Hybrid Input-output Algorithms Research Articles

Phase-retrieval algorithm based on Kramers–Kronig relations in coherent diffraction imaging

Coherent diffraction imaging (CDI) is a high-resolution technique that does not require x-ray lenses. With advances in scientific technology, such as synchrotron radiation, x-ray free-electron lasers, and coherent electron sources, CDI has been applied to diverse fields, such as biology, medicine, and semiconductors, as a high-resolution, nondestructive measure. With the rapid increase in demand for these applications, enhancing the efficiency of processing high-volume data has become a significant challenge for promotion. In this study, we proposed an algorithm that combines Kramers–Kronig (KK) relations with oversampling smoothness (OSS). The results were evaluated by introducing an error coefficient. We found that the error of the KK-OSS algorithm is always reduced by approximately 50% compared with the error reduction (ER) algorithm, hybrid input–output (HIO) algorithm, and OSS in real space. In the diffraction space, the error in the KK-OSS can be decreased to 0.11. With 100 iterations, KK-OSS spent 218.3 s on reconstructing most of the sample information, while ER was 258.1 s, HIO algorithm took 296.7 s and the reconstruction was still a random value. In Fraunhofer diffraction, it cost KK-OSS 58.8 s to reconstruct, while OSS took 61.9 s. Therefore, this method can reduce the reconstruction error, shorten the reconstruction time, and improve the efficiency compared with the ER, HIO, and OSS algorithms using a random phase as the initial value.

Modified Gerchberg-Saxton algorithm-based probe-free wavefront distortion compensation of an OAM beam

In the free space optical communication system, atmospheric turbulence (AT) will distort the helical phase of the orbital angular momentum (OAM) beam, resulting in inter-mode crosstalk and system performance degradation. Based on modified Gerchberg-Saxton (GS) algorithm, this paper proposes a wavefront sensorless adaptive optics system to compensate the wavefront distortion of an OAM beam. For the modified GS algorithm, at each iteration, the spatial phase is re-modulated to descend along a new phase gradient direction to improve algorithm convergence. At the same time, in the spatial domain, negative feedback is generated by nonlinear operation to optimize the beam amplitude, so as to speed up the convergence speed of the algorithm. Simulation results show that our modified GS algorithm can better compensate the wavefront distortion of an OAM beam caused by atmospheric turbulence than the traditional GS and hybrid input–output algorithms, which provides a new idea to solve the problem of atmospheric turbulence in free space optical communication system.

Improved Error Reduction and Hybrid Input Output Algorithms for Phase Retrieval by including a Sparse Dictionary Learning-Based Inpainting Method

The phase retrieval (PR), reconstructing an object from its Fourier magnitudes, is equivalent to a nonlinear inverse problem. In this paper, we proposed a two-step algorithm that traditional ER/HIO iteration plays as the coarse feature reconstruction, whereas the KSVD-based inpainting technique deals with the fine feature set accordingly. Since the KSVD allows the content of oversampled dictionary with sparse representation to adaptively fit a given set of object examples, as long as the ER/HIO algorithms provide decent object estimation at early stage, the pixels violating the object constraint can be restored with superior image quality. The numerical analyses demonstrated the effectiveness of ER + KSVD and HIO + KSVD through multiple independent initial Fourier phases. With its versatility and simplicity, the proposed method can be generalized to be implemented with more PR state-of-the-arts.

Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities.

Coherent diffraction imaging (CDI) is high-resolution lensless microscopy that has been applied to image a wide range of specimens using synchrotron radiation, X-ray free-electron lasers, high harmonic generation, soft X-ray lasers and electrons. Despite recent rapid advances, it remains a challenge to reconstruct fine features in weakly scattering objects such as biological specimens from noisy data. Here an effective iterative algorithm, termed oversampling smoothness (OSS), for phase retrieval of noisy diffraction intensities is presented. OSS exploits the correlation information among the pixels or voxels in the region outside of a support in real space. By properly applying spatial frequency filters to the pixels or voxels outside the support at different stages of the iterative process (i.e. a smoothness constraint), OSS finds a balance between the hybrid input-output (HIO) and error reduction (ER) algorithms to search for a global minimum in solution space, while reducing the oscillations in the reconstruction. Both numerical simulations with Poisson noise and experimental data from a biological cell indicate that OSS consistently outperforms the HIO, ER-HIO and noise robust (NR)-HIO algorithms at all noise levels in terms of accuracy and consistency of the reconstructions. It is expected that OSS will find application in the rapidly growing CDI field, as well as other disciplines where phase retrieval from noisy Fourier magnitudes is needed. The MATLAB (The MathWorks Inc., Natick, MA, USA) source code of the OSS algorithm is freely available from http://www.physics.ucla.edu/research/imaging.

Determination of polymer morphology from the intensity measurement in the reciprocal space

Reciprocal space techniques based on scattering have been widely used for structure analysis of polymer materials. It would be more useful if the reciprocal methods can determine simultaneously the morphology of the polymer as those in the real space. However, the inverse problem, i.e., determining polymer morphology from the available scattering pattern, is very difficult because of the absence of the relative phase spectrum in the experimental measurements. Therefore, the retrieval of the missing phase spectrum plays a crucial role in this problem. Herein we theoretically propose that this can be accomplished by using phase-spectrum retrieval technique. We combined the Gerchberg–Saxton and hybrid input–output algorithms to determine the morphology of block copolymers from the corresponding scattering patterns in the reciprocal space. Numerical computations have been carried out to demonstrate the concept. It has also been shown that the algorithm is immune to noise. The demonstrated method could provide an extensive methodology for the investigation of polymer behaviors in the micro- and nano-scales by interconnecting real and reciprocal spaces.

Phasing diffraction data from a stream of hydrated proteins

We consider the problem of phase determination for continuous diffraction patterns obtained from a beam of identical, aligned large molecules (such as proteins), each coated with a layer of water or vitreous ice. Many laser-aligned molecules are assumed to lie within a wide continuous x-ray beam at any instant. An iterative phasing method is developed to extract the common target structure in three dimensions from diffraction patterns of these doped ice balls. Several measurements of the diffraction intensity in reciprocal space are needed. We found iteratively two boundaries (supports) (between protein and ice and the outer iceball support) by using the charge-flipping and multiple hybrid input-output algorithms, working with multiple sets of measured data. The approach is applied to simulated data from hydrated lysozyme proteins generated by the serial crystallography method of laser-aligned protein-beam diffraction proposed by Spence and Doak [Phys. Rev. Lett. 92, 198102 (2004)]. We consider also the effect of empty ice balls on the patterns. The algorithm can also be used to align images with different randomly chosen origins, so that the same embedded subunits overlap.

Hybrid projection-reflection method for phase retrieval.

The phase-retrieval problem, fundamental in applied physics and engineering, addresses the question of how to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently we identified two important methods for phase retrieval, namely, Fienup's basic input-output and hybrid input-output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the hybrid projection-reflection (HPR) algorithm, for solving phase-retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than 20 years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.