## Abstract

Near-field X-ray ptychography has recently been proposed and shown to be able to retrieve a sample’s complex-valued transmission function from multiple near-field diffraction images each with a lateral shift of the sample and with a structured (by a diffuser) illumination [Stockmar *et al.* Sci Rep. **3** (2013)]. In this paper, we undertake the first investigation - via numerical simulation - of the influence of the sampling and step size of the lateral shifts, the diffuser structure size, and the propagation distance on the reconstruction of the sample’s transmission function. We find that for a gold Siemens star of thickness 750 nm with typical experimental parameters, for a successful reconstruction - given a theoretical minimum of four required measurements per imaged pixel - at least six diffraction images are required.

© 2015 Optical Society of America

## 1. Introduction

X-ray phase-contrast imaging is an increasingly popular technique used to investigate weakly absorbing samples, across a broad variety of fields, for instance in materials science [1] and biomedicine [2]. Phase contrast imaging can be achieved via a number of different methods such as grating interferometry [3], crystal analyzer [4] or through free-space propagation [5, 6].

In propagation-based phase-contrast imaging (also known as in-line holography or lensless imaging), the sample is placed in the X-ray beam without any further optics between the sample and the detector, and the recorded intensities are diffraction images. For shorter propagation distances, the intensity will directly show the contours of the sample, which is sufficient in some imaging applications. However, in many other applications, it is necessary to retrieve the phase of the sample from these intensity measurements in the detector plane.

Ptychography provides a solution to the phase problem that utilizes lateral shifts of the sample (alternatively of the illumination function or probe) to create multiple overlapping diffraction images [7, 8]. This shifting process creates images with varying information about the sample, and the overlap provides sufficient redundancy in the recorded diffraction images. The sample’s transmission function and the illumination function can then be recovered simultaneously in an iterative process [9].

In near-field ptychography, the probe illuminates most of the reconstructed field of view (FOV), i.e the probe illuminates an area larger than the extent of the sample in a global tomography case, and thus the Fresnel numbers are significantly higher than in far-field ptychography. Additionally, in near-field ptychography, in order to provide sufficient diversity in the measurements, the wavefront is disturbed by the use of a diffuser [10]. Near-field ptychography offers the advantages over far-field ptychography that larger FOVs can be imaged with fewer diffraction images, and with weaker requirements on the detector dynamic range and beam coherence [10].

In [10], the authors propose the method of near-field ptychography and demonstrate its use in the reconstruction of a Siemens star with a synchrotron source. This method was subsequently used in [11] to reconstruct a strongly absorbing and phase shifting sample, a uranium sphere. In [12], the authors use near-field ptychography in conjunction with computed tomography to access the three-dimensional nano-structure of a solid oxide fuel cell. In this paper, we consider a number of important aspects of near-field ptychography that have not been investigated in these previous papers. We quantify the performance of near-field ptychography via numerical simulation in terms of the number of diffraction images, and consequently total dose required, and also the sampling arrangement of these diffraction images. Similarly, we consider the performance with respect to the propagation distance between the sample and the detector, as well as the distance between the diffuser and the sample, and the structure size of the diffuser.

The remainder of this paper is structured as follows: in Section 2, the near-field ptychography algorithms and parameters as well as the error metrics are defined. The simulation results for these algorithms and conditions are presented in Section 3. Finally, conclusions are drawn in Section 4.

## 2. Methodology

In near-field ptychography, we aim to estimate both the probe or illumination function *p*(*x*, *y*) and the transmission function of the sample *T*(*x*, *y*). The exit wave *ψ _{m}*(

*x*,

*y*) after the sample for the m

*diffraction image is the product of the probe*

^{th}*p*(

*x*,

*y*) and the transmission function of the sample

*T*(

*x*,

*y*) [8]

*x*,

_{m}*y*) is the shift of the sample for the

_{m}*m*diffraction image. The complex-valued transmission function

^{th}*T*(

*x*,

*y*) can be expressed as where

*A*(

*x*,

*y*) is the magnitude and

*ϕ*(

*x*,

*y*) the phase of the transmission function, and $j=\sqrt{-1}$.

The measured intensities in the near-field *I _{m}*(

*x*,

*y*) are calculated from the Fresnel integral such that [13]

*D*is the Fresnel free-space propagator for a distance

_{z}*z*, and is defined by

*u*,

*v*) is the reciprocal space co-ordinate,

*ℱ*is the Fourier transform operator, and

*k*= 2

*π*/

*λ*where

*λ*is the X-ray wavelength. With Fresnel propagation, the intensity at a given point (

*x*,

*y*) in the detector plane is predominantly determined by the wavefront within a Fresnel zone $\sqrt{\lambda z}$ of (

*x*,

*y*) in the sample plane [14].

The outputs of the ptychographic reconstruction are the estimates of the probe *p̂*(*x*, *y*) and the transmission function *T̂*(*x*, *y*) = *Â*(*x*, *y*)exp[*jϕ̂*(*x*, *y*)]. In order to evaluate the fidelity of the reconstruction, we calculate the error in the estimate of the transmission function *T̂*(*x*, *y*) with respect to the known transmission function *T*(*x*, *y*), although a number of numerical steps have to be implemented before we can compare the two. Firstly, a residual phase ramp may arise in the reconstructed sample transmission function [10], and this is numerically removed. Secondly, the reconstructed and true sample transmission functions may be offset with respect to each other, and we calculate this relative shift (*x*_{0}, *y*_{0}) by finding the position of the maximum of the cross-correlation function *C*(*x*, *y*)

*x*

_{0},

*y*

_{0}) using the Fourier shift theorem. Finally, the reconstructed

*T̂*(

*x*,

*y*) may differ from

*T*(

*x*,

*y*) by a complex scaling factor, here

*α*. We determine this complex scaling factor using the methodology of Fienup [15] where the sum here (and subsequently in Eq. 7) is over all the pixels of the transmission function. We then remove this factor by letting

*T̂*(

*x*,

*y*) =

*T̂*(

*x*,

*y*)/

*α*. After these numerical preliminaries, we are able to calculate the error in the reconstruction of the sample transmission function, both of the magnitude

*Â*(

*x*,

*y*) and phase

*ϕ̂*(

*x*,

*y*). We use as a performance criterion in this paper the normalized root-mean-square error (rmse)

*E*and

_{A}*E*for the reconstructed magnitude and phase respectively, given by [15]

_{ϕ}*E*is invariant to phase offsets [16]. The quadrature sum rmse

_{ϕ}*E*is then defined as

As a starting point for our simulations, we use the experimental conditions and reconstruction parameters of [11], which are summarized in Table 1 and are used as the default in all simulations in this paper unless indicated otherwise. This experiment was undertaken using a magnified cone beam geometry. We convert this experimental configuration into a parallel beam geometry, and use the parallel beam equivalent parameters in the simulations. The geometric magnification is given by [13]

where*z*

_{1}is the distance between the source and the sample, and

*z*

_{2}is the distance between the sample and detector. The effective propagation distance

*z*for a parallel beam between the sample and the detector is given by [13] The effective pixel size Δ

*x*due to the cone beam magnification

*M*is where Δ

*s*is the pixel size of the detector. The propagation distance and pixel scale listed in Table 1 are the effective parallel beam equivalents.

The parallel beam near-field ptychography set-up we use in this paper is shown in Fig. 1. The parallel beam X-rays enter the diffuser, and propagate a distance *z _{d−s}* to the sample, which is shifted by a known amount in

*x*and

*y*. The X-rays propagate a further distance

*z*between the sample and the detector, where the

*m*near-field intensity

^{th}*I*(

_{m}*x*,

*y*) is recorded for a lateral shift (

*x*,

_{m}*y*). The exact parallel beam equivalent diffuser-sample distance from the experiment is unknown, since the diffuser appears before the mirrors used to focus. Unless otherwise stated, the diffuser-sample distance is set to 0.5 m in all the simulations. We only consider in this paper the fully coherent case (i.e. monochromatic radiation and full spatial coherence), with energy of 16.96 keV.

_{m}We perform the numerical calculation of the near-field diffraction images in this paper with the pyXSFW simulation framework [17], an object-oriented Python code that has been used extensively for grating-based phase-contrast X-ray imaging (eg [17, 18]) as well as speckle tracking [19]. In the simulations, the flux is simulated as a Poisson process, with the mean number of expected photons per pixel (for a flat field image) as shown in Table 1. The detector PSF is assumed to be Gaussian with a full width at half maximum (FWHM) of 2 pixels.

The diffuser used in previous experiments for near-field ptychography has been cardboard [11, 12]. Here we use sandpaper as diffuser as previously done for speckle-based imaging [19] because the structure (and consequently speckle) size can be controlled by selecting the grit designation. The sandpaper diffuser is simulated as a 2-layer structure of total thickness 300 μm. The first layer consists of cellulose, chemical formula C_{6}H_{14}O_{5} (density 1.50 g cm^{−2}), which is 243 μm thick. The second layer consists of randomly distributed SiC spheres (density 3.21 g cm^{−2}), with radii defined by the grit designation of the sandpaper as shown in Table 2, and the remainder of the layer is again cellulose. The volume of the SiC and cellulose in this layer are assumed to be equal in these simulations, for all grit designations. Also, we assume here for each grit designation that all of the SiC grains have radii equal to the mean radii as shown in Table 2. In addition to these grit designations, we also simulate some even smaller grain sizes to establish the relationship between the rmse and grain size.

The flat field speckle images, i.e. with the diffuser but without any sample, are shown in Fig. 2 for three different sandpaper grit sizes for the default parameters. The sample in the experiment was a 44 μm diameter uranium-molybdenum sphere (7% by weight is Mo), with a 1 μm thick UO_{2} shell [11]. The diffraction images with P3000 sandpaper are shown in Fig. 3(a) for the uranium sphere and in 3(b) for the gold Siemens star used in Section 3.

The speckle visibility *v* in the monochromatic case here can be defined for these flat field images in the detector plane as [19]

*σ*is the standard deviation and

*Ī*(

*x*,

*y*) the mean of the flat field image. The visibility of the flat field image is largely insensitive to the propagation distance

*z*(with the default diffuser-sample distance of 0.5 m) as shown in Fig. 4(a) (blue curve), whereas the visibility is greatly increased with the larger diffuser-sample distance as seen in the green curve. The visibility of the flat field images increases with increasing radius for radii less than 5 μm. However, for larger grain sizes, we have fewer grains, and we no longer have speckle images but are in effect imaging the individual SiC grains (e.g. the 50.0 μm radius only has 2 SiC grains in the FOV of the detector), and consequently the visibility is greatly reduced.

The size of the speckles we generate can be calculated from the autocorrelation of the flat field image [19, 21]. We choose to use the FWHM of a Gaussian fit to this autocorrelation function. The FWHM of the autocorrelation of the speckles for the P3000 diffuser, which we use as the default diffuser in this paper, is 3.6 μm for the default propagation distance of 0.0468 m. This is approximately equal to the radius of the grains (3.5 μm) for this grit designation, consistent with [22] that the size of the speckles is the size of the scatterers.

The probe function can be decomposed into a number *n* of mutually incoherent (probe) modes to incorporate partial coherence, such that [23]

As per [11], the magnitude of the sample transmission function is forced to remain within the interval [0.1, 1.2] to avoid numerical instabilities. The flat field image is used to form the initial estimate of the probe, while the initial estimate of *T*(*x*, *y*) is an array of zeros.

The experiment of [11] used sixteen diffraction images, pseudo-randomly arranged into four quadrants with random offsets as shown in Fig. 5(a), and we use these exact positions as the default in the simulations here unless stated otherwise. We do also consider the following scan patterns that are commonly used in far-field ptychography: raster scan 5(b), the round scan pattern 5(c) [26], the Fermat spiral 5(d) [27], and a purely random scan pattern 5(e). For these four additional scans, the 16 positions are arranged within an extent of 15 μm × 15 μm, as per the experimental scan.

In Fig. 7(a), we show the magnitude, and in 7(b), the phase of the reconstructed uranium sphere, as well as the reconstructed probe magnitude 7(c) and unwrapped phase 7(d). We do not unwrap the phase (which is bound between *π* and −*π*) before calculating the rmse. The sample transmission function’s magnitude and phase compare visibly with the experimental results (Fig. 2 of Ref [11]). The magnitude of the reconstructed probe |*p*(*x*, *y*)| shows the expected near-field speckles seen in the flat field images of Fig. 2. The errors in reconstructing the sample transmission function magnitude and phase for the uranium sphere are shown in Fig. 7(e) and 7(f) respectively. The error in the magnitude reconstruction primarily occurs at the edge of the sphere and the edge of the UO_{2} shell, which shows that we cannot reconstruct the highest spatial frequencies (the edges). The phase reconstruction, as well as at the edges of the sphere and UO_{2} shell, has its largest error at the center of the sphere, where the phase is almost flat and therefore less well constrained. The magnitude rmse is 0.0340 and the phase rmse 0.0316 in this case. In single-shot speckle tracking, spatial frequencies higher than the speckle size will not typically be resolvable [28]. With the shifting of the sample in near-field ptychography, higher spatial frequencies can be resolved. The highest resolved spatial frequency is a complex relationship of the speckle size, Fresnel zone width (through the propagation distances and energy) and the step sizes of the sampling pattern, and is still under investigation. Ultimately, the PSF of the detector and the effective pixel size will limit the spatial resolution.

## 3. Simulation results

#### 3.1. Siemens star as sample

The uranium sphere was used to compare the numerical simulations against the experimental results of [11]. However, the convergence of the reconstruction with the uranium sphere is not assured due to the introduction of phase vortices [11]. Instead, we choose as our sample in the remainder of this paper a gold Siemens star (density 19.30 g cm^{−2}) of diameter 70 μm, of thickness 750 nm with three rings and a minimum feature size of four pixels (300 nm), which yields more stable convergence and hence gives a clearer understanding of the relationship of the rmse to the parameters under consideration. The reconstructed magnitude and phase for the Siemens star are shown in Fig. 8(a) and 8(b) respectively. The rmse of the magnitude is 0.0126, and for the phase 0.0085.

#### 3.2. Convergence

We evaluate the convergence of the DM algorithm by calculating the rmse in both amplitude and phase versus the number of iterations as shown in Fig. 9. The first 1000 iterations are with one probe mode, and the subsequent 300 iterations are with three probe modes. We see that the 1000 iterations in the first step that we have adopted from [11] are more than sufficient for convergence (in fact, 500 iterations would be sufficient), and we use 1000 iterations for the first step in all the simulations in this paper. It is interesting to note that the magnitude rmse actually increases after 200 iterations, although the phase rmse is still decreasing at this point. The addition of the extra probe modes gives a significant improvement to the sample transmission function estimate, and we use an additional step of 300 iterations with three probe modes for all the simulations in this paper.

#### 3.3. Need for the diffuser

We quantify the improvement to near-field ptychography with a diffuser, compared to without, for the default parameters. Without the diffuser and no beam artifacts, each diffraction image is exactly the same except that it is laterally shifted. Since the diffraction images are the same (except due to photon noise), the ptychography problem here reduces to a single phase retrieval loop. Without additional constraints, this problem is underdetermined and a successful reconstruction cannot be expected. The results of an attempt are shown in Fig. 10(a) and 10(b) for the magnitude and phase respectively, and are visibly much worse than with the diffuser (Fig. 8(a) and 8(b)). The rmse without the diffuser is 0.1904 in magnitude and 0.0947 in phase, an order of magnitude worse in both cases than with the diffuser, confirming the criticality of the modulation of the incident illumination for successful phase retrieval. With the addition of the diffuser, each diffraction image is different - it is modulated by the speckle pattern - and therefore adds diversity to the measurements. Figure 10 shows that there are certain spatial frequencies (the two inner rings of the Siemens star) that cannot be recovered with this single (underdetermined) phase retrieval loop, and which can with the use of the diffuser as seen in Fig. 8. The result in Fig. 10(b) is typical of near-field free-space propagation imaging, due to the zero crossings of the contrast transfer function, certain spatial frequencies are not transported well [29].

#### 3.4. Step size

We vary the maximum step size (which results in a varying extent of the sampling pattern) of the random sampling pattern of Fig. 5(e) whilst keeping the same relative pattern (asterism), and calculate the rmse in magnitude, phase and quadrature sum, which is shown in Fig. 11. Given that the wavefront at the sample plane is more-or-less constant over the speckle size, lateral shifts of less than the speckle size will provide little additional diversity to the measurements. From Eq. 1, the probe and sample are in principle symmetric, so, analogously, step sizes less than feature sizes in the sample would also provide little additional diversity to the measurements. However, real samples will invariably have edges, which introduce diversity into the measurements. For the smallest extent plotted (2 μm), the step sizes between the images are all less than the speckle size (the radius of the grains of the P3000 diffuser here is 3.5 μm and the FWHM of the autocorrelation of the flat field image is 3.6 μm), and the rmse is high. As the step sizes increase, we get more diversity in the measurements and the rmse decreases. There is a very broad minimum to the curve between approximately 16 and 80 μm. For maximum step sizes larger than 80 μm, the rmse increases as the diffraction images start to become truncated by the finite extent of the detector (154 μm × 154 μm). The minimum extent of 16 μm is approximately four times the speckle size.

#### 3.5. Scan pattern

The magnitude, phase and quadrature sum rmse are shown for the five considered scan patterns in Table 3. The pseudo-random scan pattern provides a noticeably better sample transmission function reconstruction than the other four patterns in both magnitude and phase. The histograms of the distance between each pair of sampling positions for each of the the five sampling patterns are shown in Fig. 6. It is apparent that only the experimental scan pattern contains multiple instances of separations of the order of the diffuser structure size (the radius of the grains of the P3000 diffuser here is 3.5 μm and the FWHM of the autocorrelation of the flat field image is 3.6 μm). All four other scan patterns provide a more uniform distribution of distances between points than the experimental pattern. These lateral shifts of the order of the speckle size in the default (pseudo-random) scan pattern give the maximum overlap, whilst still providing diversity.

#### 3.6. Number of images required

As noted in [10], the ptychographic problem involves the estimation of four unknowns (magnitude and phase of both the probe and sample transmission functions) and so more than four diffraction images are required for the system to be over-constrained. In practice, the authors used sixteen diffraction images. We increase the number of images with the default (pseduo-random) scan pattern by alternating between each of the four quadrants of the scan pattern (i.e for four images, we have one image from each quadrant, and for eight images, we have two images from each quadrant). In Fig. 12, we see that the reconstruction is very poor in both magnitude and phase for five or fewer images, and there is a large improvement as we go to six images. There is a small incremental improvement in rmse for each extra image used in the reconstruction. For each additional image used however, there is an increase in the dose (total number of photons that the sample is exposed to) and we investigate the optimal distribution of dose in the next subsection.

#### 3.7. Dose distribution

In many applications with X-ray imaging, we are interested in the performance with respect to the total number of photons that the sample is exposed to. This total number of photons here is defined as the number of diffraction images used times the number of photons per image. As well as the default flux of 6.0 × 10^{10} photons per image (which corresponds to a flux per unit area of 2.55 photons /nm^{2}), we also consider flux levels per image of one thirty-second, one sixteenth, one eighth, one quarter, one half, two times, four times and eight times the default value. The rmse (quadrature sum of the magnitude and phase) versus the total flux is shown in Fig. 13. As per Fig. 12, we increase the number of images in the reconstruction by alternating between the four quadrants of the scan pattern. All nine curves of Fig. 13 have a similar shape to Fig. 12, in that with fewer than six images the reconstruction essentially fails, but with six or more the reconstruction succeeds. This shows that six images is a reasonable choice for the reconstruction regardless of the flux of the system. At the higher end of total flux, it is better to have more images with fewer photons, than fewer images with more photons. For example, the reconstruction with the default flux with all sixteen images is superior to that of twice the flux per image but with only eight images. The increased diversity of the extra images outweighs the reduction in signal-to-noise ratio (SNR) in each image. The converse is true at the lower end of total flux; it is better to have fewer images with more photons, than more images with fewer photons. For example, the reconstruction with eight images with one eighth the default flux is better than sixteen images with one sixteenth the default flux. In this case, the increased diversity of the extra images does not outweigh the reduction in SNR in each image. In a practical implementation, the time to record the extra diffraction images would have to be traded-off against any improved image quality.

#### 3.8. Propagation distances and structure size

We compute in Fig. 14(a) the rmse for the sandpaper grain radii of Table 2, and also include smaller radius grains of 0.6 μm and 0.9 μm. In this default case (*z*=0.0468 m), the rmse shows a broad minimum at radii between approximately 2 μm and 10 μm. Figure 14(a) and Fig. 4(b) show that the most visible speckles correspond to the lowest rmse. For radii much larger than this broad minima, we no longer have a true speckle pattern, but instead start imaging the grains of the sandpaper themselves. For example, for the 50 μm radius grains, we only have two grains in the FOV of the detector, hence the poor visibility of the speckle image and consequently of the ptychographic reconstruction.

In Fig. 14(b), we display the rmse verus the propagation distance *z* between the sample and detector with diffusers of three different grain radius, and consequently different speckle sizes. We see that the rmse is at a minimum for larger propagation distances, and this is also a function of the diffuser structure size: a larger structure (speckle) size requires a larger propagation distance to reach its optimum performance. We recall that a point in the sample (*x*, *y*) is mostly affected by the wavefront within a Fresnel zone (
$\sqrt{\lambda z}$) of that point. Given that the wavefront is essentially constant over the size of the speckle, we need the Fresnel zone to be larger than the speckle size to have diversity in the measurements when we shift the sample. Consequently, for larger speckles we need to propagate further to incorporate a larger Fresnel zone, in order to create more diversity in the measurements and hence achieve the optimum performance.

Figure 14(c) shows that the rmse improves marginally with increasing *z _{d−s}*, and this can be attributed to the increasing visibility of the speckles with increasing

*z*as seen in Fig. 4(a). The chosen value of

_{d−s}*z*of 0.5 m used throughout this paper is therefore reasonable.

_{d−s}## 4. Conclusion

In this paper, we have shown via numerical simulation that at least six diffraction images are required for successful reconstruction with near-field ptychography, a pseudo-random sampling pattern incorporating a combination of step-sizes of approximately the speckle size as well as much larger step-sizes is superior to raster, spiral or round scanning patterns. In high total flux (dose) applications, for a given total flux level, it is preferable to have more images each with less flux than fewer images each with more flux. The converse is true at lower total flux levels. The structure size of the diffuser element should be chosen to give the highest visibility of the near-field speckles. The propagation distance between the sample and detector should be sufficient to ensure that the Fresnel zone is larger than the near field speckles. The propagation distance between the diffuser and sample should also ensure high speckle visibility.

A number of aspects of near-field ptychography still need to be clarified in future studies including the effect of temporal and transversal coherence on the reconstruction.

## Acknowledgments

We acknowledge financial support through the German Research Foundation (DFG) Cluster of Excellence Munich-Centre for Advanced Photonics (MAP) (EXC 158), the DFG Gottfried Wilhelm Leibniz program, and the Technische Universität München within the funding program Open Access Publishing.

## References and links

**1. **A. W. Stevenson, T. E. Gureyev, D. Paganin, S. W. Wilkins, T. Weitkamp, A. Snigirev, C. Rau, I. Snigireva, H. S. Youn, I. P. Dolbnya, W. Yun, B. Lai, R. F. Garrett, D. J. Cookson, K. Hyodo, and M. Ando, “Phase-contrast X-ray imaging with synchrotron radiation for materials science applications,” Nucl. Instrum. Meth. B **199**, 427–435 (2003). [CrossRef]

**2. **R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today **53**, 23–26 (2000). [CrossRef]

**3. **F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. **2**, 258–261 (2006). [CrossRef]

**4. **T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature , **373**, 595–598 (1995). [CrossRef]

**5. **S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature **384**, 335–338 (1996). [CrossRef]

**6. **P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. , **29**, 133–146 (1996). [CrossRef]

**7. **H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

**8. **J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illimuination,” App. Phys. Lett. **85**, 4795–4797 (2004). [CrossRef]

**9. **A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1262 (2009). [CrossRef] [PubMed]

**10. **M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. **3**, 1927 (2013). [CrossRef] [PubMed]

**11. **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**, 014005 (2015). [CrossRef]

**12. **M. Stockmar, M. Hubert, M. Dierolf, B. Enders, R. Clare, S. Allner, I. Zanette, J. Villanova, J. Laurencin, P. Cloetens, F. Pfeiffer, and P. Thibault, “X-ray nanotomography using near-field ptychography,” Opt. Express **23**, 12720–12731 (2015). [CrossRef] [PubMed]

**13. **D. M. Paganin, *Coherent X-ray Optics (Oxford series on synchrotron radiation)*, (Oxford University, 2006). [CrossRef]

**14. **M. Born and E. Wolf, eds., *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light*, 7th ed. (Cambridge University, 1999). [CrossRef]

**15. **J. R. Fieunp, “Invariant error metrics for image reconstruction,” Appl. Opt. **36**, 8352–8357 (1997). [CrossRef]

**16. **O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical engine,” Ultramicroscopy **108**, 481–487 (2008). [CrossRef]

**17. **A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” Europhys. Lett. **99**, 48001 (2012). [CrossRef]

**18. **J. Wolf, A. Malecki, J. Sperl, M. Chabior, M. Schüttler, D. Berqué, C. Cozzini, and P. Pfeiffer, “Fast one-dimensional wave-front propagation for x-ray differential phase-contrast imaging,” Biomed. Opt. Express **5**, 3739–3747 (2014). [PubMed]

**19. **M.-C Zdora, “X-ray multimodal imaging using near-field speckles,” Master’s Thesis, Technische Universität München (2014).

**20. ** Federation of European Producers of Abrasives, “FEPA P-grit sizes coated abrasives,” http://www.fepa-abrasives.org/Abrasiveproducts/Grains/Pgritsizescoated.aspx

**21. **A. Hamed, “Recognition of direction of new apertures from the eloganted speckle images: simulation,” J. Opt. Photon. **3**, 250–258 (2013). [CrossRef]

**22. **R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. **4**, 238–243 (2008). [CrossRef]

**23. **P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature **494**, 68 (2013). [CrossRef] [PubMed]

**24. **P. Thibault, M. Deirolf, A. Menzel, O. Bunk, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy **109**, 338–343 (2009). [CrossRef] [PubMed]

**25. **P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New. J. Phys. **14**, 1256–1262 (2012). [CrossRef]

**26. **M. Dierolf, P. Thibault, A. Menzel, C. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. **12**, 35017 (2010). [CrossRef]

**27. **H. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express **22**, 12634–12644 (2014). [CrossRef] [PubMed]

**28. **I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Herz, “Speckle-base X-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. **112**, 253903 (2014). [CrossRef]

**29. **S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, “Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. **207**, 79–96 (2002). [CrossRef] [PubMed]