[论文解读] Phase Retrieval for Sparse Signals: Uniqueness Conditions

Juri Ranieri, Amina Chebira+2 more|arXiv (Cornell University)|Aug 14, 2013

Advanced X-ray Imaging Techniques参考文献 28被引用 57

一句话总结

本文通过将相位恢复问题与组合数学中的换向问题（turnpike problem）建立联系，确立了稀疏信号相位恢复的唯一性条件。研究证明，当自相关函数无碰撞时，一维情况下解几乎必然唯一，并将该结论推广至多维信号，显著提升了先前对稀疏相位恢复唯一性的保证。

ABSTRACT

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?" In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.

研究动机与目标

• 确定稀疏信号相位恢复问题在何种充分条件下可获得唯一解。
• 将相位恢复与组合数学中的经典换向问题相连接，推导新的唯一性准则。
• 证明自相关函数中无碰撞可确保一维稀疏信号的几乎必然唯一性。
• 通过将问题分解为坐标方向上的一维子问题，将一维唯一性结果推广至多维信号。
• 提供一个适用于离散与连续域的统一框架，仅依赖于稀疏性与支撑结构。

提出的方法

• 作者将信号建模为在空间域中稀疏，非零元素被限制在已知或可估计的支撑集内。
• 他们建立了相位恢复与组合重构中的经典换向问题之间的强数学联系。
• 他们证明自相关函数中的碰撞会阻碍唯一恢复，因此识别出自相关无碰撞是唯一性的必要条件。
• 对于一维信号，他们证明若自相关无碰撞，则解几乎必然唯一，基于对信号支撑的概率性论证。
• 通过沿坐标方向将问题简化为一组一维相位恢复问题，将一维结果推广至多维信号。
• 该方法适用于离散与连续域，关键条件为自相关无碰撞且信号具有稀疏性。

实验结果

研究问题

• RQ1 在一维域中，稀疏信号的相位恢复问题在何种条件下可唯一求解？
• RQ2 换向问题与相位恢复解的唯一性有何关联？
• RQ3 在一维信号中推导出的唯一性条件能否推广至多维稀疏信号？
• RQ4 自相关函数中的碰撞在阻止唯一恢复中起何种作用？
• RQ5 在离散与连续域中，满足唯一性条件的概率有何差异？

主要发现

• 当自相关函数中无碰撞时，一维稀疏信号的相位恢复问题解几乎必然唯一。
• 自相关无碰撞在 1D 稀疏相位恢复中既是必要条件也是充分条件。
• 通过求解一组 1D 问题，1D 信号中推导出的唯一性条件可推广至多维信号，当多维自相关无碰撞时可确保唯一性。
• 所提出的唯一性条件通过提供更强且更通用的准则，优于先前已知结果，尤其在多维相位恢复中。
• 该框架在离散与连续域中均适用，唯一差异在于由于域结构导致满足唯一性条件的概率不同。