Sparse ACEKF for phase reconstruction

Abstract

Abstract: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 log N and storage requirement of O(N), compared with the original ACEKF method, which has a computational complexity of O(N z N 3 ) and storage requirement of O(N 2 ), where N z 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.

Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes

“…Methods that use a series of through-focus intensity images (e.g., [3][4][5][6][7][8][9]) are especially popular due to their experimental simplicity. In-focus intensity images contain no phase information; however, defocus introduces phase contrast.…”

Partially coherent phase imaging with simultaneous source recovery

“…Unfortunately, this means that one must store and manipulate a 4D covariance matrix of size N 2 (where N is the total number of pixels), so memory requirements are prohibitive. We recently solved this problem by proving that the covariance matrix should be sparse [17], reducing both memory and computational complexity by many orders of magnitude. Note that the sparsity of the covariance matrix does not imply sparsity of the object [18][19][20].…”

Solution to the Boundary problem for Fourier and Multigrid transport equation of intensity based solvers

“…However, in order to have efficient minimization of LFA, a large number of planes have to be captured. In a recent publication [8], it was shown that the exponential separa-* E-mail: j.martinez@mchtr.pw.edu.pl tion strategy [8] can mitigate the LFAs on TIE methods by using only a few captured intensities [9].…”

“…where I 0 is the on-focus intensity, φ is the phase and ∂ z I is the intensity axial derivative that can be estimated using different methods [7,9]. Equation (1) …”

“…2b are mirrored and multiplied by minus one. In order to impose the required NBC in the FT-TIE solver, it is necessary to apply the mirror padding shown in Fig 2b to captured intensities, and thus, these images will be used to estimate the axial intensity derivative [7,9]. Later on, the padded I 0 and ∂ z I will be used as inputs of the FT-TIE method [14].…”