We address the phase retrieval problem with errors in the sensing vectors. A number of recent methods for phase retrieval are based on least squares (LS) formulations which assume errors in the quadratic measurements. We extend this approach to handle errors in the sensing vectors by adopting the total least squares (TLS) framework that is used in linear inverse problems with operator errors. We show how gradient descent and the specific geometry of the phase retrieval problem can be used to obtain a simple and efficient TLS solution. Additionally, we derive the gradients of the TLS and LS solutions with respect to the sensing vectors and measurements which enables us to calculate the solution errors. By analyzing these error expressions we determine conditions under which each method should outperform the other. We run simulations to demonstrate that our method can lead to more accurate solutions. We further demonstrate the effectiveness of our approach by performing phase retrieval experiments on real optical hardware which naturally contains both sensing vector and measurement errors.
翻译:我们用感应矢量错误来解决阶段检索问题。一些最近的阶段检索方法基于最小平方(LS)的配方,这些配方假定了二次测量中的错误。我们推广了这种方法,通过采用在操作员错误的线性反向问题中使用的总最小平方(TLS)框架来处理感应矢量中的错误。我们展示了如何使用梯度下降和阶段检索问题的具体几何来获得简单而有效的TLS解决方案。此外,我们从感应矢量和测量中得出TLS和LS解决方案的梯度,这些梯度使我们得以计算解答错误。通过分析这些错误的表达方式,我们确定每种方法应优于另一种方法的条件。我们进行模拟,以证明我们的方法可以导致更准确的解决办法。我们进一步展示了我们的方法的有效性,对自然包含感应矢量和测量错误的真正光学硬件进行阶段检索实验。