Collaborative Research: Computational Conformal Mapping and Scientific Visualization

这个重点研究小组由纯数学家、计算数学家和神经学家组成。他们开发了用于多学科使用的离散保形映射的实现，既在数学本身中，复杂的分析正被新的离散技术重振，也在更大的科学背景下与科学数据的可视化和分析相结合。黎曼映射定理保证了任何一对共形2-圆盘(或共形2-球面)之间的唯一保形映射；由这种映射保持的保形几何具有有价值的数学结构。这样的表面在许多科学背景下自然出现，比如三维空间中的分段平坦(来自数据)或平滑嵌入(来自理论)的表面。最近，圆填充的新计算技术允许对这些保形映射进行计算逼近。在大型科学数据集上实现这样的近似面临着理论和计算上的挑战。这位研究人员和他的同事们致力于三个相关主题：圆填充技术的理论超结构，用于大数据集的圆填充算法的精化和并行化，以及这些保形映射在科学数据可视化和分析中的应用。主要应用集中在人脑皮质表面的共形展平。研究人员利用构象图的唯一性将基于表面的坐标系安装在这些表面上；这些坐标系允许在正电子发射断层扫描(PET)和功能磁共振成像(FMRI)中定位激活焦点。保形平面化作为可视化和图形嵌入技术具有更广泛的适用性，这些联系为研究提供了指导。这个专注的研究小组开发算法，将经典的数学定理(Riemann映射定理，1854年)应用于数据可视化问题。黎曼映射定理保证了曲面之间存在唯一的共形(保角)映射，但说明了如何计算这些映射。现代计算机和新的算法改变了这一切，因为我们新的计算能力可以为经典的数学存在定理注入活力，将理论转化为计算工具。这个项目开发了一个算法来实现复杂空间曲面上的保角映射的计算。它的主要应用是人脑皮质表面的平面图。大脑表面在空间中非常曲折和折叠，大部分大脑表面被折叠起来，隐藏在人们的视线之外。如果一个人把表面变平，就可以同时看到所有的褶皱。该算法产生的数学上唯一的保角映射允许在大脑表面上计算基于表面的坐标系，以便可以精确地确定表面位置。此外，如果一个人把功能激活的焦点放在平坦的表面上，那么他就可以可视化和测量大脑功能和大脑解剖之间的关系。新的表面映射技术及其在脑表面的应用使生物医学研究人员和临床医生能够快速而准确地绘制和比较研究对象以及患有各种神经和精神疾病的患者大脑中生理和病理“事件”的位置。该项目得到了计算数学、应用数学和几何分析项目和MPS多学科活动办公室以及生物计算神经科学项目的支持。

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.