Phase retrieval using the Hessian operator

Phase retrieval is a fundamental problem in optics, particularly in ptychography, where reconstructing an unknown wavefield from intensity-only measurements poses significant computational challenges. In this work, we introduce a novel phase retrieval method using the bilinear Hessian and Hessian operator to improve optimization efficiency. By formulating the problem in a differential calculus framework over linear spaces, we derive gradient and Hessian expressions without explicitly differentiating, enabling efficient second-order based solvers. This enables to implement and analyze conjugate gradient, quasi-Newton, and gradient descent methods with Newton step size, demonstrating a one-order-of-magnitude improvement in convergence speed compared to traditional approaches. Numerical experiments on near-field ptychography data validate the method’s robustness and computational efficiency. Our approach significantly reduces reconstruction time, making it highly relevant for real-time imaging applications, such as in-situ experiments.