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Abstract
We study by numerical simulation how spatial coherence affects the reconstruction quality of images in coherent diffractive x-ray imaging. Using a conceptually simple, but computationally demanding approach, we have simulated diffraction data recorded under partial coherence, and then use the data for iterative reconstruction algorithms using a support constraint. By comparison of experimental regimes and parameters, we observe a significantly higher robustness against partially coherent illumination in the near-field compared to the far-field setting.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The assumption of a fully coherent illumination (probe) is ubiquitous for lens-less coherent x-ray imaging [1–3]. However, this assumption can never be fully met. One would therefore like to know how the false assumption of full coherence affects the reconstruction quality in coherent imaging? After all, the coherence properties describe the ability of an x-ray wave field to interfere – a prerequisite for coherent x-ray imaging. More generally, all x-ray techniques based on diffraction, require the partial coherence to be sufficiently high [4]. Beyond imaging, this has been prominently discussed for coherent spectroscopic techniques such as x-ray photon correlation spectroscopy (XPCS) [5]. Considerable efforts have been undertaken to measure coherence properties of different x-ray sources for example by oscillation visibility [6,7], Young’s double slit [8,9] or grating interferometry [10], as well as to numerically track the propagation of partially coherent x-ray wavefields [11–13]. For coherent (lens-less) diffractive imaging (CDI) in the far-field setting, the criterion of minimal spatial (lateral) coherence depends on object size L, while the criterion for temporal (longitudinal) coherence depends on the desired resolution [14]. As shown experimentally in [15], these criteria can be relaxed by about a factor of two, if information on the coherence state is available, exploiting modal expansion. The projection operators of iterative algorithms have also been adapted to the state of partial coherence [16]. However, in most experiments, little is known about the coherence state, and images are reconstructed from data recorded under partial coherence with an approach derived or justified for the case of full coherence. This introduces an inconsistency in the constraints applied in an iterative reconstruction scheme.
In this paper we study the effects of partially coherent illumination on the image quality, if the reconstruction is carried out under the false assumption of full coherence. We compare the two main optical regimes, in which coherent x-ray imaging can be implemented, namely near-field holography (NFH) and far-field based coherent diffractive imaging (CDI). NFH is also often denoted by propagation-based phase contrast imaging. To this end, we use numerical simulations of partially coherent and noisy data acquisition. Experimentally, NFH is often implemented using highly focused beams in order to achieve sufficient magnification. Note that near-field imaging in highly focused fields is also sometimes denoted also as keyhole or Fresnel CDI [17,18].
However, using the Fresnel scaling theorem [1], this geometry can always be transformed to an equivalent parallel beam geometry, so that it is sufficient here to implement parallel beam propagation. As in most experiments, we thus assume full coherence in the numerical reconstruction, but in contrast to experiments dealing with unknown objects, we can fully quantify the associated errors.
In principle, there is a possibility to avoid this information deficit by reconstructing the modal structure of a partially coherent probe, as first shown by [19] for the far-field and later extended to the near-field [20], but this requires multiple recordings, i.e. a high number of images at different object or detector position. If this amount of data is available, even temporal instabilities of the probing illumination P can be treated by a localized relaxation scheme [21]. Even if the extended measurement time and dose are tolerable, this may, however, not be possible in practice due to temporal instabilities in the object O. Very recently, an interesting method for CDI has been proposed [22] to recover the object and the coherence properties of the illumination from a single measurement. The only experimental requirement with respect to full coherence was an increased oversampling ratio. Previous methods relied on an independent characterization of coherence properties [23]. While this may indeed help to remove the coherence deficits of CDI data in future applications, we here address the problem as it applies to the status quo of the vast majority of lens-less reconstructions which lack any ability to identify and to compensate for the effect of partial coherence. While temporal coherence has been considered previously, i.e. how monochromaticity affects resolution and reconstruction quality [24], we consider the effects of a reduced spatial degree of coherence.
In particular we study in detail the deterioration of image quality with decreasing coherence length ξ and line out how this depends on image parameters and in particular the optical regime. Previous studies compared the reconstruction effectiveness in the presence of noise [25,26]. As in our previous study on the fluence-resolution relationship [26], we compare two prototypical configuration, namely near-field holography (NFH) [27–31] and far-field CDI [32,33]. Further, we assume that only a single intensity recording is available and that therefore coherent modes of the probe cannot be reconstructed, i.e. that the true coherence state is unknown. Data generation and object reconstruction is carried out by numerical simulation, with the transverse coherence length ξ as the main control parameter. The reconstruction is carried out by iterative algorithms using canonical constraints, and resolution and reconstruction error are quantified with respect to a phantom as a function of ξ. As in [26], we choose a pure phase object. By ‘reconstructed object’, we hence always denote the phases of the reconstructed object.
The conceptually very simple – but for the purpose sufficient – coherence model is briefly described in the next section (Sec. 2), followed by the description and illustration of the numerical simulations in Sec. 3. The results are presented and discussed in Sec. 4, before the paper closes with a brief conclusion in Sec. 5.
2. Coherence model
To model image formation in NFH and CDI with a partially coherent probe, we make use of the model of a completely incoherent but monochromatic source with the wavelength λ. In the detector plane, we then simulate the incoherent superposition of ergodic realizations of plane waves with random wave vector [34]. To this end, we consider the illumination of an object O placed at a distance R behind a completely incoherent source of size d, with all source points emitting uncorrelated radiation, as illustrated in Fig. 1. We assume the angle θc subtended by the source when observed from O to be sufficiently small so that the paraxial assumption holds. The lateral or spatial coherence length ξ is then given by
i.e. the length at which two plane waves intersecting with θc have a phase shift of π/2. The lateral momentum of a wave impinging on O under angle α ∈ [−θc/2, θc/2] is With θc = λ/(2ξ) and writing α = ζθc/2, ζ ∈ [−1, 1], we have for angles α ≈ 0 i.e. the lateral momentum can be expressed as a function of ξ, independently of λ. This allows us to simulate the model in a dimensionless setting with ξ given in units of pixels, as the only control parameter. Note, that the numerical implementation recourses to the use of α to choose the realizations of incident plane waves.
Fig. 1 Schematic and notation for the coherence model. Plane waves are emitted by an incoherent source with extension d. An object is placed at distance R behind the source. This setting defines the lateral coherence length ξ = λR/(2d) in the object plane, expressed in pixel units for the simulation. Plane waves are then simulated to impinge on the object with random wave vectors out of a cone with opening angle θc. For each realisation, the diffraction pattern of the phantom object (see Fig. 2) is calculated both for the near- and far-field, i.e. for NFH and CDI, respectively. The diffraction patterns are then added up incoherently, providing the input data for the reconstruction. Note that this model of partial coherence essentially corresponds to a convolution of the diffraction pattern with an effective point-spread function in the detection plane. In the simulation however, the data is not generated by a convolution but by different stochastic realisations.
3. Numerical simulation
Figure 2 illustrates the setup of the numerical simulation in the limit of full coherence, both for NFH and CDI. For both cases, the phantom of two adhering cells shown in (a) has been used and treated as pure phase-contrast object with phases ϕx,y ∈ [0, −1] rad. The phantom has a size of L = 512 × 512 pixel2 embedded in 1024 × 1024 (Nx × Ny) pixel2. Ideal measurements are generated for NFH (b) at a Fresnel number Fr = Δx2/(λz) = 10−3 and CDI (c) at Fr = 0, where z denotes the propagation distance and Δx the pixel size. To ensure artifact-free propagation a padded size of 2048 × 2048 pixel2 has been used. Note that all length scales are measured in units of pixels, hence the pixel size is set to Δx = 1, and the Fresnel number Fr is defined with respect to the pixel size. This is in contrast to the often used definition based on the object extension L. Of course, this definition entails smaller values for the transition from optical near- to far-field, when judged by visual inspection which is dominated by the characteristic length scales of the object, much larger than Δx.
Fig. 2 Setup for the simulation: limit of full coherence. (a) The pure phase phantom of a cell with maximum phase shift of −1 rad. The red line indicates the support used in the reconstruction. (b) The corresponding ideal near-field hologram for Fr = 10−3. (c) The corresponding ideal far-field measurement. The diffraction pattern simulated under partially settings are presented in Fig. 3. Scalebar: 50 px.
For the incoherent source model, in particular for small ξ, a large number of realizations NR for the plane waves is required. The measurements shown in the following have been obtained by the superposition of NR = 5000 realizations of the source, chosen high enough to ensure convergence of the generated data for partial coherence. The generation of a measurement takes 5 s for NR = 5000. For large ξ, NR could have been reduced, but for simplicity the value was kept for the entire study. Each realization of the exit wave Ψα is then computed by
where Pα is a canted plane wave emitted by S at angle α. α is hence the direction of the wave vector corresponding to the plane wave interacting with the object O. Next, the propagator 𝒳 was applied, i.e. the Fresnel propagator for a given Fr 𝒟Fr or the Fourier transformation ℱ for NFH or CDI, respectively. The partially coherent data, i.e. the measurement M is simulated by 𝒟Fr is given by where kx = 2 nx/Nx and ky = 2 ny/Ny are spatial frequencies in Fourier space with nx,y ∈ [−Nx,y/2 ... Nx,y/2]. The measurements for CDI have been generated by discrete Fourier transformation ℱ of the corresponding exit wave.To simulate the two dimensionally extended source, the incidence angles of the plane waves were uniformly distributed over a circle with radius θc/2. The uniform distribution was generated in polar coordinates and then transformed to Cartesian coordinates:
- Generate 2 uniform distributed random numbers n1, n2 ∈ [0, 1].
- Calculate the polar angle ϑ = 2πn1 and radius . The root is for normalization on the circle surface.
- Transform in Cartesian coordinates αx = ρ cos(ϑ) and αy = ρ sin(ϑ).
With that the canted plane wave Pα is
where rx,y denote coordinates in the plane of O. While this scheme provides an intuitive way to simulate M, it has two drawbacks: (i) it is numerical extremely costly and (ii) for CDI the multiplication with Pα leads to numerous numerical artifacts in ℱ(Pα · O). In order to remove these artifacts it is necessary to precisely choose padding and windowing functions [35]. A more practical approach can be obtained by analyzing the properties of the propagators. Multiplication with Pα leads to a simple shift of the individual realization of the measurement. For the far field, α shifts the ideal (central) measurement by For the near field case with the finite propagation distance z has to be taken into account, yielding a shift ΔsNF with ΔsNF = q(α) · z. Plugging in the definitions for q and Fr this can be written as Based on these shifts, the simulation of M can be efficiently implemented using shifting operations, as detailed in Alg. 1. The shifting operation in line 8 is carried out with sub-pixel precision using a linear interpolation in real space. The simulated measurements for varied ξ are illustrated in Fig. 3 (a,c). Note that the simulation of M can also be generated in form of a convolution with the Fourier transformation of the source profile (cf. Van Zittert-Zernike-theorem [34]), which can be interpreted as a coherence envelope. This is an approach often used in high resolution electron microscopy [36]. The present method is a more bottom-up Monte-Carlo like approach, in which also more complicated upstream optical schemes could be included in future, such as different ray-tracing, propagation and monochromatisation steps in a beamline.Fig. 3 Reconstructions from measurements with finite coherence length ξ. The measurements are presented for (a) NFH and (c) CDI, with each panel showing on the left half the fully coherent measurement and on the right half the measurement obtained for the finite ξ as given in the title of the panel. The partially coherent CDI measurements have a virtual beam stop covering the zeroth order. The NFH measurement are shown on linear scale and the CDI measurement on logarithmic scale. The corresponding reconstructions are shown in (b) for NFH and (d) for CDI. The number in the upper right corner of each panel denotes the resolution Δr in 1/px. The scalebar indicates 50 px in all panels.
Algorithm 1. Generation of of partially coherent measurements using sub-pixel shifts.
After the M has been simulated, the phase reconstruction process has to be carried out. The reconstructions from the partially coherent data have been obtained by the Relaxed Averaged Alternating Reflections (RAAR)-algorithm [37]. The iterates are given by
where RS/M (Ψ) = 2PS/M (Ψ) − Ψ denotes a (mirror) reflection by a given constraint set and n the iteration index. The parameter βn controls the relaxation and is chosen by the function where β0 denotes the starting value, βmax the final value of βn and βs the iteration number when the relaxation is switched. This relaxation strategy follows [37] (Eq. 37). The parameters have been set for to β0 = 0.99, βm = 0.75, βs = 150 iterations all reconstructions. The projection on the measurements PM is the standard magnitude projector where 𝒳 is either ℱ or 𝒟Fr for far field or near field propagation, respectively.The operator PS is used to enforce the support S which is assumed to be perfectly known and the pure phase constraint in the object plane i.e.
Note that the phase reconstruction is of course carried out under the assumption of full coherence, as in experiments in absence of better knowledge. The corresponding inconsistency in the constraints results in reconstruction errors, notably a loss of resolution. To quantify this effect, we use Fourier ring correlation [38,39] (FRC) with respect to the phantom. Note that FRC is based on the fact that high spatial frequencies of the image are less correlated to the object than small spatial frequencies, when noise is uncorrelated. The FRC criteria for resolution thus quantify the spatial frequency at which the correlation falls below a pre-defined 1 bit threshold, as explained in [38,39]. For partial coherence, we observe a similar behaviour, which makes the FRC a well-suited tool also in this case to quantify the resolution.
4. Results
First the influence of ξ < ∞ on the measurement M is surveyed in Fig. 3. The M for varied ξ are shown in (a) for NFH and (c) for CDI. Note the different choice of ξ for NFH and CDI. The left half of the measurement panels shows the ideal fully coherent M, while the right half shows M for the ξ given in the title with NR = 5000.
The NFH measurements are shown on linear scale in a diverging cool-warm colormap [40], which illustrates the oscillation of fringes. By increasing ξ, the fringes spread out wider and sub-structure in the fringes becomes visible. The recovered objects (b) used the respective measurement from (a). The increase in ξ is accompanied by an increase in resolution Δr, given in the upper right corner of the reconstruction panels. For ξ = 225 px, which is still smaller than the full horizontal 306 px extension of of the object, NFH already reaches full resolution and the recovered object exhibits no visible deviations from the phantom. Interestingly, the corresponding hologram still shows noticeable deviations to the ideal hologram.
The CDI measurements (c) are depicted on a logarithmic scale. For small ξ they show a strong smearing. With increasing ξ the speckles become sharper, but the fine lines between the speckles do not go to zero as in the ideal measurement. The corresponding reconstructions in (d) show a total reconstruction failure for ξ = 20 px and 100 px. The reconstruction for ξ = 325 px still shows low frequency and stripe artifacts but the smallest features are quite clearly visible. For ξ = 450 px the reconstruction has almost reached full resolution and is nearly artifact free. The reader may have noticed the white circles in (c). These depict virtual beam stops to block the zeroth order in the diffraction patterns and to exclude these values during the reconstruction process. The beam stops’ radii have been chosen as 2 · Δsmax, with Δsmax the maximum shift of the measurement for a given θc. This value seems prohibitively high but ensures stable reconstructions without introducing noticeable reconstruction artifacts. Figure 4 (a) shows a measurement for ξ = 500 px without beam stop and the corresponding reconstruction in (b). The high coherence length results in Δsmax = 0.5 px. Despite this low value, the reconstruction fails without the beam stop. The artifacts encountered are not just low frequency deviations, but prevent reconstruction of the entire structure. Only the support shows a strong imprint in the reconstruction.
Fig. 4 CDI reconstruction without the virtual beam stop. (a) The simulated measurement for ξ = 500 px (a) and (b) the corresponding reconstructed objet.
Next, we turn to the coherence-resolution relationships which are computed by performing the automatized reconstruction and FRC analysis for measurements of systematically varied coherence length ξ. For each ξ covering the range from 20 to 900 px, 20 realizations have been generated and reconstructed, each with the same parameters. Figure 5 shows the results. The ξ-resolution curves shown in (a) exhibits a different behavior for NFH and CDI. For NFH, the resolution increases linearly until it reaches full resolution at ξ = 225 px. For CDI, the resolution does not improve over the range of ξ = 20 px to 250 px, then a steep increase in resolution is observed which saturates at ξ = 475 px. In (b), the reconstruction error Δ is quantified by the ℓ2-norm error of the reconstruction with respect to the phantom, according to
For NFH we observe a steep decrease up to ξ = 225 px followed by a slower reduction in Δ. The CDI curve shows again improvements, i.e. error reduction only at larger ξ. For all ξ, the error is roughly an order of magnitude larger than for NFH.Fig. 5 Image quality as a function of coherence length ξ. (a) Resolution as a function of ξ. (b) The ℓ2 error (Eq. (14)) with respect to the original phantom, as a function of ξ. Curves are shown for NFH (blue) and CDI (red). ξ was varied between 20 to 900 px. Results were obtained by averaging of 20 reconstructions for each ξ. Each reconstruction started from a new measurement generated from 5000 source realizations.
The results shown so far have been obtained at fixed simulation parameters, notably Fr and object size L. The change of these parameters has different effects on NFH and CDI, as presented in Fig. 6. The algorithm and realization parameters are set as before. (a) shows the effect of changing Fr on the NFH result. The plot shows variation of Fr from 0.7 · 10−3 to 5 · 10−3. Smaller Fr require a larger ξ to be properly reconstructed. This can be explained by the fact, that at smaller Fr, the fringes of the hologram are propagated further out. Consider two features in the object that have a large separation distance. In the small Fr case, the fringes emanated from these features come in contact. The fringes have to interfere coherently in order to form the correct hologram. This requires a larger ξ. (b) shows the effect of the object size L on the CDI result. The plot shows the ξ-resolution curves for varied L = {256, 512, 700, 800} px, corresponding to maximum extension of the phantom cells of {153, 306, 418, 478} px. As expected, as larger coherence length is required to reconstruct larger objects.
Fig. 6 ξ-resolution relationship for different setup parameters, namely (a) Fresnel number Fr and (b) object size L, for (a) NFH and (b) CDI, respectively The NFH results in (a) show that a smaller Fr requires a larger ξ for a high resolution reconstruction. The CDI result in (b) show that objects with larger size L require a higher degree of coherence in the illumination to reach high resolution in the reconstruction. In both settings, NR = 5000 and 20 realizations per ξ have been used.
Next, we study the influence of Poissonian noise on the partially coherent measurements, using the expectation value μ for the photon fluence (photons per pixel) as additional control parameter. The numerical implementation was based on the Matlab function imnoise, as described in more detail in [26]. The results of this experiment are presented in Fig. 7. The parameters have been kept the same as for Fig. 5(a), i.e. Fr = 1 · 10−3 and L = 512. Again we have used NR = 5000 and each ξ has been repeated 30 times. The noise has been added after the simulation of the partially coherent measurement, following the approach from [26]. Both NFH in (a) and CDI in (b) show stagnation of the resolution below 0.5 1/px if μ is too small, e.g. up to μ = 399 ph/px for NFH and μ = 5000 ph/px for CDI. For NFH we observe the interesting case μ = 2512 ph/px where the increase in ξ increases the contrast in the measurement so that the fringes have sufficient contrast, yielding full resolution for the reconstructed object. As in our previous fluence-resolution study (for full coherence) [26], we note roughly 5 times higher requirements on the fluence for CDI then for NFH, before the reconstruction reaches full resolution.
Fig. 7 Resolution as function of ξ at varied fluence μ (photons per pixel). (a) results for NFH and (b) for CDI. Note that (a) and (b) do not show exactly the same choice of μ. Also the sampling of ξ’s was adapted to the respective method. The dashed lines indicate the noise free result for comparison, from Fig. 5(a).
5. Conclusion and outlook
In conclusion, the spatial coherence requirements for near- and far-field coherent diffractive x-ray imaging, i.e. for NFH and CDI respectively, can be quantified by simulation of the experimental parameters with a simple stochastic model and are found to differ significantly between the two different optical regimes. While the coherence length ξ has to equal or exceed the object size L for CDI, as expected based on the criteria formulated in [14], full resolution in NFH could already be observed at a coherence length of half of this value ξ ≃ L/2 (Fig. 5). Note, however, that even for ξ > L, high quality CDI reconstructions required a exclusion of the pixels in the direct vicinity of the origin, i.e. the introduction of a numerical beamstop. The interpretation of this effect seems rather straightforward. Small spatial frequency data corresponds to large distances in the object which ’explore’ the decay of coherence. Interference of the low spatial frequency mode(s) is therefore hampered, and excluding this data from the subsequent reconstruction is better than including the corresponding flawed values.
For NFH, the resolution increases linearly with ξ, with a slope depending on Fr. For the entire range of ξ investigated the reconstruction error was smaller for NFH than for CDI. The comparison was carried out first for the noiseless setting, and then in addition for simulated Poissonian noise. NFH showed higher robustness than CDI with respect to both photon noise and finite coherence (Fig. 7). While this was expected based on the fact that the interference is more predominantly local in NFH, but global, i.e. over the entire object, the quantification of the effect was hitherto missing, and is important for the design of CDI and NFH experiments.
The results also confirm the standard recommendation for CDI to use “compact and isolated” objects. Small objects require significantly smaller coherence length than larger objects. In fact, for CDI the natural parameter is the coherence length in units of the object size ξ/L. For NFH, the resolution dependence on Fr can be simply regarded as a manifestation of an effective coherent numerical aperture, resulting in resolution scaling with λ/(ξ/z). In planning NFH experiments, higher contrast for small phase differences at smaller Fr has to be balanced versus the available coherence which favors higher Fr.
The higher robustness of near-field coherent imaging (NFH) lends itself for imaging with sources of low and moderate brightness, such as bending magnets, compact synchrotrons, liquid metal jet and other laboratory sources. In fact, this surely explains why it has been possible at all to achieve the recent successful translation of propagation imaging from synchroton radiation to laboratory sources [41–44].
Funding
Deutsche Forschungsgemeinschaft (DFG) (SFB 755); Bundesministerium für Bildung und Forschung (BMBF) (05K16MGB).
Acknowledgments
We thank Jan Goeman for keeping the number crunching machines up and running.
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