEEE Transactions on Aerospace and Electronic Systems.