基于矩阵分解的相位恢复非凸优化算法研究

The problem of phase retrieval arises in the process of optical imaging, it means to recover the signal itself from the intensity of Fourier transform of the signal. The diffraction effect of light can be described by the Fourier transform. However, under the experimental conditions, the detector can only record the intensity distribution of the light field and the phase information of the light field is lost. By solving the phase recovery problem, the structure of the target object can be obtained. Efficient and stable numerical solution of phase retrieval for the structure of the sample is of great analytical significance. Due to incomplete measurement data (lack of phase information), the solution of phase retrieval is seriously non-unique. In general, multiple measurements make the solution of the problem unique up to a phase constant difference. The phase retrieval algorithm based on multiple measurement models is mainly divided into convex optimization algorithm and non-convex optimization algorithm. The storage complexity of the non-convex algorithm based on the original decision vector is optimal, but the algorithm only has local convergence. The existing convex algorithm PhaseLift lifts the decision vector to a decision matrix, which has high computational complexity and storage complexity, which is not suitable for high-dimensional phase retrieval. However, the convex optimization algorithm can guarantee global convergence and is more stable. The key to solve the high dimensional phase retrieval algorithm based on convex optimization model is to reduce the computational and storage complexity of the algorithm. Assuming the relative uniqueness of solution, based on the connection between convex model and non-convex model, this project solves convex model indirectly by solving a series of non-convex models. The new algorithm combines the advantages of the global convergence of convex models and the low storage requirements of non-convex models and can be used to solve the phase retrieval problem in practice. The phase retrieval convex model is actually a rank-one positive definite matrix recovery problem. For the more general problem of low-rank matrix recovery and matrix complemention, we can get a similar algorithm based on the connection of full-matrix convex model and matrix decomposition non-convex model based on low-rank information. The advantage of this algorithm is that it avoids the estimation of prior rank, and guarantees the algorithm's convergence and low-rank solution.

相位恢复的求解是光学成像科学和工程中的关键问题。由于实验条件的限制，光场的相位不能像强度一样被测量到，通过光场强度来确定相位就称为相位恢复。相位恢复解的不唯一性以及正问题的非线性使得该反问题数值求解困难。通常我们用多次测量的方式来确保其解的相对唯一性。现有的非凸优化方法是基于优化相位恢复正问题导出的损失函数来重建原信号。现有非凸算法的缺点是算法的收敛依赖于一个靠近真解的初值以及较多的测量次数。为了克服非凸优化的局部收敛性，研究者提出求解基于向量提升的凸松弛优化模型来近似求解原非凸优化模型，但算法的计算复杂度和存储复杂度都太高，对高维相位恢复问题不适用。基于凸优化模型的全局收敛性以及提升矩阵的秩一性，本项目计划结合凸优化模型和非凸模型两者各自的优点，研究相位恢复的非凸优化算法。算法不仅具有全局收敛性且算法的存储复杂度较低，其能更稳定高效地重建相位恢复的原信号。

本项目研究相位恢复问题的高效非凸优化算法。相位恢复问题在物理学和成像科学领域里，如X射线衍射成像、电子显微镜等有着广泛的应用。在这些应用中，确定由于物理测量仪器的限制而损失的相位信息是成像的关键。求解相位恢复问题意味着求解一个非凸的优化问题，而优化一个非凸问题是困难的。关于相位恢复问题的数值算法研究仍是光学领域的热点问题。由于非凸优化易陷入局部最优点，且优化算法依赖于初始点的选取，现存的数值算法研究要么缺少收敛性的理论保证，要么需要算法使用者提供一个靠近真解的初始值以保证算法的收敛性。为了解决该问题的非凸性带来的优化困难，Candès 提出了相位恢复的凸松弛求解算法PhaseLift，其中我们需要求解一个高维的半定矩阵优化问题。现存的凸求解器不能很好地求解高维半定矩阵优化问题，所以该凸优化算法不能用于求解实际的工程问题。本项目的主要研究内容就是设计基于PhaseLift的和基于矩阵分解的，适用于实际问题的数值算法。本项目设计的算法继承了不依赖于初始点选取的凸优化算法和低计算复杂度的非凸算法两种算法的优点，为相位恢复问题的求解创造性地提出了新的解决方案。本项目发展的方法在理论上能给出PhaseLift问题的全局最优解，除了理论上的保证，算法对于Fourier相位恢复问题的求解能获得更好的恢复效果，且算法具有对噪音测量数据的成像鲁棒性的优点。本项目发展的方法除了适用于相位恢复这一特定的非凸问题，我们相信发展的方法对于其他类似的非凸问题也是适用的。

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