掩模傅立叶测量下相位恢复问题的理论与算法

近几十年，相位恢复问题因其在X射线晶体学、光学、天文成像、语音处理等领域的重要应用而引起了各界的广泛关注。系统研究其理论及算法，有重要的理论和实际意义。本项目将利用函数逼近论、傅立叶分析、优化理论、随机矩阵理论、代数学和组合数论等研究工具，研究傅立叶测量或掩模傅立叶测量下的相位恢复问题。目前，大多数相位恢复问题的理论分析建立在随机测量的前提下，该假设在实际问题中具有很强的局限性。申请人将基于傅立叶测量或带掩模的傅立叶测量，分析信号唯一恢复的条件。进一步，设计高效的算法来求解此类测量下的相位恢复问题，分析信号恢复的稳定性、算法收敛性及采样复杂度。在此基础上，针对此类测量下的稀疏相位恢复问题，进行算法设计和理论分析。本项目将傅立叶分析，压缩感知、优化理论和图像处理等热门领域结合，建立更深刻的联系。旨在为相位恢复问题提供更完善的理论支撑，克服现有技术和理论上的不足，推动理论和实际应用发展。

In recent decades, the problem of phase retrieval has attracted wide attention due to its important applications in X-ray crystallography, optics, astronomical imaging, speech processing and so on. Therefore, it is of great theoretical and practical significance to systematically study its theory and algorithms. This project will use the methods of function approximation theory, Fourier analysis, optimization theory, random matrix theory, algebra and combinational number theory to study the phase retrieval problem from Fourier measurements with masks or not. So far, most of theoretical analysis in literature has been established under some random measurements. This assumption is strongly limited in real applications. The applicant will analyze the conditions required for signal recovery under Fourier measurements with masks or not. Furthermore, the project will design an efficient algorithm to solve the phase recovery problem under such measurement, and analyze the stability, convergence and sampling complexity of the related algorithm. On this basis, the algorithm design and theoretical analysis are carried out for the sparse phase recovery problem under such measurements. The project combines Fourier analysis, compressed sensing, optimization theory and image processing to establish a deeper connection. It aims to provide more complete theoretical support for phase retrieval problem, overcome the shortcomings of the prior art, and promote the development of both theory and real applications.