Multiscale Data Representations in Geometric and Nonlinear Settings

主要研究者和他的合作者计划在各种非线性和几何设置中构建、简化和分析一套多尺度数据表示方法，以及它们的应用。其中一些多尺度表示包括：自由形式射流细分曲面，自由形式细分曲面上矢量和张量场的多尺度表示，非线性距离数据的类小波变换，例如在流形上取值的数据的时间序列或空间数组。PI计划探索以下理论问题：当传统的样条或细分方法被修改以应用于满足特定非线性约束的数据时，底层非线性流形的曲率如何影响原始方法的稳定性、逼近性和平滑性？PI和他的合作者也在开发软件工具，用于这些新颖的多尺度方法的快速原型和计算分析，以及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）进步的推动下，新形式的数据类型开始出现在科学和工程的许多不同重要领域，数学家的任务是帮助世界做好准备，充分利用这些数据。明显的应用领域包括材料科学，扩散张量成像，高光谱成像，计算机辅助设计/动画，机器人（运动规划），仅举几例。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、结构重建等任务的必要条件。

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.