Open Access
Add to BibSonomy
Abstract
X-ray ptychography is a cutting edge imaging technique providing ultra-high spatial resolutions. In ptychography, phase retrieval, i.e., the recovery of a complex valued signal from intensity-only measurements, is enabled by exploiting a redundancy of information contained in diffraction patterns measured with overlapping illuminations. For samples that are considerably larger than the probe we show that during the iteration the bulk information has to propagate from the sample edges to the center. This constitutes an inherent limitation of reconstruction speed for algorithms that use a flat initialization. Here, we experimentally demonstrate that a considerable improvement of computational speed can be achieved by utilizing a low resolution sample wavefront retrieved from measured diffraction patterns as object initialization. In addition, we show that this approach avoids phase artifacts associated with large phase gradients and may alleviate the requirements on phase structure within the probe. Object initialization is computationally fast, potentially beneficial for bulky sample and compatible with flat samples. Therefore, the presented approach is readily adaptable with established ptychographic reconstruction algorithms implying a wide spread use.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
X-ray ptychography can be regarded as a combination of scanning X-ray transmission microscopy (STXM) and coherent diffraction imaging (CDI). STXM utilizes a lateral scan of the sample through a focused X-ray beam, which provides the transmission function of the specimen [1]. However, spatial resolution is limited by the focus size. CDI, on the other hand, employs an extended X-ray beam larger than the sample and exploits effective oversampling contained in diffraction patterns with additional constrains during algorithmic retrieval [2]. While spatial resolution provided by CDI is at least in principle wavelength limited, the availability of high quality, large X-ray beams practically limits the sample size.
X-ray ptychography integrates the benefits of STXM and CDI by scanning the specimen through a focal spot with overlapping illuminations, which realizes the oversampling required for stable phase retrieval [3–5]. This approach allows for samples that are larger than the focal spot while the achieved spatial resolutions are smaller than the focus size. Data analysis is performed by ptychographic reconstruction algorithms, these are able to retrieve the complex wave field of the sample as well as of the illuminating beam, usually referred to as the probe. Therefore, the quality of ptychographic reconstructions is nearly independent from the quality of the X-ray optics rendering this a lensless technique.
Evidently, the success of ptychographic experiments depends on reliable phase retrieval [6] and, thus, it does not come as a surprise that a wide variety of algorithms have been published. Examples include the extended ptychographic engine (ePIE) [7], 3PIE [8], maximum likelihood refinement [9], scaled gradient ptychography [10], multi-modal ptychography [11] or momentum accelerated ptychography [12]. These algorithms tend to reconstruct the complex wave field associated with the sample (and the probe), whereat the phase is wrapped to values modulus $2\pi$. Thus, appropriate phase unwrapping procedures [13,14] have to be used in order to combine 2D ptychography with 3D tomography [13,15,16]. Refractive ptychography [17,18], on the other hand, reconstructs a refractive object function directly and, thus, avoids corresponding phase wrapping artifacts.
These iterative algorithms start with initial guesses for the probe as well as the object and refine these guesses during the ptychographic reconstruction. Commonly, the probe can be either directly taken from previous scans or well estimated from experimental parameters. As this is generally not the case for the object, a flat initialization is usually used for the object’s complex wave field. Recently, spectral methods for ptychographic reconstruction were introduced and it was demonstrated that initial spectral estimates can be used to speed up convergence and avoid local minima [19,20].
Here, we show that the STXM information inherent to ptychography measurements can be exploited to construct a low-resolution estimate of the object’s wave field. This estimate is quick to compute and closely resembles the actual object’s wave field. We will demonstrate that using such an estimate as the initial guess is especially beneficial for bulky samples increasing the convergence speed of ptychographic reconstruction while avoiding artifacts associated with large phase gradients.
We will first collect elements of the theoretical basis for ptychography and reconstruct the phase signal of a bulky sample. Thereby, we will demonstrate that the bulk information, i.e., the information that a particular pixel is located in the center and not outside of the sample, propagates from the sample edges to the center during iterative reconstruction. Then, we will describe a procedure for the retrieval of the sample’s complex wave field directly from the measured diffraction patterns. This low resolution representation of the sample will then be used for object initialization in ptychographic reconstruction, which improves convergence speed and avoid phase artifacts associated with large phase gradients. Finally, we will show the reconstruction of a worst case sample (i.e., a non-bulky sample with structures smaller than the probe, which does not benefit from the avoided information propagation from the edges to the center of the sample) is not impaired by utilizing object initialization.
2. Ptychographic reconstruction
In the following, we will re-iterate elements of the frame work for refractive ptychography [18] for the convenience of the reader. In the X-ray regime the complex refractive index $n$ of a material is commonly expressed as
with $\delta$, the refractive index decrement and $\beta$, the absorption index. For samples that are sufficiently thin to avoid internal diffraction, the complex wave field $O({{\mathbf r}})$ at the object plane point ${{\mathbf r}} = (x,y)$ (Fig. 1) after transmission is given byThen the modeled amplitude $| \hat \psi _j ({{\mathbf q}})|$ is replaced with the amplitude of the measurements $\sqrt {\hat D_j({{\mathbf q}})}$ and the resulting wave field is propagated back to the object plane
The update strength is tuned by the parameters $\alpha$ and $\beta$. Initialization of the iterative procedure refers to the starting values for the object function $\tilde O({{\mathbf r}})$ and the probe $P({{\mathbf r}})$. Usually, a flat initialization for the object function is chosen, i.e.
The complex wave field of the probe $P({{\mathbf r}})$ is usually well characterized, which allows a realistic initialization.
For the experimental demonstration of our proposed approach we will reuse a previously published ptychographic scan of a micrometeorite [18]. The essential experimental parameters were as follows. The scan was carried out at the P06 beamline of PETRA III (DESY, Hamburg) [21,22] using the combined Micro- and Nanoprobe setup [23]. The X-ray beam with a photon energy of 18 keV was focused by two orthogonal Kirkpatrick–Baez mirrors to a probe size of 300 nm $\times$ 200 nm at the position of the sample. The micrometeorite with a diameter of about 80 $\mu$m was scanned in fly-mode over a field of view of 100 $\mu$m $\times$ 100 $\mu$m in steps of 200 nm resulting in 250,000 scan points. The diffraction patterns were recorded with an EIGER 500k (Dectris, Switzerland) located 8.75 m downstream of the sample with a dwell time of 1 ms. The central 128 $\times$ 128 pixels were used for ptychographic reconstruction yielding a effective pixel size of 62.8 nm. Iterative reconstruction was performed based on refractive ptychography as described above and by scaled gradient descent [10] with an additional momentum accelerated update [12] every second iteration step.
Figure 2 shows the result of refractive ptychographic reconstruction with flat initialization (Eq. (10)). Artifacts at the left and the top edges of the sample appear in the reconstruction of the phase (see Fig. 5(a) below), which can be attributed to large phase gradients. Clearly, the micrometeorite constitutes a bulky sample, which is about 400 times larger than the probe.
Fig. 2. Modulus (a) and phase (b) of the micrometeorite as retrieved by refractive ptychography. The field of view is 100 $\mu$m $\times$ 100 $\mu$m. The horizontal line in (b) indicates the position of the lines profiles in Fig. 3.
Figure 3 illustrates the behaviour of the reconstructed phase signal as a function of iteration steps for this bulky sample. It is evident that during iteration the bulk information propagates from the edges of the sample to the center, which can be explained as follows. Phase information is predominately encoded in the lateral offset of the diffraction patterns due to refraction or, equivalently, phase gradients. Thus, there is little difference between the exterior of the sample and its center as the phase gradients in both regions are negligible. Therefore, bulk phase information is most prevalent only at the edges of the sample. In each iteration step this information can only be shared between neighboring scan points, which leads to a large number of required iteration steps for bulky samples. Initialising the ptychographic reconstruction with a sample wave field that already carries bulk information can improve reconstruction speed considerably in contrast to a flat initialization.
Fig. 3. Horizontal line profiles (Fig. 2(b)) of the reconstructed phase signal through the middle of the micrometeorite as a function of iteration steps. With each iteration, the reconstructed phase improves from the edges of the sample towards the center.
3. Moment analysis
The low resolution estimate for the object’s complex wavefront $\mathscr{O}({{\mathbf R}_j}) = \mathscr{O}({{\mathbf R}_j}) e^{\mathrm {i}\phi ({{\mathbf R}_j})}$ with the object’s transmission $\mathscr{O}({{\mathbf R}_j})$ and the object’s phase $\phi ({{\mathbf R}_j})$ will be constructed from the moments $M_{uv}$ of the measured diffraction patterns $\hat D_j({{\mathbf q}})$. Please note that the low resolution object wavefront $\mathscr{O}({{\mathbf r}})$ relates to a convolution of the true object wavefront $O({{\mathbf r}})$ with the probe and that only the blurred object wavefront $\mathscr{O}({{\mathbf r}})$ can be directly retrieved from the diffraction patterns by moment analysis, which is given as [24–26]
In the following, we will be only interested in the three moments up to the first order. The transmission signal of the object corresponds to $M_{00}$ in a straight forward way:
Assuming that the second term in Eq. (14) can be neglected, the differential phase signals of the object in the area of illumination defined by the probe $P({{\mathbf r}}-{{\mathbf R}_j})$ at the scan point ${{\mathbf R}_j}$ can be estimated as
If the second term in Eq. (14) cannot be neglected, the above equations have to be corrected. For this, the influence of the object’s transmission and the probe’s differential phase signal must be determined. This can be done by either using the second term directly or – more conveniently – by calculating virtual diffraction patterns $\hat V_j({{\mathbf q}})$ provided by the pure absorption signal of the object according to
The moments $M_{10}$ and $M_{01}$ of these virtual diffraction pattern correspond exactly to the second term in Eq. (14). Thus, the differential phase signals can be estimated in this case, if a reasonably approximation of the probe $P$ is available from previous reconstructions or focus parameters, as
4. Wavefront retrieval
In order to retrieve the phase image $\phi ({{\mathbf R}_j})$ from the estimated differential phase images $\phi _x({{\mathbf R}_j})$ and $\phi _y({{\mathbf R}_j})$ we use the non-iterative, boundary-artifact-free wavefront reconstruction presented in [28]. This approach starts with constructing an antisymmetric extension of the inputs
Then the integrated image is retrieved by calculating
Fig. 4. Non-iterative wavefront retrieval of the sample’s complex wave field used for subsequent initialization. The moment analysis of diffraction patterns provide the absorption (a), horizontal (b) and vertical phase gradients (c). The combination of the absorption image (a) and the non-iteratively retrieved phase (d) provide a complex wave field of the sample with low spatial resolution.
5. Ptychographic reconstruction with object initialization
With the availability of the low resolution versions of the absorption and the phase image the initialization for the sample’s complex wave field is given by
Figure 5 shows the resulting phase signal of the micrometeorite after 1000 iteration steps for flat initialization in panel (a) and object initialization in panel (b). As expected the images are fairly similar. However, in the reconstruction with flat initialization (a) phase artifacts are present at the left and the top border of the sample coinciding with the locations of the largest phase gradients visible in Fig. 4 (b) and (c). These artifacts are absent for ptychographic reconstruction with object initialization (insets in Fig. 5(b)) rendering the use of phase unwrapping procedure unnecessary.
Fig. 5. The retrieved phase signals of the micrometeorite with flat initialization (a; identical to data shown in Fig. 2(b)) and object initialization (b) are broadly similar as expected. The insets show magnified regions, where phase artifacts occur with flat initialization and are located at positions of the largest phase gradients (cmp. to Fig. 4(b) and (c)). These artifacts are avoided by using object initialization as shown by the inset in (b).
In order to demonstrate the versatility of the object initialization for ptychographic reconstructions, the proposed approach was further applied to a ptychographic data set which was acquired at the cSAXS beamline of the Swiss Light Source, PSI, Switzerland [31]. Here a catalytic particle with a diameter of 20 $\mu$m was scanned with a photon energy of 6.2 keV over 2344 positions in a spiral trajectory. The average step size was 0.8 $\mu$m covering a field of view of 50 $\mu$m by 30 $\mu$m. The pixel size of the reconstruction was 29 nm. More details of the setup and scan procedure can be found in the original work [32] and the data set is available online [30]. One of the goals of Odstrčil et al was to show that purposefully structured probes can improve ptychographic reconstructions. However, the data set used in the following (named "FCC_particle_FZP_11_dataset_id1.mat") was acquired with a Fresnel zone plate without additional wavefront modulation, but since the object was placed out of the beam focus, the probe still showed a noticeable phase gradient.
Figure 6 summarizes the results of ptychographic reconstruction of this data set in the context at hand. Figure 6(a) shows the estimation of the object’s phase without appropriate correction for the probe’s phase gradient (i.e., using Eqs. (15) and (16)). Figure 6(b) demonstrates the utilization of the appropriate correction (i.e., using Eqs. (18) and (19)). Figure 6(c) displays the result of refractive pytchographic reconstruction and flat initialization after 400 iteration steps (equivalent to Fig. 5(a) in [32]). Figure 6(d) shows the result of ptychographic reconstruction with object initialization, i.e. the phase estimation in panel (b) plus the corresponding absorption signal was used to initialize the iterative minimization. Figure 6(d) constitutes the final reconstruction with the lowest value of the cost function among the phase signals shown in Fig. 6 and, thus, can be taken as the gold standard for judging the accuracies of the low resolution phase signals in Figs. 6(a) and (b). Apparently, the object’s phase shift was significantly overestimated without using the appropriate correction (Fig. 6(a)), which is congruent with the expectation from Eq. (14) as a non negligible structured probe phase derivative ($\partial _x \xi ({{\mathbf r}}$) only adds to the measured moment leading to an overestimation. Further, the low resolution object’s phase estimation with appropriate correction shown in Figure 6(b) matches the final phase reconstruction in Figure 6(d) much better than the phase signal shown in Figure 6(a). In addition, the resulting ptychographic reconstruction with object initialization (Fig. 6(d)) has a remarkable visual similarity with Fig. 5(d) in [32] even without using a structured probe. While we make no claim about the spread of convergence between repeated measurements as in [32], the results here hint a potential benefit for combining ptychography with other X-ray techniques that would suffer from structured probes, such as X-ray fluorescence.
Fig. 6. Estimated phase and ptychographic reconstruction of a fluid catalytic cracking catalyst particle. (a) Estimated phase without the necessary correction for the probe influence (i.e., using Eqs. (15) and 16). (b) Estimated phase with the necessary correction for the probe influence (i.e., using Eqs. (18) and 19). (c) Ptychographic reconstruction of the object’s phase signal with flat initialization. (d) Ptychographic reconstruction of the object’s phase signal with object initialization, which can also be taken as a gold standard for judging the accuracies of panels (a) and (b). The data set was taken from [30].
Using object initialization (Eq. (23)) implies that the bulk phase information associated with the sample is already present and iteration steps that are associated with the propagation of information from the edges to the center of the sample are skipped during ptychographic reconstruction. Figure 7 demonstrates that this saves several hundred iteration steps in case of the micrometeorite or even provides superior convergence over the first 400 iteration steps in case of the fluid catalytic cracking catalyst particle. Therefore, ptychographic reconstruction with object initialization can improve iteration speed considerably.
Fig. 7. Comparison of the logarithmic value of the cost function $L$ (Eq. (5)) between flat and object initialization as a function of iteration steps for (a) the micrometeorite data and (b) the fluid catalytic cracking catalyst particle. Ptychographic reconstruction with object initialization converges about 200 iteration steps faster in case of (a) and provides superior convergence over the course of 400 iterations in case of (b).
Up to now, the samples included in this study were considerably bulky and, thus, well suited to benefit from ptychographic reconstruction with object initialization. In order to investigate the performance for samples that are more challenging in the present context, we have used numerical simulations of a Siemens star pattern. In this case, the sample is flat with structures smaller than the probe and, thus, there is no bulk information to propagate during iteration.
The simulated Siemens star was sampled on a $213\times 213$ grid and provided a minimum transmission of 0.8 and a phase shift of -1.2 rad (Fig. 8(b)), which corresponds to Au structures with a height of 650 nm imaged at a photon energy of 8.2 keV. The probe had a Gaussian-like shape with a full width half maximum of 7 pixels, which is markedly larger than the smallest sample features. The observable diffraction patterns were calculated according to Eq. (4) and wavefront retrieval was performed as described above.
Fig. 8. Ptychographic reconstruction of a Siemens star with numerical simulations. (a) Low-resolution phase estimate. (b) Retrieved phase distribution using object initialization. (c) Comparison of the logarithmic value of the cost function $L$ (Eq. (5)) between flat and object initialization as a function of iteration steps. Thick lines are the average values of the cost function of 6 repeated reconstructions (thin lines). While object initialization performs better than flat initialization in the beginning, both take about the same number of iteration steps to converge.
Although the estimated wave field for the object used for initialization has insufficient resolution to sample the Siemens star (Fig. 8(a)), the reconstructed object’s phase distribution (Fig. 8(b)) shows that structures smaller than the probe are still reliably retrieved using object initialization. This demonstrates that object initialization is compatible even with challenging samples. However, the comparison of the cost function $L$ between flat and object initialization (Fig. 8(c)) illustrates only a negligible difference in terms of convergence speed.
6. Conclusion
We have demonstrated that the computational speed of ptychographic reconstruction approaches, which use a flat initialization for the object’s complex wave field, is inherently limited for bulky samples. This is due to the fact that during the iterations the bulk phase information has to propagate from the edges of the object to its center. By instead using object initialization, the reconstruction speed is considerably increased as the bulk phase information is already present. In addition, we have shown that object initialization can avoid phase artifacts associated with large phase gradients and lessens the necessity for phase unwrapping or structured probes.
The input data for constructing the object initialization is readily accessible via moment analysis of the measured diffraction patterns in most ptychographic scans. The corresponding algorithm for retrieving the complex wave field is – compared to the ptychographic reconstruction – fast. In addition to its speed, object initialization is readily compatible with a broad range of ptychographic reconstruction algorithms. Taken all together, this renders the presented approach attractive for most ptychography applications.
Acknowledgments
We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research was carried out at the PETRA III beamline P06. We thank Stijn van Malderen and Jan Garrevoet for assistance in using P06. This research was supported in part through the Maxwell computational resources operated at Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data for the micrometeorite experiment and for the Siemens star simulations are not publicly available at this time but may be obtained from the authors upon reasonable request. Data underlying the fluid catalytic cracking catalyst particle experiments are not owned by the authors but are available online [30]. A notebook demonstrating how to calculate the object estimate is available under [33].
References
1. A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010). [CrossRef]
2. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010). [CrossRef]
3. H. M. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004). [CrossRef]
4. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008). [CrossRef]
5. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12(1), 9–17 (2018). [CrossRef]
6. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2014). [CrossRef]
7. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef]
8. A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29(8), 1606–1614 (2012). [CrossRef]
9. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14(6), 063004 (2012). [CrossRef]
10. P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20(23), 25914–25934 (2012). [CrossRef]
11. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013). [CrossRef]
12. A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4(7), 736–745 (2017). [CrossRef]
13. M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 21345–21357 (2011). [CrossRef]
14. M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3(1), 014005 (2015). [CrossRef]
15. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010). [CrossRef]
16. A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85(2), 020104 (2012). [CrossRef]
17. S. Chowdhury, M. Chen, R. Eckert, D. Ren, F. Wu, N. Repina, and L. Waller, “High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images,” Optica 6(9), 1211–1219 (2019). [CrossRef]
18. F. Wittwer, J. Hagemann, D. Brückner, S. Flenner, and C. G. Schroer, “Phase retrieval framework for direct reconstruction of the projected refractive index applied to ptychography and holography,” Optica 9(3), 295–302 (2022). [CrossRef]
19. S. Marchesini, Y.-C. Tu, and H.-T. Wu, “Alternating projection, ptychographic imaging and phase synchronization,” Appl. Comput. Harmon. Analysis 41(3), 815–851 (2016). [CrossRef]
20. L. Valzania, J. Dong, and S. Gigan, “Accelerating ptychographic reconstructions using spectral initializations,” Opt. Lett. 46(6), 1357–1360 (2021). [CrossRef]
21. C. G. Schroer, M. Seyrich, M. Kahnt, S. Botta, R. Döhrmann, G. Falkenberg, J. Garrevoet, M. Lyubomirskiy, M. Scholz, A. Schropp, and F. Wittwer, “PtyNAMi: Ptychographic Nano-Analytical Microscope at PETRA III: interferometrically tracking positions for 3D x-ray scanning microscopy using a ball-lens retroreflector,” in SPIE Opt. Eng. + Appl., (2017), September 2017, p. 13.
22. A. Schropp, R. Dohrmann, S. Botta, D. Bruckner, M. Kahnt, M. Lyubomirskiy, C. Ossig, M. Scholz, M. Seyrich, M. E. Stuckelberger, P. Wiljes, F. Wittwer, J. Garrevoet, G. Falkenberg, Y. Fam, T. L. Sheppard, J. D. Grunwaldtd, and C. G. Schroer, “PtyNAMi: Ptychographic nano-analytical microscope,” J. Appl. Crystallogr. 53(4), 957–971 (2020). [CrossRef]
23. A. Schropp, D. Brückner, J. Bulda, G. Falkenberg, J. Garrevoet, J. Hagemann, F. Seiboth, K. Spiers, F. Koch, C. David, M. Gambino, M. Veselý, F. Meirer, and C. G. Schroer, “Full-field hard X-ray microscopy based on aberration-corrected Be CRLs,” Proc. SPIE 11112, 1111208 (2019). [CrossRef]
24. P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80(4), 043813 (2009). [CrossRef]
25. O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’L, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11(12), 123016 (2009). [CrossRef]
26. P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105(2), 024102 (2014). [CrossRef]
27. P. Modregger, M. Kagias, S. C. Irvine, R. Brönnimann, K. Jefimovs, M. Endrizzi, and A. Olivo, “Interpretation and utility of the moments of small-angle x-ray scattering distributions,” Phys. Rev. Lett. 118(26), 265501 (2017). [CrossRef]
28. P. Bon, S. Monneret, and B. Wattellier, “Noniterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt. 51(23), 5698–5704 (2012). [CrossRef]
29. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007). [CrossRef]
30. M. Odstrčil and M. Holler, “X-ray dataset for Illumination improvements for high-resolution ptychography,” http://dx.doi.org/10.5281/zenodo.2639759, (2019).
31. M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4(1), 3857 (2014). [CrossRef]
32. M. Odstrčil, M. Lebugle, M. Guizar-Sicairos, C. David, and M. Holler, “Towards optimized illumination for high-resolution ptychography,” Opt. Express 27(10), 14981–14997 (2019). [CrossRef]
33. F. Wittwer, D. Brueckner, and P. Modregger, “Notebook for-- Ptychographic reconstruction with object initialization,” Zenodo (2022), https://doi.org/10.5281/zenodo.2639759
