CHS: Small: Novel Data-adaptive Analytics for Manifold Informatics: Theory, Algorithms, and Applications

这项研究将对完全自适应数据的空间频率分析做出重大贡献。项目成果还将影响数据类型/领域的信息处理和分析，如三维体中的标量/矢量/张量场，四维时变医疗数据，具有弯曲和规则结构的高维数据，或不规则结构的图形。该项目的总体目标是在一维信号的Hilbert-Huang变换（HHT）的启发下，为流形信息建模和分析开创一种数据自适应理论，并将新理论应用于流形几何/纹理/外观分析、综合和可视化，重点是数据驱动的分析和信息学。HHT在处理一维空间中的非线性和非平稳时间序列方面取得了初步成功。它包括经验模态分解（EMD）和希尔伯特谱分析。尽管HHT在许多科学和工程领域（包括地球物理、海洋科学和气候研究）越来越受欢迎，但在试图弥合HHT与数据自适应空间频率分析之间的巨大差距时，技术挑战和未解决的研究问题仍然普遍存在。此外，希尔伯特谱分析（基于瞬时频率、局部振幅和每个点的局部相位）仍然是流形数据处理和分析的一个开放的研究问题。本研究将探索流形上数据自适应空间频率分析的新理论，并阐明一种新的数据自适应分析框架，使多尺度空间频率分析/过程成为可能，这是以前从未尝试过的。研究内容包括：(1)定义一种新的基于自适应尺度的纹理几何分解计算理论，并发现EMD与流形压缩感知理论之间的内在联系；(2)通过Riesz变换进行空频分析，计算流形上任意位置的每个IMF的局部瞬时频率、幅度和相位，从而在全新的高维内禀特征空间中实现更精确的特征描述和定量特征分析；(3)创建新的、高效的算法来计算流形和高维数据集上（多通道）信号的imf和Riesz变换；(4)进行全面的实验验证，包括特征描述、噪声与特征解耦、特征感知形状补全与编辑、结构敏感变形、显著性可视化等。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research will make significant contributions towards completely data-adaptive space-frequency analytics. Project outcomes will also impact information processing and analytics of data types/domains such as scalar/vector/tensor fields in 3D volume, 4D time-varying medical data, higher-dimensional data with curved and regular structure, or graphs of irregular structure. This project's overarching objective is to trailblaze a data-adaptive theory for manifold information modeling and analysis, inspired by the Hilbert-Huang transform (HHT) on one-dimensional signals, and to apply the new theory to manifold geometry/texture/appearance analysis, synthesis, and visualization, with an emphasis on data-driven analytics and informatics. HHT has exhibited initial success in handling nonlinear and nonstationary time series in one-dimensional space. It comprises empirical mode decomposition (EMD) and Hilbert spectral analysis. Despite its growing popularity in many scientific and engineering fields (including geophysics, marine science, and climate studies), technical challenges and unsolved research issues still prevail when trying to bridge the large gap between HHT and data-adaptive space-frequency analytics for manifold data. Additionally, Hilbert spectral analysis (based on instantaneous frequencies, local amplitude, and local phase at each point) remains an open research problem for data processing and analysis on manifolds. This research will explore a new theory of data-adaptive space-frequency analysis on manifolds, and articulate a novel data-adaptive analytics framework enabling multi-scale space-frequency analysis/process, which has never been attempted before. Research activities will include: (1) definition of a new computational theory based on texture-geometry decomposition with adaptive scales, and discovery of the intrinsic connection between the EMD and compressed sensing theory on manifolds; (2) space-frequency analysis via Riesz transforms, to enable computation of local instantaneous frequency, amplitude, and phase for each IMF anywhere on a manifold so as to achieve more accurate feature description and quantitative feature analysis in a brand new, high-dimensional, intrinsic feature space; (3) creation of new, efficient algorithms to compute IMFs and Riesz transforms of (multi-channel) signals on manifolds as well as in high-dimensional datasets; and (4) conducing comprehensive experimental validation, including feature description, noise and feature decoupling, feature-aware shape completion and editing, structure-sensitive deformation, and saliency visualization.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.