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Phase Retrieval – A Deconvolution Perspective

机译：相位检索–反卷积的观点

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Phase retrieval finds applications in various optical imaging modalities such as X-ray crystallography, holography, frequency-domain optical-coherence tomography, etc. The sensors used in optical imaging can measure only the magnitudes of incoming wavefronts and the phase information is not measured directly. This necessitates developing appropriate phase retrieval algorithms to reconstruct the object as phase contains most of the structural information. The phase retrieval problem naturally arises in the Fourier imaging context, where the measurement is the Fourier magnitude/intensity spectrum. Reconstruction from the Fourier intensity results in the autocorrelation and not the signal. We therefore address the equivalent problem of signal retrieval from the autocorrelation. Since the signal autocorrelation can be expressed as a convolution of the signal with its flipped version, we propose to solve the phase retrieval problem within a deconvolution framework. We consider a non-convex cost in two vector variables, the signal and its flipped version. An alternating minimization (Alt. Min.) strategy is employed to arrive at an optimal estimate of the signal, given the autocorrelation. Due to non-convexity of the cost function, the accuracy of the estimation is critically dependent on the initialization. We establish that the Alt. Min. iterates ensure that the cost is nonincreasing. For the specific case of causal, delta-dominant signals, the proposed framework results in exact reconstruction with an all zero-phase initialization. We shall also consider the effect of random initialization on the estimation accuracy.
机译：相位检索可用于各种光学成像模式，例如X射线晶体学，全息术，频域光学相干断层扫描等。光学成像中使用的传感器只能测量入射波前的幅度，而不能直接测量相位信息。由于相位包含大多数结构信息，因此有必要开发适当的相位检索算法来重建对象。在傅立叶成像环境中自然会出现相位恢复问题，其中测量值为傅立叶幅度/强度谱。从傅立叶强度重建将导致自相关而不是信号。因此，我们解决了自相关中信号检索的等效问题。由于信号自相关可以表示为信号与其翻转版本的卷积，因此我们建议在反卷积框架内解决相位检索问题。我们考虑两个向量变量（信号及其翻转形式）的非凸代价。给定自相关，采用交替最小化（Alt。Min。）策略来获得信号的最佳估计。由于成本函数的非凸性，估计的准确性主要取决于初始化。我们建立了Alt。最小反复进行，以确保成本不会增加。对于因果的，占主导地位的信号的特定情况，提出的框架可通过全零相位初始化实现精确的重构。我们还将考虑随机初始化对估计精度的影响。

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