Total Variation--Based Phase Retrieval for Poisson Noise Removal

Abstract

Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of Poisson noise model, which is referred to as "TV-PoiPR". We also propose an efficient numerical algorithm based on alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real and complex-valued images to demonstrate the effectiveness of our proposed methods.

Advanced denoising for X-ray ptychography

“…Several researchers have also studied techniques to remove the random noise from photoncounting or read-out noises by employing a sparse regularization technique to further improve the image quality for conventional ptychography, .e.g. Tikhonov regularization with nonlinear conjugate gradient method [17], total variation regularization with ADMM [28] and dictionary learning method with proximal algorithm [29]. Specially, for the more practical noises, like a mix of Poisson and Gaussian noises, several variational methods have been proposed for linear inverse problems, e.g.…”

Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers

“…The alternating direction method of multipliers (ADMM) [17,42,4] was adopted to solve non-blind PR problems in [41,6], where it demonstrated fast convergence and robustness. However, it has never been applied to BP-PR problems (1.1).…”

“…It is standard to show the existence of a minimizer to Model I, and we omit the details. The generalized derivative for complex-valued variables are adopted as in [21,6] by separating the real and imaginary parts of the variables and operators defined in the complex space. Below we denote the Lipschitz property of the function G defined on C m .…”

“…One readily knows that noise level gets higher if η gets smaller. For the noisy measurements, SNR intensity [6] is used to measure the noise level, which is denoted below: SNR intensity := −10 log 10 f − (a η ) 2 2 / (a η ) 2 2 .…”

Multigrid Optimization for Large-Scale Ptychographic Phase Retrieval

“…Popular methods that solve this problem include classical gradient-based algorithms [38,52] and Wirtinger flow [7,51]. Variants of the intensity Gaussian error metric include the amplitude Gaussian metric [9,38,50], intensity Poisson metric [8,9,46], and the weighted intensity Gaussian metric [38], all of which measure some variants of the misfit between the forward model and the observed data. Another formulation of the inverse problem involves solving a feasibility problem.…”