Multiscale Geometric Analysis for Higher Dimensional Signal Processing

在过去的二十年里，像离散小波变换(DWT)这样的多尺度方法发展了信号处理；例如，小波是FBI指纹数据库和新的数字照片JPEG2000图像压缩标准的核心。虽然小波在分析和处理一维(1-D)信号(例如利率的价格波动)方面可以说是理想的或近乎理想的，但过去几年一个令人惊讶的发现是，它们无法以类似的方式利用包含“奇点”的2-D、3-D和更高维信号(例如，2-D数码照片或3-D视频中的边缘和脊线)。新的、令人困惑的方面是“几何”：奇点通常位于光滑、低维的流形上。迫切需要新的理论和工具来开发这些几何结构。然而，除了一些零散的有希望的结果外，这样的理论和工具目前还不存在。本项目旨在建立一个统一的理论和实用工具集，用于分析和处理具有几何流形结构的高维分段平滑信号。具体来说，这项研究涉及(1)基于局部定向原子的新的多尺度信号表示，这些原子相对于下面的流形具有明确的几何意义；(2)这些表示的新的统计模型；(3)利用新的表示和模型的新的多尺度处理算法；以及(4)在这一及时的研究领域中吸引本科生、研究生和其他研究人员的教育推广。鉴于最近在2-D和3-D方面取得的成功的初步结果，在这些方向上取得进展的巨大潜力最终将在更高维度的实际应用中产生数量级更好的压缩、逼近、建模和去噪性能。

Over the past twenty years multiscale methods like the discrete wavelet transform (DWT) have evolutionized signal processing; for example, wavelets lie at the core of the FBI fingerprint data base and the new JPEG 2000 image compression standard for digital photos. While wavelets are arguably ideal or near-ideal for analyzing and processing 1-dimensional (1-D) signals (price fluctuations of interest rates, for example), a surprising realization of the past few years is their inability to capitalize in a similar way on 2-D, 3-D, and higher-D signals containing "singularities" (edges and ridges in 2-D digital photos or 3-D videos, for example). The new, confounding aspect is "geometry": the singularities are typically localized along smooth, lower-dimensional manifolds. There is a great need for new theory and tools to exploit these geometric structures. Other than a few scattered promising results, however, such theory and tools do not exist today.This project aims toward a unified theory and practical toolset for the analysis and processing of higher-dimensional piecewise smooth signals that feature geometric manifold structures. In particular,the research involves (1) new multiscale signal representations based on local, directional atoms having a clear geometric meaning with respect to the underlying manifolds; (2) new statistical models for these representations; (3) new multiscale processing algorithms that exploit the new representations and models; and (4) educational outreach to engage undergraduates, graduate students, and other researchers in this timely research area. Given recent successful preliminary results in 2-D and 3-D, there is significant potential for progress in these directions to eventually yield orders of magnitude better compression, approximation, modeling, and denoising performance in real applications in higher dimensions.