KEYWORDS: Reconstruction algorithms, Phase retrieval, Chemical elements, Fourier transforms, Visualization, Signal processing, Optimization (mathematics), Network architectures, Inverse problems, Signal to noise ratio
Fourier phase retrieval (FPR) is to recover a signal from its Fourier magnitude measurement. Due to the ill-posedness of the problem, it is often necessary to introduce prior information. Recently, replacing hand-crafted priors with data-learned priors, such as deep generative priors, has received a lot of attention in inverse problems. Note that the reconstruction performance of trained generative priors relies on a large amount of training data, in this paper we solve FPR problem with an untrained generative network, which approximates the unknown signal only with a fixed seed. We propose an algorithm that combines the alternating direction method of multipliers (ADMM) with an untrained generative network. Specifically, we model the problem as a constrained optimization problem, the ADMM is adopted to solve it alternately, and then an untrained generative network is embedded into the iterative process to constrain the estimated signal. The effectiveness of the proposed algorithm is demonstrated through experiments on grayscale images. Both PSNR, SSIM, and the visual quality of the reconstructed images are superior to that of the state-of-the-art algorithms, especially when the measurement is incomplete.
In this paper, we propose an end-to-end neural network abbreviated as TCNN to solve the blind phase retrieval problem in multiple scattering imaging. TCNN is a kind of auto-encoder with a transform layer, which acts as a bridge between transforming domains. Compared to double phase retrieval method, TCNN can directly estimate the image from those phaseless measurements through the nonlinear network structure. During training, the parameters of TCNN are updated by the adaptive moment estimation algorithm Adam. Numerical experiments show that TCNN can recover images with comparable quality to that of state-of-the-art methods. Moreover, TCNN hugely reduces the time cost for recovering images once the training procedure is completed.
Fourier Ptychographic Microscopy (FPM) has achieved large field of view and high-resolution microscopy, which has attracted widespread attention. However, whether FPM can be restored accurately has not been able to provide a theoretical guarantee. The corresponding measurement system of FPM is the phase retrieval problem of the 2-dimensional inverse short-time Fourier transform (2-D ISTFT). In this paper, aiming at the defect that there are multiple global optimal solutions (ambiguities) for the phase retrieval problem, it is proved that FPM has almost the unique global optimal solution in the sense of removing the global phase solution. Based on different overlap ratios (the ratio of the area of the overlapped part to the window area), the upper bound estimation of the number of the ambiguities is given, and it is proved that FPM can eliminate the conjugate flip solution. In the simulations, the sequential Gerchberg-Saxton method is used to update under different overlap ratio conditions. It is verified that under the condition of low overlap ratio, FPM has poor convergence; under the condition of high overlap ratio, FPM reaches the global optimal solution rapidly.
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