Collaborative Research: Computational Conformal Mapping and Scientific Visualization

这个重点研究小组由纯粹数学家、计算数学家和神经科学家组成。他们开发了用于多学科应用的离散共形映射的实现，无论是在数学本身，复杂的分析被新的离散技术重新激活，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意共形2-盘（或共形2-球）对之间的唯一共形映射；这种映射所保留的共形几何学具有宝贵的数学结构。在许多科学背景下，这样的表面自然地出现在三维空间中，如分段平坦（来自数据）或平滑嵌入（来自理论）的表面。最近，新的圆填充计算技术使得这些共形映射的计算近似成为可能。对大型科学数据集实施这种近似面临理论和计算方面的挑战。研究者和他的同事们研究了三个相关的课题：圆填充技术的理论上层结构，用于大型数据集的圆填充算法的改进和并行化，以及这些保形图在科学数据可视化和分析中的应用。主要应用于人类大脑皮层表面的适形平坦化。研究者使用保形映射的唯一性在这些表面上安装基于表面的坐标系；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）脑部扫描中定位激活病灶。保形平坦作为一种可视化和图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，将经典数学定理（黎曼映射定理，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保角映射，但并没有说明如何计算这些映射。现代计算机和新的算法已经改变了这一切，因为我们新的计算能力可以为经典的数学存在定理注入生命，把理论变成计算工具。本项目开发了实现复杂空间表面保形映射计算的算法。其主要应用是人脑皮质表面的平面映射。大脑表面在空间上高度卷曲和折叠，大部分大脑表面折叠起来，从视线中隐藏起来。如果把表面压平，就能同时看到所有的褶皱。由算法产生的数学上独特的保角图允许在大脑表面计算基于表面的坐标系统，以便可以精确确定表面位置。此外，如果把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和大脑解剖结构之间的关系。这些新的表面绘图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绘制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项目和多学科活动办公室以及BIO的计算神经科学项目支持。

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.