Multiscale Analysis, Modeling, and Processing of Higher-Dimensional Geometric Data.

摘要

The wavelet transform has emerged over the last decade as a powerful new tool for statistical signal processing. The wavelet domain provides a natural setting for many applications involving real-world signals and images, especially those rich in singularities (edges, ridges, and other transients). In this project, we extended wavelet transform modeling and processing algorithms to handle multidimensional signals that are smooth save for singularities along lower-dimensional manifolds. The key building block is a new quaternion wavelet transform (QWT) that generalizes the complex wavelet transform to higher dimensions using a multidimensional Hilbert transform. The QWT has a quaternion magnitude-phase representation that encodes image shifts in an absolute (x,y)-coordinate system and thus provides a theoretical framework for analyzing the phase behavior of 2-D image shifts. We conducted a thorough analysis of the QWT phase around edge regions and thereby developed efficient multiscale edge localization and flow/motion estimation algorithms for image registration based on the QWT phase.