We investigate both the theoretical and algorithmic aspects of likelihood-based methods for recovering a complex-valued signal from multiple sets of measurements, referred to as looks, affected by speckle (multiplicative) noise. Our theoretical contributions include establishing the first existing theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our theoretical results capture the dependence of MSE upon the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we introduce the concept of bagged Deep Image Priors (Bagged-DIP) and integrate them with projected gradient descent. Furthermore, we show how employing Newton-Schulz algorithm for calculating matrix inverses within the iterations of PGD reduces the computational complexity of the algorithm. We will show that this method achieves the state-of-the-art performance.

翻译：本文研究了基于似然方法恢复复值信号的理论与算法方面，该信号从多组测量（称为“视数”）中获取，且受到散斑（乘性）噪声的影响。在理论贡献方面，我们首次在深度图像先验假设下建立了最大似然估计器均方误差（MSE）的理论上界。理论结果揭示了MSE对深度图像先验参数数量、视数、信号维度以及每视测量数量的依赖关系。在算法方面，我们引入了打包深度图像先验（Bagged-DIP）的概念，并将其与投影梯度下降法相结合。此外，我们展示了在投影梯度下降的迭代过程中采用牛顿-舒尔茨算法计算矩阵逆如何降低算法的计算复杂度。实验表明，该方法达到了当前最优性能。