Time-frequency analysis in biomedicine: mathematical, computational, and application aspects

医学影像技术的快速发展带来了许多重要的应用，包括计算机辅助疾病检测，这是临床常规的重要组成部分。由于生物医学数据固有的时间/空间可变性，产生用于疾病诊断的高质量信息在数据分析和处理方面提出了挑战。这些挑战反过来又要求开发创新的理论和方法，以实现最佳的数据表示。这一领域的一个关键方法是时频分析。它提供了一个数据表示，同时揭示了时间和频率的变化，是优秀的数据分析和特征detection. My研究计划的最终目标是进一步发展时频分析技术的生物医学应用。本研究计画将继续对Stockwell变换与Hilbert-Huang变换的研究。这两种变换在信号和图像处理领域越来越受欢迎，在物理学、工程学、海洋学和生物医学中有着广泛的应用。 然而，由于这两种变换都代表了最近的研究，起源于应用，大量的基本问题仍然存在于其数学基础，数值计算的改进，应用程序的开发。因此，本研究的目标有三个：1）更深入地了解斯托克韦尔变换和希尔伯特-黄变换的数学框架，2）开发高效且鲁棒的方法来计算变换，以便它们更加实用，以及3）确定可以从这些先进技术中受益的重要生物医学应用，并设计为特定应用量身定制的新算法。该方案是新颖的、多组分的和跨学科的，其结果将独特地联合收割机结合数学和数值计算的理论进步，为生物医学和其他领域的广泛挑战性问题提供更好的解决方案。

The rapid development of medical imaging technologies has led to many important applications, including computer-aided disease detection, an important component of clinical routine. Due to the inherent time/space variability of biomedical data, producing high quality information for disease diagnosis presents challenges in terms of data analysis and processing. These challenges in turn require the development of innovative theories and methods for optimal data representation. One key methodology in this area is time-frequency analysis. It provides a data representation that simultaneously reveals time and frequency variations and is excellent for data analysis and feature detection.The ultimate goal of my research program is to further develop time-frequency analysis techniques for biomedical applications. This research proposal is to continue the studies on the Stockwell and Hilbert-Huang transforms. Both transforms has had increasing popularity in signal and image processing community for applications in geophysics, engineering, oceanology, and biomedicine.  However, since both transforms represent recent research that originated from applications, a large number of fundamental questions remain in terms of their mathematical foundations, improvement of numerical computations, development of applications. Therefore, the objectives of this research are threefold: 1) provide a more thorough understanding of mathematical framework of the Stockwell and Hilbert-Huang transforms, 2) develop efficient and robust means to compute the transforms so that they can be more practical, and 3) identify important biomedical applications that can benefit from these advanced techniques and design new algorithms tailored for specific applications. This proposal is novel, multi-components and interdisciplinary, the results of which will uniquely combine the theoretical advancements in mathematics and numerical computation to provide better solutions for a wide range of challenging problems in biomedicine and other fields.