Denoising Poisson phaseless measurements via orthogonal dictionary learning

Abstract

Phaseless diffraction measurements recorded by CCD detectors are often affected by Poisson noise. In this paper, we propose a dictionary learning model by employing patches based sparsity in order to denoise such Poisson phaseless measurements. The model consists of three terms: (i) A representation term by an orthogonal dictionary, (ii) an L pseudo norm of the coefficient matrix, and (iii) a Kullback-Leibler divergence term to fit phaseless Poisson data. Fast alternating minimization method (AMM) and proximal alternating linearized minimization (PALM) are adopted to solve the proposed model, and especially the theoretical guarantee of the convergence of PALM is provided. The subproblems for these two algorithms both have fast solvers, and indeed, the solutions for the sparse coding and dictionary updating both have closed forms due to the orthogonality of learned dictionaries. Numerical experiments for phase retrieval using coded diffraction and ptychographic patterns are conducted to show the efficiency and robustness of proposed methods, which, by preserving texture features, produce visually and quantitatively improved restored images compared with other phase retrieval algorithms without regularization and local sparsity promoting algorithms.

Poisson Phase Retrieval With Wirtinger Flow

“…Similar problems for the case b i = 0 have been considered previously [16][17][18][19][20][21]. Many optical sensors also have Gaussian readout noise [22,23]; the log likelihood for a Poisson + Gaussian distribution is complicated, so a common approximation is to use a shifted Poisson model that also leads to the cost function in (3).…”

A nonlocal low rank model for poisson noise removal

“…These methods include Newton's method and fixed point iteration method. In this paper, we will use the fixed iteration method to compute the solution: start from an initial solution u (0) and use the recursive process (12) u (k+1) = T (u (k) ).…”

“…Gaussian distribution has many convenient mathematical properties, and many methods have been developed to remove additive Gaussian noise. In recent decades, by using the redundancy and similarity among the patches in the image, non-local patch based denoising methods [9,14,12,24,37,39,36,49] have drawn a lot of attention. Usually, these methods consist of three procedures: patch extraction and matching according to some similarity measure, patch denoising utilizing certain structures that live inside the patch blocks, and patch aggregation to recover the clean image.…”

PDMM: A novel Primal-Dual Majorization-Minimization algorithm for Poisson Phase-Retrieval problem

“…a diagonal matrix formed by setting elements of d as its principal diagonal. It is to be noted that A H A −1 A H is the pseudo-inverse of matrix A and hence (35) can be compactly written as:…”

“…In the works of [31], [32], [33], [34], [35], [36], various authors have considered data models similar to (5) for phaseretrieval for the case of background signal b i = 0. However, background signal is rarely zero in real world applications.…”