Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

Rabi Bhattacharya; Lizhen Lin

doi:10.1007/978-981-15-0298-9_1

Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

Rabi Bhattacharya, Lizhen Lin

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fréchet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fréchet minimizer and the asymptotic distribution of the sample Fréchet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds.

Original language	English (US)
Title of host publication	Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman
Editors	Vladas Sidoravicius
Publisher	Springer
Pages	1-43
Number of pages	43
ISBN (Print)	9789811502972
DOIs	https://doi.org/10.1007/978-981-15-0298-9_1
State	Published - 2019
Event	International Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China Duration: Mar 25 2016 → Mar 27 2016

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	299
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Probability Theory and Statistical Physics, 2016
Country/Territory	China
City	Shanghai
Period	3/25/16 → 3/27/16

Keywords

Fréchet means
Image analysis
Nonparametric Bayes
Nonparametric inference on manifolds
Statistics on manifolds
Stratified spaces

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-981-15-0298-9_1

Cite this

Bhattacharya, R., & Lin, L. (2019). Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments. In V. Sidoravicius (Ed.), Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman (pp. 1-43). (Springer Proceedings in Mathematics and Statistics; Vol. 299). Springer. https://doi.org/10.1007/978-981-15-0298-9_1

Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments. / Bhattacharya, Rabi; Lin, Lizhen.
Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman. ed. / Vladas Sidoravicius. Springer, 2019. p. 1-43 (Springer Proceedings in Mathematics and Statistics; Vol. 299).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bhattacharya, R & Lin, L 2019, Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments. in V Sidoravicius (ed.), Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman. Springer Proceedings in Mathematics and Statistics, vol. 299, Springer, pp. 1-43, International Conference on Probability Theory and Statistical Physics, 2016, Shanghai, China, 3/25/16. https://doi.org/10.1007/978-981-15-0298-9_1

Bhattacharya R, Lin L. Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments. In Sidoravicius V, editor, Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman. Springer. 2019. p. 1-43. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-981-15-0298-9_1

Bhattacharya, Rabi ; Lin, Lizhen. / Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data : Recent Developments. Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman. editor / Vladas Sidoravicius. Springer, 2019. pp. 1-43 (Springer Proceedings in Mathematics and Statistics).

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abstract = "This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fr{\'e}chet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fr{\'e}chet minimizer and the asymptotic distribution of the sample Fr{\'e}chet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds.",

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Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

Abstract

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Keywords

ASJC Scopus subject areas

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