Collaborative Research: Complex-Valued Signal Processing and its Application to Analysis of Brain Imaging Data

合作研究：复值信号处理及其在脑成像数据分析中的应用复值信号经常出现在通信、雷达和生物医学等各种应用中，因为大多数实用的调制格式都是复杂类型的，而雷达和磁共振成像等应用导致数据本质上是复值的。复数域不仅为这些信号提供了方便的表示，而且还提供了一种自然的方式来保持信号的物理特征以及它们所经历的变换。然而，复数领域也给信号处理算法的推导和分析带来了许多挑战，因此，绝大多数为复数领域开发的算法都走了捷径，限制了它们的应用。本研究建立了复值信号处理的框架，以便充分发挥复值信号处理的潜力。它允许所有计算在复数域中进行，在算法的推导和分析中消除了许多简化假设的需要，例如信号的圆形性。它还允许使用完全复杂的函数，而不是更常用的有界但非解析函数。这些函数通过有效地生成高阶统计信息，为执行独立分量分析(ICA)提供了有吸引力的替代方案。利用这一框架，导出了一类新的高效算法，用于在复数域中执行独立分量分析，特别是用于使用原始的复杂形式的医学成像数据来研究大脑功能。

Collaborative Research: Complex-Valued Signal Processing and its Application to Analysis of Brain Imaging DataComplex-valued signals arise frequently in applications as diverse as communications, radar, and biomedicine, as most practical modulation formats are of complex type and applications such as radar and magnetic resonance imaging lead to data that are inherently complex valued. The complex domain not only provides a convenient representation for these signals but also a natural way to preserve the physical characteristics of the signals and the transformations they go though. The complex domain, however, also presents a number of challenges in the derivation and analysis of signal processing algorithms, and as a result, the vast majority of algorithms developed for the complex domain have taken shortcuts limiting their usefulness.This research establishes a framework for complex-valued signal processing such that the full potential of complex-valued signal processing can be realized. It allows for all computations to be carried out in the complex domain eliminating the need for many simplifying assumptions, such as the circularity of signal, both in the derivation and the analysis of the algorithms. It also allows for the use of fully complex functions rather than the more commonly utilized bounded but non-analytic functions. These functions provide attractive alternatives for performing independent component analysis(ICA) by efficiently generating higher-order statistical information. Using this framework, a new class of efficient algorithms are derived for performing ICA in the complex domain, in particular, for studying brain function using the medical imaging data in its native, complex form.