Award Abstract #0101339
FRG: Collaborative Research-Computational Conformal Mapping and Scientific Visualization

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	September 12, 2001
Latest Amendment Date:	September 12, 2001
Award Number:	0101339
Award Instrument:	Standard Grant
Program Manager:	Junping Wang DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	September 15, 2001
Expires:	August 31, 2004 (Estimated)
Awarded Amount to Date:	$170000
Investigator(s):	David Rottenberg dar@pet.med.va.gov (Principal Investigator)
Sponsor:	University of Minnesota-Twin Cities 200 OAK ST SE MINNEAPOLIS, MN 55455 612/624-5599
NSF Program(s):	COMPUTATIONAL MATHEMATICS, APPLIED MATHEMATICS, COMPUTATIONAL NEUROSCIENCE
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR, 9263, 1616, 0000
Program Element Code(s):	1271, 1266, 1162

ABSTRACT

This Focused Research Group is composed of pure

mathematicians, computational mathematicians, and

neuroscientists. They develop implementations of discrete

conformal mapping for multidisciplinary use, both within

mathematics itself where complex analysis is being reinvigorated

by new discrete techniques, and in the larger scientific context

with visualization and analysis of scientific data. The Riemann

Mapping Theorem guarantees unique conformal maps between any pair

of conformal 2-discs (or conformal 2-spheres); the conformal

geometry preserved by such maps carries valuable mathematical

structure. Such surfaces arise naturally in many scientific

contexts as piecewise flat (from data) or smoothly embedded (from

theory) surfaces in 3-space. Recently the new computational

technique of circle packing has allowed computational

approximations to these conformal maps. Implementing such

approximations for large scientific datasets faces both

theoretical and computational challenges. The investigator and

his colleagues work on three related topics: theoretical

superstructure of the circle packing technique, refinement and

parallelization of the circle packing algorithm for use on large

datasets, and the application of these conformal maps to

visualization and analysis of scientific data. The main

application focuses on conformal flattening of human brain

cortical surfaces. The investigators use uniqueness of conformal

maps to install surface-based coordinate systems on these

surfaces; these coordinate systems allow localization of

activation foci in Positron Emission Tomography (PET) and

functional Magnetic Resonance Imaging (fMRI) brain scans.

Conformal flattening has wider applicability as a visualization

and graph embedding technique, and these connections inform the

research.

This Focused Research Group develops algorithms to bring a

classical mathematics theorem (the Riemann Mapping Theorem, 1854)

to bear on problems of visualization of data. The Riemann Mapping

Theorem guarantees the existence of unique conformal

(angle-preserving) maps between surfaces, but does say how to

compute these maps. Modern computers and new algorithms have

changed all that, because our new computational ability can

breathe life into classical existence theorems of mathematics,

turning theory into computational tools. This project develops

algorithms to implement the computation of conformal maps on

complex spatial surfaces. The main application is the flat

mapping of human brain cortical surfaces. The brain surface is

highly convoluted and folded in space, and most of the brain

surface is folded up and hidden from view. If one flattens the

surface, one can simultaneously see down into all the folds. The

mathematically unique conformal maps produced by the algorithms

allow surface-based coordinate systems to be computed on the

brain surface so that surface positions can be precisely

determined. Moreover, if one puts foci of functional activation

onto the flattened surface, one can then visualize and measure

the relationship between brain function and brain anatomy. These

new surface-mapping techniques and their application to the brain

surface permit biomedical researchers and clinicians to rapidly

and accurately map and compare the locations of physiological and

pathological "events" in the brains of research subjects and of

patients with a variety of neurological and psychiatric

disorders. The project is supported by the Computational

Mathematics, Applied Mathematics, and Geometric Analysis programs

and the Office of Multidisciplinary Activities in MPS and by the

Computational Neuroscience program in BIO.

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