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Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints

机译：包含稀疏约束的正则化的单距离相位检索问题的非线性方法

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摘要

The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Frechet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp-Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.
机译：相位检索过程是一个非线性不适定问题。用硬X射线同步加速器光束获得的菲涅耳衍射图可用于检索相衬。在这项工作中，我们提出了几种针对相位检索问题的非线性方法的收敛性比较，这些方法涉及具有稀疏约束的正则化。假设相位解相对于正交小波基具有稀疏表示。一种方法可替代地使用基于弗雷切特导数的非线性问题的解决方案和具有迭代阈值的小波坐标中的线性问题的解决方案。第二种方法是Ramlau和Teschke提出的方法，该方法将经典阈值算法推广到非线性问题。该算法在被白高斯噪声破坏的3D Shepp-Logan幻象上进行了测试。对于所研究的各种噪声水平和初始化，通过第一种方法可以获得最佳的仿真结果。与经典线性相位检索方法所给出的误差相比，重构误差显着降低。

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