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.