Award Abstract # 1406872
Nonparametric Statistics and Riemannian Geometry in Image Analysis: New Perspectives with Applications in Biology, Medicine, Neuroscience and Machine Vision

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ARIZONA
Initial Amendment Date: July 19, 2014
Latest Amendment Date: May 5, 2016
Award Number: 1406872
Award Instrument: Continuing Grant
Program Manager: Gabor Szekely
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: August 1, 2014
End Date: July 31, 2017 (Estimated)
Total Intended Award Amount: $120,000.00
Total Awarded Amount to Date: $120,000.00
Funds Obligated to Date: FY 2014 = $55,000.00
FY 2015 = $35,000.00

FY 2016 = $30,000.00
History of Investigator:
  • Rabindra Bhattacharya (Principal Investigator)
    rabi@math.arizona.edu
Recipient Sponsored Research Office: University of Arizona
845 N PARK AVE RM 538
TUCSON
AZ  US  85721
(520)626-6000
Sponsor Congressional District: 07
Primary Place of Performance: Department of Mathematics/University of Arizona
617 N. Santa Rita Ave
Tucson
AZ  US  85721-0089
Primary Place of Performance
Congressional District:
07
Unique Entity Identifier (UEI): ED44Y3W6P7B9
Parent UEI:
NSF Program(s): STATISTICS
Primary Program Source: 01001415DB NSF RESEARCH & RELATED ACTIVIT
01001516DB NSF RESEARCH & RELATED ACTIVIT

01001617DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s): 126900
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

This project aims at (1) precise geometric depictions of digital images arising in biology, medicine, machine vision and other fields of science and engineering and (2) providing their model-independent statistical analysis for purposes of identification, discrimination and diagnostics. One specific application is to discriminate between a normal organ and a diseased one in the human body. Among examples, one may refer to the diagnosis of glaucoma and certain types of schizophrenia based on shape changes. A subject that the project will especially look at and analyze in depth, concerns changes in the geometric structure of the white matter in the brain's cortex brought about by Parkinson's disease, Alzheimers, schizophrenia, autism, etc., and their progression. Important applications in the fields of graphics, robotics, etc., will be explored as well.

Advancements in imaging technology enable scientists and medical professionals today to view the inner functioning of organs at the cell level and beyond. For example, in the white matter in the cortex, the coefficients of the 3x3 diffusion matrix of water molecules can be measured. In the absence of a disease or trauma, these matrices show pronounced anisotropy along well organized neural structures, while perturbations due to a disease lead to a decrease in anisotropy in each such location. This is one aspect of the structural change due to a disease that is visible in the diffusion tensor imaging scans. There are others. So far there is no statistical methodology that can precisely associate such a decrease in anisotropy with the particular disease that causes it. The present project will represent the main neural structures in the white matter in terms of elements of a Riemannian manifold and their geodesics. As one specific task, the project will choose appropriate metric tensors on the space of alignments of positive definite matrices along neural structures. The broad goal is to provide a nonparametric statistical methodology based on Fre'chet means for discrimination and diagnostics, extending much further and in novel directions the research that was carried out under earlier NSF supports. In a completely different direction, one theoretical objective of the project is to provide broad conditions for uniqueness of the Fre'chet mean under a geodesic distance. Such conditions are required for statistical applications but are unavailable in adequate generality for Riemannian manifolds with positive curvature. This matter of uniqueness also has surprising implications, for graphics and robotics.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Rabi Bhattacharya and Lizhen Lin "Omnibus CLTs for Fre'chet means and nonparametric inference on non-Euclidean spaces" Proceedings of the American Mathematical Society , 2017
Rabi Bhattacharya and Mukul Majumdar "Ruin probabilities in models of resource management and insurance: A synthesis" International Journal of Economic Theory , v.11 , 2015
R.Bhattacharya and M.Majumdar "Ruin probabilities in models of resource management and insurance: A synthesis" International Journal of Economic Theory , v.11 , 2015 , p.59
R.Bhattacharya and M.Majumdar "Ruin probabilities in models of resource management and insurance: A synthesis" International Journal of Economic Theory , v.11 , 2015 , p.59
R.Bhattacharya, H. Kim and M. Majumdar "Sustainability in stochastic Ramsey model" Journal of Quantitative Economics , v.13 , 2015 , p.169
R.Bhattacharya, H.Kim and M. Majumdar "Sustainability in stochastic Ramsey model" Journal of Quantitative Economics , v.13 , 2015 , p.169
R. Bhattacharya, L.Lin and W. Piegorsch "Nonparametric benchmark dose estimation with continuous dose-response data" Scandinavian Journal of Statistics , v.42 , 2015 , p.713
R.Bhattacharya, L.Lin and W. Piegorsch "Nonparametric benchmark dose estimation with continuous dose-response data" Scandinavian Journal of Statistics , v.42 , 2015 , p.713

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

The main objective of the project was to advance the model independent theory of statistical inference on non-Euclidean spaces developed by the PI and his students during the past fifteen years, published in top journals in the field. Such spaces arise, for example,  in scanning images (MRI, DTI,etc.) for purposes of medical diagnostics, and in machine vision. The model independence of the theory ensures that the results are robust and do not suffer from model misspecification which often plague so called parametric inference. The successes of this theory have been broadly demonstrated in data analysis presented in many articles and a 2012 research monograph. The present project was aimed at broadening the scope of the theory to more general spaces which arise in practice but are not covered by the earlier theory mentioned above. In several recent publications the PI and a former Ph.D student of his have (1) removed some of the restrictive assumptions in the earlier work which required, e.g., that the data distribution be concentrated (i.e., have a rather small support), and (2) extended the theory to spaces which are not smooth geometric objects such as manifolds, but are manifolds of different dimensions glued together. Examples of these latter spaces include certain classes of 3d images and also models of phylogenetic trees, etc.  

New applications have been made in this project especially in the analysis of neuroimages, based on DTI (diffusion tensor imaging) data, e.g.,  on HIV patients.

The PI has been guiding two female Ph.D. students on this project,. At several US and international conferences and workshops the thoeory developed has been disseminated broadly.by the PI.


Last Modified: 09/24/2017
Modified by: Rabindra Bhattacharya

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