Coded aperture design for solving the phase retrieval problem in X-ray crystallography

Pinilla, Samuel; García, Hans; Díaz, Luis A.; Poveda, Juan Carlos; Argüello, Henry

doi:10.1016/j.cam.2018.02.002

Cited by 16 publications

(7 citation statements)

References 15 publications

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“…The simulated coded aperture was a block-unblock ensemble as used in our previous works [23,30]. All simulations were implemented in Matlab R2019a on an Intel Core i7 3.41Ghz CPU with 32 GB RAM.…”

Section: Simulations and Resultsmentioning

confidence: 99%

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Exact Crystalline Structure Recovery in X-ray Crystallography from Coded Diffraction Patterns

Pinilla,

Bacca,

Vargas

et al. 2019

Preprint

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X-ray crystallography (XC) is an experimental technique used to determine three-dimensional crystalline structures. The acquired data in XC, called diffraction patterns, is the Fourier magnitudes of the unknown crystalline structure. To estimate the crystalline structure from its diffraction patterns, we propose to modify the traditional system by including an optical element called coded aperture which modulates the diffracted field to acquire coded diffraction patterns (CDP). For the proposed coded system, in contrast with the traditional, we derive exact reconstruction guarantees for the crystalline structure from CDP (up to a global shift phase). Additionally, exploiting the fact that the crystalline structure can be sparsely represented in the Fourier domain, we develop an algorithm to estimate the crystal structure from CDP. We show that this method requires 50% fewer measurements to estimate the crystal structure in comparison with its competitive alternatives. Specifically, the proposed method is able to reduce the exposition time of the crystal, implying that under the proposed setup, its structural integrity is less affected in comparison with the traditional. We discuss further implementation of imaging devices that exploits this theoretical coded system.

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Section: Simulations and Resultsmentioning

confidence: 99%

“…To implement the proposed acquisition system in Fig. 2 a block-unblock coded aperture is feasible [23,30]. Specifically, the blocking elements of these coded apertures can be fabricated using tungsten, since this material can stop an x-ray beam, resulting in low fabrication costs [31][32][33].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Exact Crystalline Structure Recovery in X-ray Crystallography from Coded Diffraction Patterns

Pinilla,

Bacca,

Vargas

et al. 2019

Preprint

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“…In this section, we develop a stochastic algorithm, named Stochastic Smoothing Phase Retrieval (SSPR). This algorithm results from a stochastic gradient descent method based on the Wirtinger derivative, which smoothes the stochastic (non-smooth) cost function g(•) in (4). The concept of Wirtinger derivative and smoothing function were introduced in [16,17] and are recalled below Definition 3.1.…”

Section: Stochastic Smoothing Phase Retrieval Algorithmmentioning

confidence: 99%

“…Phase retrieval is an inverse problem that consists of recovering a signal from the squared modulus of some linear transforms, which has proved efficient in in various applications such as, optics [1], astronomy [2] and X-ray crystallography [3,4,5,6]. Recent works [7,8,9] have been proposed to solve the phase retrieval problem by optimizing a non-convex and non-smooth objective function with a gradient descent algorithm based on the Wirtinger derivative with an appropriate initialization.…”

Section: Introductionmentioning

confidence: 99%

A Smoothing Stochastic Phase Retrieval Algorithm for Solving Random Quadratic Systems

Pinilla

Bacca

Tourneret

et al. 2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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A novel Stochastic Smoothing Phase Retrieval (SSPR) algorithm is studied to reconstruct an unknown signal x ∈ R n or C n from a set of absolute square projections y k = | a k , x | 2. This inverse problem is known in the literature as Phase Retrieval (PR). Recent works have shown that the PR problem can be solved by optimizing a nonconvex and non-smooth cost function. Contrary to the recent truncated gradient descend methods developed to solve the PR problem (using truncation parameters to bypass the non-smoothness of the cost function), the proposed algorithm approximates the cost function of interest by a smooth function. Optimizing this smooth function involves a single equation per iteration, which leads to a simple scalable and fast method especially for large sample sizes. Extensive simulations suggest that SSPR requires a reduced number of measurements for recovering the signal x, when compared to recently developed stochastic algorithms. Our experiments also demonstrate that SSPR is robust to the presence of additive noise and has a speed of convergence comparable with that of state-of-the-art algorithms.

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“…In many applications of science and engineering, it is required to recover a signal from the squared modulus of any linear transform, which is known as phase retrieval (PR). Such a task is present in optics [1], astronomical imaging [2], microscopy [3] and x-ray crystallography [4,5,6], where the optical sensors measure the intensities of the reflection, but they are not able to measure the phase of the signal. For example, in x-ray crystallography [4], PR is used to determine the atomic position of a crystal in a three-dimensional (3D) space [7].…”

Section: Introductionmentioning

confidence: 99%

SPRSF: Sparse Phase Retrieval via Smoothing Function

Pinilla,

Bacca,

Arguello

2018

Preprint

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Phase retrieval (PR) is an ill-conditioned inverse problem which can be found in various science and engineering applications. Assuming sparse priority over the signal of interest, recent algorithms have been developed to solve the phase retrieval problem. Some examples include SparseAltMinPhase (SAMP), Sparse Wirtinger flow (SWF) and Sparse Truncated Amplitude flow (SPARTA). However, the optimization cost functions of the mentioned algorithms are nonconvex and non-smooth. In order to fix the non-smoothness of the cost function, the SPARTA method uses truncation thresholds to calculate a truncated step update direction. In practice, the truncation procedure requires calculating more parameters to obtain a desired performance in the phase recovery. Therefore, this paper proposes an algorithm called SPRSF (Sparse Phase retrieval via Smoothing Function) to solve the sparse PR problem by introducing a smoothing function. SPRSF is an iterative algorithm where the update step is obtained by a hard thresholding over a gradient descent direction. Theoretical analyses show that the smoothing function uniformly approximates the non-convex and non-smooth sparse PR optimization problem. Moreover, SPRSF does not require the truncation procedure used in SPARTA. Numerical tests demonstrate that SPRSF performs better than state-of-the-art methods, especially when there is no knowledge about the sparsity k. In particular, SPRSF attains a higher mean recovery rate in comparison with SPARTA, SAMP and SWF methods, when the sparsity varies for the real and complex cases. Further, in terms of the sampling complexity, the SPRSF method outperforms its competitive alternatives.

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Coded aperture design for solving the phase retrieval problem in X-ray crystallography

Cited by 16 publications

References 15 publications

Exact Crystalline Structure Recovery in X-ray Crystallography from Coded Diffraction Patterns

Exact Crystalline Structure Recovery in X-ray Crystallography from Coded Diffraction Patterns

A Smoothing Stochastic Phase Retrieval Algorithm for Solving Random Quadratic Systems

SPRSF: Sparse Phase Retrieval via Smoothing Function

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