Collaborative Research: Computational Conformal Mapping and Scientific Visualization

这个研究小组由纯数学家、计算数学家和神经科学家组成。 他们开发了多学科使用的离散保角映射的实现，无论是在数学本身，复杂的分析正在通过新的离散技术重新焕发活力，还是在更大的科学背景下，科学数据的可视化和分析。 黎曼映射定理保证了任何一对共形2-圆盘（或共形2-球面）之间的唯一的共形映射;这种映射所保持的共形几何具有有价值的几何结构。 这样的表面自然出现在许多scientificcontext作为分段平坦（从数据）或光滑嵌入（从理论）表面在3空间。 最近，新的计算技术的圆包装允许计算近似这些保角映射。 实现这样的近似大型科学数据集面临着理论和计算的挑战。 研究人员和他的同事们致力于三个相关的主题：圆包装技术的理论上层结构，用于大型数据集的圆包装算法的改进和并行化，以及这些保形映射在科学数据可视化和分析中的应用。 主要应用集中在人体大脑表面的适形展平。 研究人员利用共形映射的独特性在这些表面上安装基于表面的坐标系统;这些坐标系统允许在正电子发射断层扫描（PET）和功能性磁共振成像（fMRI）脑扫描中定位激活灶。共形平坦化作为一种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信息。 这个重点研究小组开发算法，使经典数学定理（黎曼映射定理，1854年）承担数据可视化的问题。 黎曼映射定理保证了曲面之间唯一的保角映射的存在，但也说明了如何计算这些映射。 现代计算机和新算法改变了这一切，因为我们新的计算能力可以为数学的经典存在定理注入生命，将理论转化为计算工具。 本计画发展出租软体以实现复杂空间曲面上之共形映射计算。 主要应用是人脑皮层表面的平面映射。 大脑表面在空间上是高度卷曲和折叠的，大部分的大脑表面都折叠起来，隐藏在视野之外。 如果一个人看到表面，他可以同时看到所有的褶皱。 该算法产生的数学上独特的共形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，如果将功能激活的焦点放在平坦的表面上，就可以可视化和测量大脑功能与大脑解剖结构之间的关系。 这些新的表面测绘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准确地测绘和比较研究对象和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。 该项目由MPS的计算数学，应用数学和几何分析计划和多学科活动办公室以及BIO的计算神经科学计划支持。

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.