Multiscale Data Representations in Geometric and Nonlinear Settings

主要研究人员和他的合作者计划建立、简化和分析一套在各种非线性和几何环境下的多尺度数据表示方法及其应用。这些多尺度表示包括：自由形式的喷射细分曲面、自由形式细分曲面上的矢量场和张量场的多尺度表示、非线性范围数据的小波样变换，例如取流形上的值的时间序列或空间数组。PI旨在探索如下理论问题：当传统的样条或细分方法被修改为适用于满足特定非线性约束的数据时，潜在的非线性流形的曲率如何影响稳定性，原方法的逼近和光滑性？PI和他的合作者还在开发软件工具，用于这些新的多尺度方法的快速原型和计算分析，并将它们应用于实际数据集。在各种传感技术(光学、合成孔径雷达等)的推动下，科学和工程中许多不同的重要领域开始出现新的数据类型，数学家的任务是帮助世界做好充分利用这些数据的准备。明显的应用领域包括材料科学、扩散张量成像、高光谱图像、计算机辅助设计/动画、机器人(运动规划)等，在这些技术中，高效的多尺度方法是处理诸如压缩、配准、快速搜索、浏览、结构重建等任务所必需的。

The principle investigator and his collaborators plan to construct, streamline and analyze a suite of multiscale data representation methods in various nonlinear and geometric settings, as well as their applications.Some of these multiscale representations include: free-form jet subdivision surfaces,multiscale representation of vector and tensor fields on free-form subdivision surfaces, wavelet-like transform of nonlinear range data, e.g. time series or spatial arrays of data taking values at a manifold.The PI intends to explore theoretical questions such as:When a traditional spline or subdivision method is modified to apply to data which satisfy specific nonlinear constraints, how would the curviness of the underlying nonlinear manifold affect the stability, approximation and smoothness behavior of the original method? The PI and his collaborators are alsodeveloping software tools for the fast prototyping and computational analysis of these novel multiscale methods, as well as applying them to real datasets. Fueled by the advances in various sensing technologies (in optics, syntheture aperture radar, etc.), new forms of data type begin to arise in many different significant fields in science and engineering, and it is the task of mathematicians to help preparing the world to make good use of such data. Obvious application areas include material sciences, diffusion tensor imaging, hyperspectral imagery, computer aided design/animation, robotics (motion planning), just to name a few.In each of these technologies, efficient multiscale methods is a must for processing tasks such as the compression, registration, fast search, browsing, structure reconstruction, etc.