Sparse ACEKF for phase reconstruction

We propose a novel low-complexity recursive filter to efficiently recover quantitative phase from a series of noisy intensity images taken through focus. We first transform the wave propagation equation and nonlinear observation model (intensity measurement) into a complex augmented state space model. From the state space model, we derive a sparse augmented complex extended Kalman filter (ACEKF) to infer the complex optical field (amplitude and phase), and find that it converges under mild conditions. Our proposed method has a computational complexity of N(z)N logN and storage requirement of O(N), compared with the original ACEKF method, which has a computational complexity of O(NzN(3)) and storage requirement of O(N(2)), where Nz is the number of images and N is the number of pixels in each image. Thus, it is efficient, robust and recursive, and may be feasible for real-time phase recovery applications with high resolution images

Efficient Gaussian inference algorithms for phase imaging

Novel efficient algorithms are developed to infer the phase of a complex optical field from a sequence of intensity images taken at different defocus distances. The non-linear observation model is approximated by a linear model. The complex optical field is inferred by iterative Kalman smoothing in the Fourier domain: forward and backward sweeps of Kalman recursions are alternated, and in each such sweep, the approximate linear model is refined. By limiting the number of iterations, one can trade off accuracy vs. complexity. The complexity of each iteration in the proposed algorithm is in the order of N logN, where N is the number of pixels per image. The storage required scales linearly with N. In contrast, the complexity of existing phase inference algorithms scales with N3 and the required storage with N2. The proposed algorithms may enable real-time estimation of optical fields from noisy intensity images