
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
|
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 2015 = $35,000.00 FY 2016 = $30,000.00 |
History of Investigator: |
|
Recipient Sponsored Research Office: |
845 N PARK AVE RM 538 TUCSON AZ US 85721 (520)626-6000 |
Sponsor Congressional District: |
|
Primary Place of Performance: |
617 N. Santa Rita Ave Tucson AZ US 85721-0089 |
Primary Place of
Performance Congressional District: |
|
Unique Entity Identifier (UEI): |
|
Parent UEI: |
|
NSF Program(s): | STATISTICS |
Primary Program Source: |
01001516DB NSF RESEARCH & RELATED ACTIVIT 01001617DB NSF RESEARCH & RELATED ACTIVIT |
Program Reference Code(s): | |
Program Element Code(s): |
|
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
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
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
Please report errors in award information by writing to: awardsearch@nsf.gov.