On inverse problem with phase retrieval for an inclined plane in the parabolic approximation

X-ray phase contrast tomography (XPCT) is widely used for 3D imaging of objects with weak contrast in X-ray absorption index but strong contrast in refractive index decrement. To reconstruct an object imaged using XPCT, phase retrieval algorithms are first used to estimate the X-ray phase projections, which is the 2D projection of the refractive index decrement, at each view. Phase retrieval is followed by refractive index decrement reconstruction from the phase projections using an algorithm such as filtered back projection (FBP). In practice, phase retrieval is most commonly solved by approximating it as a linear inverse problem. However, this linear approximation often results in artifacts and blurring when the conditions for the approximation are violated. In this paper, we formulate phase retrieval as a non-linear inverse problem, where we solve for the transmission function, which is the negative exponential of the projections, from XPCT measurements. We use a constraint to enforce proportionality between phase and absorption projections. We do not use constraints such as large Fresnel number, slowly varying phase, or Born/Rytov approximations. Our approach also does not require any regularization parameter tuning since there is no explicit sparsity enforcing regularization function. We validate the performance of our non-linear phase retrieval (NLPR) method using both simulated and real synchrotron datasets. We compare NLPR with a popular linear phase retrieval (LPR) approach and show that NLPR achieves sharper reconstructions with higher quantitative accuracy.

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

International Journal of Optics ◽

10.1155/2020/3481830 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Jian-Jia Su ◽

Chung-Hao Tien

Keyword(s):

Inverse Problem ◽

Phase Retrieval ◽

Early Stage ◽

Error Reduction ◽

Input Output ◽

Nonlinear Inverse Problem ◽

Sparse Dictionary Learning ◽

The Arts ◽

Step Algorithm ◽

Superior Image Quality

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.

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Lensless Reflection Imaging of Obliquely Illuminated Objects I: Choosing a Domain for Phase Retrieval and Ptychography

Symmetry ◽

10.3390/sym13081439 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1439

Author(s):

Igor A. Artyukov ◽

Nikolay L. Popov ◽

Alexander V. Vinogradov

Keyword(s):

Inverse Problem ◽

Phase Retrieval ◽

Extreme Ultraviolet ◽

X Rays ◽

Reflection Mode ◽

Optical Elements ◽

Diffraction Integral ◽

Diffraction Patterns ◽

Set Up ◽

Object Material

Ptychography is a lensless imaging technology that is validated from hard X-rays to terahertz spectral range. It is most attractive for extreme ultraviolet (EUV) and X-rays as optical elements are expensive and often not available. Typically, the set up involves coherently illuminated object that directs the scattered radiation normally to detector which is parallel to the object plane. Computer processing of diffraction patterns obtained when scanning the object gives the image, more precisely, the distribution of intensity and phase on its surface. However, this scheme is inefficient for EUV and X-rays due to poor reflectivity and low penetration in all materials. Reflection mode ptychography solves the problem if illumination angles do not exceed the critical angle of object material. Changing the geometry of experiment changes physical and mathematical model of image formation. Including: diffraction integral describing beam propagation from object to detector, inverse problem, optimization of object illumination angle, position and orientation of detector, choosing size and grid of coordinate and frequency computer domains. This paper considers the wavefield scattered to detector by obliquely illuminated object and determines a domain for processing of obtained scans. Solution of inverse problem with phase retrieval and resulting numerical images will be presented in the next paper.

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Quantitative analysis by reconstruction of HREM focal series

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100171432 ◽

1994 ◽

Vol 52 ◽

pp. 740-741

Author(s):

W. Coene ◽

A. Thust ◽

M. Op de Beeck ◽

D. Van Dyck

Keyword(s):

Phase Retrieval ◽

Image Plane ◽

Initial Guess ◽

Recursive Least Squares ◽

Objective Lens ◽

Electron Wave ◽

Electron Sources ◽

Original Algorithm ◽

Coherent Field ◽

Direct Interpretation

Compared to conventional electron sources, the use of a highly coherent field-emission gun (FEG) in TEM improves the information resolution considerably. A direct interpretation of this extra information, however, is hampered since amplitude and phase of the electron wave are scrambled in a complicated way upon transfer from the specimen exit plane through the objective lens towards the image plane. In order to make the additional high-resolution information interpretable, a phase retrieval procedure is applied, which yields the aberration-corrected electron wave from a focal series of HRTEM images (Coene et al, 1992).Kirkland (1984) tackled non-linear image reconstruction using a recursive least-squares formalism in which the electron wave is modified stepwise towards the solution which optimally matches the contrast features in the experimental through-focus series. The original algorithm suffers from two major drawbacks : first, the result depends strongly on the quality of the initial guess of the first step, second, the processing time is impractically high.

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Limitations to the Spacing Effect

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie) ◽

10.1027/1618-3169.52.4.257 ◽

2005 ◽

Vol 52 (4) ◽

pp. 257-263 ◽

Cited By ~ 22

Author(s):

Peter P. J. L. Verkoeijen ◽

Remy M. J. P. Rikers ◽

Henk G. Schmidt

Keyword(s):

Free Recall ◽

Phase Retrieval ◽

Incidental Learning ◽

Factor Model ◽

Recall Performance ◽

Spacing Effect ◽

Word List ◽

Intentional Learning ◽

Study Phase ◽

Learning Condition

Abstract. The spacing effect refers to the finding that memory for repeated items improves when the interrepetition interval increases. To explain the spacing effect in free-recall tasks, a two-factor model has been put forward that combines mechanisms of contextual variability and study-phase retrieval (e.g., Raaijmakers, 2003 ; Verkoeijen, Rikers, & Schmidt, 2004 ). An important, yet untested, implication of this model is that free recall of repetitions should follow an inverted u-shaped relationship with interrepetition spacing. To demonstrate the suggested relationship an experiment was conducted. Participants studied a word list, consisting of items repeated at different interrepetition intervals, either under incidental or under intentional learning instructions. Subsequently, participants received a free-recall test. The results revealed an inverted u-shaped relationship between free recall and interrepetition spacing in both the incidental-learning condition and the intentional-learning condition. Moreover, for intentionally learned repetitions, the maximum free-recall performance was located at a longer interrepetition interval than for incidentally learned repetitions. These findings are interpreted in terms of the two-factor model of spacing effects in free-recall tasks.

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