Collaborative Research: Computational Conformal Mapping and Scientific Visualization

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This Focused Research Group is composed of puremathematicians, computational mathematicians, andneuroscientists. They develop implementations of discreteconformal mapping for multidisciplinary use, both withinmathematics itself where complex analysis is being reinvigoratedby new discrete techniques, and in the larger scientific contextwith visualization and analysis of scientific data. The RiemannMapping Theorem guarantees unique conformal maps between any pairof conformal 2-discs (or conformal 2-spheres); the conformalgeometry preserved by such maps carries valuable mathematicalstructure. Such surfaces arise naturally in many scientificcontexts as piecewise flat (from data) or smoothly embedded (fromtheory) surfaces in 3-space. Recently the new computationaltechnique of circle packing has allowed computationalapproximations to these conformal maps. Implementing suchapproximations for large scientific datasets faces boththeoretical and computational challenges. The investigator andhis colleagues work on three related topics: theoreticalsuperstructure of the circle packing technique, refinement andparallelization of the circle packing algorithm for use on largedatasets, and the application of these conformal maps tovisualization and analysis of scientific data. The mainapplication focuses on conformal flattening of human braincortical surfaces. The investigators use uniqueness of conformalmaps to install surface-based coordinate systems on thesesurfaces; these coordinate systems allow localization ofactivation foci in Positron Emission Tomography (PET) andfunctional Magnetic Resonance Imaging (fMRI) brain scans.Conformal flattening has wider applicability as a visualizationand graph embedding technique, and these connections inform theresearch. This Focused Research Group develops algorithms to bring aclassical mathematics theorem (the Riemann Mapping Theorem, 1854)to bear on problems of visualization of data. The Riemann MappingTheorem guarantees the existence of unique conformal(angle-preserving) maps between surfaces, but does say how tocompute these maps. Modern computers and new algorithms havechanged all that, because our new computational ability canbreathe life into classical existence theorems of mathematics,turning theory into computational tools. This project developsalgorithms to implement the computation of conformal maps oncomplex spatial surfaces. The main application is the flatmapping of human brain cortical surfaces. The brain surface ishighly convoluted and folded in space, and most of the brainsurface is folded up and hidden from view. If one flattens thesurface, one can simultaneously see down into all the folds. Themathematically unique conformal maps produced by the algorithmsallow surface-based coordinate systems to be computed on thebrain surface so that surface positions can be preciselydetermined. Moreover, if one puts foci of functional activationonto the flattened surface, one can then visualize and measurethe relationship between brain function and brain anatomy. Thesenew surface-mapping techniques and their application to the brainsurface permit biomedical researchers and clinicians to rapidlyand accurately map and compare the locations of physiological andpathological "events" in the brains of research subjects and ofpatients with a variety of neurological and psychiatricdisorders. The project is supported by the ComputationalMathematics, Applied Mathematics, and Geometric Analysis programsand the Office of Multidisciplinary Activities in MPS and by theComputational Neuroscience program in BIO.