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Register a new userStudy of cost functionals for ptychographic phase retrieval to improve the robustness against noise, and a proposal for another noise-robust ptychographic phase retrieval scheme
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Added by
Sander Konijnenberg
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Recently, efforts have been made to improve ptychography phase retrieval algorithms so that they are more robust against noise. Often the algorithm is adapted by changing the cost functional that needs to be minimized. In particular, it has been suggested that the cost functional should be obtained using a maximum-likelihood approach that takes the noise statistics into account. Here, we consider the different choices of cost functional, and to how they affect the reconstruction results. We find that seemingly the only consistently reliable way to improve reconstruction results in the presence of noise is to reduce the step size of the update function. In addition, a noise-robust ptychographic reconstruction method has been proposed that relies on adapting the intensity constraints
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Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and improved reconstructions. Ptychographic microscopes allow for large fields to be imaged at high resolution at the cost of additional computational expense. In this work, we propose a multigrid-based optimization framework to reduce the computational burdens of large-scale ptychographic phase retrieval. Our proposed method exploits the inherent hierarchical structures in ptychography through tailored restriction and prolongation operators for the object and data domains. Our numerical results show that our proposed scheme accelerates the convergence of its underlying solver and outperforms the Ptychographic Iterative Engine (PIE), a workhorse in the optics community.
The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for general constrained optimization problems beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the derivative terms are calculated automatically through matrix-vector products, without explicitly forming any large Jacobian or Gauss-Newton matrices. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained blind ptychographic object and probe retrieval problems, under both the Gaussian and Poisson noise models, and that this method outperforms best-in-class first-order ptychographic reconstruction methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.
Ptychography has risen as a reference X-ray imaging technique: it achieves resolutions of one billionth of a meter, macroscopic field of view, or the capability to retrieve chemical or magnetic contrast, among other features. A ptychographyic reconstruction is normally formulated as a blind phase retrieval problem, where both the image (sample) and the probe (illumination) have to be recovered from phaseless measured data. In this article we address a nonlinear least squares model for the blind ptychography problem with constraints on the image and the probe by maximum likelihood estimation of the Poisson noise model. We formulate a variant model that incorporates the information of phaseless measurements of the probe to eliminate possible artifacts. Next, we propose a generalized alternating direction method of multipliers designed for the proposed nonconvex models with convergence guarantee under mild conditions, where their subproblems can be solved by fast element-wise operations. Numerically, the proposed algorithm outperforms state-of-the-art algorithms in both speed and image quality.
In recent years, the mathematical and algorithmic aspects of the phase retrieval problem have received considerable attention. Many papers in this area mention crystallography as a principal application. In crystallography, the signal to be recovered is periodic and comprised of atomic distributions arranged homogeneously in the unit cell of the crystal. The crystallographic problem is both the leading application and one of the hardest forms of phase retrieval. We have constructed a graded set of benchmark problems for evaluating algorithms that perform this type of phase retrieval. The data, publicly available online, is provided in an easily interpretable format. We also propose a simple and unambiguous success/failure criterion based on the actual needs in crystallography. Baseline runtimes were obtained with an iterative algorithm that is similar but more transparent than those used in crystallography. Empirically, the runtimes grow exponentially with respect to a new hardness parameter: the sparsity of the signal autocorrelation. We also review the algorithms used by the leading software packages. This set of benchmark problems, we hope, will encourage the development of new algorithms for the phase retrieval problem in general, and crystallography in particular.
In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: Under which conditions can we uniquely recover the signal of interest from its measured magnitudes? In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.
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