平移不变信号的相位恢复和仿射相位恢复

Phase retrieval is a new hot topic in applied harmonic analysis which is related to the time-frequency analysis and frame theory etc. And it is widely used in engineering and physics, such as speech recognition, optics. Signals in a shift-invariant space play an important role in the study of approximation theory, wavelet analysis and sampling theory since they provide good models for signals in the real world. Based on previous work by the applicant about phase retrieval for signals in a real-valued shift-invariant space, the project will focus on the following problems: (1) how to characterize the phase retrievability and affine phase retrievability for signals which take values in complex field or a real Hilbert space based on theory of Hilbert space; (2) Combined with sampling theory and time-frequency analysis, how to guarantee the uniqueness and stability for phase retrievability of signals in complex shift-invariant space, and how to design efficient algorithm for the recovery of complex shift-invariant space; (3) Together with the technique in graph theory and algebraic geometry, for shift-invariant signals which take values in a real Hilbert space, how to find suitable conditions on the generators and interference signals in the space to provide uniqueness and stability for phase retrieval and affine phase retrieval. The discussion of above problems will complement the phase retrieval of signals in a shift-invariant space, provide a new method for the study of phase retrieval of complex signals, and enrich the theory of phase retrieval of complex-valued signals.

信号的相位恢复是应用调和分析中新兴的研究方向之一，与时频分析、框架理论有着密切的联系，且广泛应用于工程应用和物理学习等领域，如语音识别、光学。而平移不变信号常被用来模拟现实信号，在逼近论、小波分析和采样理论中有着重要地位。本项目拟在申请人实平移不变信号的相位恢复的工作基础上进一步研究如下问题：(1)利用Hilbert空间相关理论刻画复线性空间和取值于实希尔伯特空间的连续时间信号的相位恢复和仿射相位恢复；(2)结合采样理论和时频分析等理论研究复平移不变信号的相位恢复和仿射相位恢复的存在性和稳定性，并找出适当的算法来实现；(3)利用图论和代数几何中的工具探讨取值于实希尔伯特空间的平移不变信号相位恢复和仿射相位恢复的存在性和稳定性。这些问题的探讨将进一步完善平移不变信号相位恢复的研究，并为复值连续信号相位恢复的研究提供新的途径，丰富复值信号相位恢复的理论。

相位恢复研究了从无相线性观测恢复未知信号的问题,是计算与应用调和分析的研究热点之一,广泛应用于物理和工程领域.平移不变空间常用来模拟现实世界的连续信号,在逼近论、小波分析等得到深入研究.依托本项目,负责人人完成了如下工作:1)对复值信号的共轭相位恢复进行了刻画,并将该刻画应用于复平移不变空间,找出了二阶B样条生成的平移不变空间中的能共轭相位恢复的信号;2)提出了向量值函数相位恢复、仿射相位恢复的新范式,并对其相位恢复及仿射相位恢复进行刻画,也将其应用于图上向量场相位恢复的研究,该结果也可应用于几何分析中欧氏距离几何中点确定的唯一性研究;3)对紧支集进行更一般的时频变换,研究了一般线性正则变换下的相位恢复.这些结果已发表在经典的分析类权威期刊泛函分析及SCI杂志Numerical Functional Analysis and Optimization上.本项目的研究成果丰富和发展了相位恢复和仿射相位恢复的理论,促进了应用调和分析与几何分析、图论等领域的融合.