For iterative phase retrieval algorithms in near field x-ray propagation imaging experiments with a single distance measurement, it is indispensable to have a strong constraint based on a priori information about the specimen; for example, information about the specimen’s support. Recently, Loock and Plonka proposed to use the a priori information that the exit wave is sparsely represented in a certain directional representation system, a so-called shearlet system. In this work, we extend this approach to complex-valued signals by applying the new shearlet constraint to amplitude and phase separately. Further, we demonstrate its applicability to experimental data.
© 2016 Optical Society of America
The first observation of propagation-based phase contrast in the x-ray regime some 20 years ago at the European Synchrotron Radiation Facility (ESRF, Grenoble) [1,2] has started a considerable activity in developing this lens-less full field imaging approach to resolve micro- and nano-structures in the interior of specimens [3–5]. Using the divergent wavefronts emitted from the quasi-point source of an x-ray waveguide channel has enabled a resolution down to 20 nm . The phase retrieval problem in this near field setting can be regarded as better posed compared to the far field counterpart, see also  for an illuminating novel view on uniqueness in near field phase retrieval. Nevertheless, a persisting challenge remains in the phase retrieval of images recorded at a single distance and without translations of the object in the beam. Without such added diversity in the data, phase and amplitude of an object’s wave cannot be recovered independently, unless ‘strong’ a priori information, such as compact support, is available. Since many relevant samples are extended, it is generally believed that phase information from single distance measurements can only be recovered for objects where phase and amplitude variations are strongly coupled, or for objects which interact only by either phase or amplitude, i.e. so-called pure phase or pure absorption objects.
As an introduction and for notational clarity, let us briefly recall the imaging process. When an x-ray wave field interacts with a specimen not only its amplitude is modified but its phase distribution as well. Both effects are determined by the specimen’s internal structure, which is reflected in its complex refractive index for x-rays, see e.g. ,Fig. 1(a) for the experimental setup. By this means, x-ray propagation imaging is able to extract and visualize information about both β and δ of the analyzed specimen. In particular, for thin biological samples of light elements, phase contrast is indispensable, as δ becomes about three orders of magnitude larger than β and absorption contrast is negligibly low .
More precisely, within the Fresnel regime, the connection between the wave field in the detector plane ψ(x,y,z) and the exit wave field ψ(x,y,0), under the paraxial and the projection approximation and assuming monochromatic plane wave illumination in z-direction, reads 
Here, ℱ⊥ and denote the Fourier transform in lateral coordinates and its inverse, τ is the specimen’s maximal thickness and k = (kx,ky,kz) is the wave vector with modulus . The operator Dz is the so-called Fresnel propagator. However, detectors can only capture the time-averaged squared amplitude, i.e. the intensity, of the propagated wave field
The loss of phase information impedes the reconstruction of the exit wave from the measurement by simply applying the inverse Fresnel propagator D−z (non - crystallographic phase problem). Many different methods to approach this inverse problem have been developed throughout the years [11–13].
For phase retrieval in (far field) coherent diffractive imaging (CDI) iterative algorithms that make use of a priori information of the signal to be reconstructed represent an established strategy [14–17]. Likewise, they have become a powerful tool in (near field) x-ray propagation imaging in the holographic regime [18–21]. The a priori information and/or information from intensity measurements at a single or several defocus distances are incorporated into constraint maps that, in turn, are combined into iterative schemes. For example, in , a regularization strategy for phase retrieval is proposed that employs sparsity in an orthogonal wavelet frame. Further remarks and references on application of wavelet sparsity constraints can be found in . Seminal algorithms, which can be applied to both the conventional far field and the optical near field case, are the Gerchberg-Saxton algorithm  and Fienups’s Error Reduction (ER) algorithm  and Hybrid Input-Output (HIO) algorithm . Aiming at more stable and fast phase retrieval algorithms new relaxation strategies have been developed that led to the Hybrid Projection Reflection (HPR) algorithm  and to the Relaxed Averaged Alternating Reflection (RAAR) algorithm . In this paper, we are interested in near field experiments with a single distance measurement and we will use the RAAR method for phase reconstruction that is especially suitable for noisy input data. The standard approach in this setting is to use a priori knowledge about the compact support of the specimen in terms of a corresponding support constraint. However, in case the object’s support cannot be identified or it occupies the whole field of view, alternative constraints of similar strength need to be found.
Recently, Loock and Plonka  proposed a new constraint for iterative x-ray phase retrieval in the Fresnel regime, which uses sparsity information of the wavefront, and they presented promising numerical results with this method. In this paper, we will extend this approach to the physical situation of a complex-valued exit wave field and apply the new sparsity constraint separately to its amplitude and phase. Further, we will demonstrate its applicability to experimental data. We start by introducing the idea behind this new constraint (Section 2), followed by the presentation of simulation results (Section 3). In Section 4 the method is applied to experimental data and we end with a short conclusion in Section 5.
2. The shearlet constraint for iterative phase retrieval
The a priori information that an object transmission function (exit wave) may be sparsely represented in a certain representation system can be advantageously used for near field phase retrieval. To this end, sparse may mean that only a few expansion coefficients are nonzero, or in a more relaxed sense that only few expansion coefficients contain the ’essential’ information. In , it was in particular shown that the so-called shearlet system is very suitable for this purpose, as demonstrated by reconstructing simulated data of real-valued object transmission functions in the presence of noise. Two-dimensional (2D) shearlet systems, introduced in  and , are constructed by applying scaling, translation and shearing operators to a set of 2D wavelet-type generating functions. By these means, the system’s elements are defined at various scales, locations and orientations, allowing them to efficiently capture anisotropic image elements like discontinuity curves. The associated shearlet transform maps a function f ∈ L2(ℝ2) to the sequence of all shearlet coefficients, defined as the L2-scalar product between the function and each element of the corresponding shearlet system. In  detailed information on the theory and applications of shearlets are given. In particular, under certain conditions the shearlet system forms a frame for the L2(ℝ2), providing stable reconstructions of elements f ∈ L2(ℝ2) from their shearlet coefficients (inverse shearlet transform). While the first shearlet systems had been constructed using band-limited generating functions, newer developments also focus on shearlet frames whose elements have compact support in space domain, see .
The class of so-called cartoon-like images is mathematically defined as a subspace of L2(ℝ2) that contains all compactly supported functions that are twice continuously differentiable apart from piecewise C2-discontinuity curves, see . Many images encountered in physical applications, like the phase or the amplitude of exit wave fields in x-ray propagation imaging, can be expected to fall into this image class. Let f be a cartoon-like image and let fN be the approximation that takes the N largest shearlet coefficients in magnitude into account for the reconstruction via the inverse transform. Then, shearlet theory ensures the following approximation rate33] as well as for compactly supported shearlets , which is nearly optimal (apart from the log-factor). Hence, shearlet frames provide sparse representations of cartoon-like functions. In other words, in many relevant experimental situations encountered in x-ray propagation imaging, we may assume the a priori knowledge that the amplitude/phase of the exit wave field is sparsely represented in a suitable shearlet frame. As shown in , this information can be used as a constraint for iterative phase retrieval algorithms.
Let be the discrete, complex-valued exit wave field and let be the discrete, propagated field in the detector plane, i.e., Dz denotes the discrete Fresnel transform. Further, SH denotes the discrete shearlet transform that transfers an image of size Nx×Ny into a vector of shearlet coefficients of length Nx · Ny · S, where S > 1 indicates the redundancy of the transform that depends on the number of different scales and shearings that are taken into account by the discrete shearlet frame. Further, SH−1 denotes the discrete inverse shearlet transform, which is a pseudo inverse that can be efficiently computed by conjugate gradient methods . In this work, the transforms are computed with the ShearLab 3D toolbox (freely accessible from shearlab.org) that employs a system of compactly supported shearlets .
The basic idea of the shearlet (sparsity) constraint for complex-valued exit waves is to enforce a sparse shearlet representation of the amplitude and phase of the exit wavefront by applying a shearlet shrinkage at each iteration step. Let us explain the approach in more detail within the framework of the Error Reduction algorithm, see Fig. 1(b). Starting with an initial guess ψ0, the current approximation ψn is alternately propagated between the object and the detector plane, where it is forced to fulfill the object plane constraint (here: the shearlet constraint) and the modulus constraint, respectively. The modulus constraint enforces the approximation to be in agreement with the measurement , see Eq. (3), on a grid in the detector plane. The corresponding projection onto the set of digital images that agree with the measurement reads, see e.g. ,
In the object plane we employ the shearlet constraint. For that purpose, we apply first the shearlet transform separately to the amplitude and to the phase of ψ, then cut off small shearlet coefficients by a threshold operator before applying the inverse shearlet transform. In practice, instead of such a strict cut off, we use a soft threshold operator for the shearlet coefficients , defined as27], since it coincides with the so-called proximity operator corresponding to the l1-norm of the sequence of shearlet coefficients. At the same time, it causes a smoothing effect, similarly as for wavelet denoising, see e.g. . The shearlet constraint operator can be written as
For the reconstructions presented in this work we use the more complex Relaxed Averaged Alternating Reflection (RAAR) algorithm . Formulated as a fixed-point iteration with the shearlet constraint as object plane constraint, this algorithm reads27]. Here, we focus on the evaluation of the numerical results.
Equation (7) can be applied to the general case of a mixed object that causes both phase shifts and absorption of the illuminating wave field, the so-called probe p. As we are dealing with monochromatic plane wave illumination we have p = 1. For so-called pure phase objects, the absorption of the probe can be neglected, i.e. the amplitude of the exit wave can be set to one and Eq. (7) becomes . Likewise, for pure absorption objects, where the phase shift can be neglected, it reads , i.e. the phase is set to zero. Another constraint based on a priori information, which is frequently used for phase retrieval in x-ray propagation imaging experiments, concerns the range of values that the phase and amplitude of the exit wave can obtain. Within the projection approximation we know that the amplitude of the exit wave field can only be diminished due to absorption inside the object, i.e. we have the a priori information |ψ| ≤ 1. Furthermore, we know that with x-ray energies the real part of the refractive index is smaller than one, i.e. δ ≥ 0 (cf. Eq. (1)) and we only observe negative phase shifts φ(ψ) ≤ 0 (cf. Eq. (2)). This knowledge can define an object plane constraint by itself (the so-called range constraint) or it can be additionally combined with other constraints. Table 1 summarizes the effects of all constraints used within this paper for different types of objects.
Figure 2 shows a simulated x-ray propagation imaging experiment for an object which causes both phase shifts and absorption of the probe (denoted here as mixed object). Figures 2(a) and 2(b) depict the phase shift and amplitude distribution of the exit wave field behind the object, which are structurally completely uncorrelated. The simulation’s setup parameters are: δ = 1.6 · 10−6, β = 8 · 10−7, τ = 10 µm, wavelength λ = 10−10 m and Fresnel number F = 4 · 10−3. Figure 2(c) shows the simulated intensity measurement, degraded by artificial Poisson noise with 50 photons per pixel. For reconstructing the exit wave field we use the RAAR algorithm (relaxation parameter α = 0.55) with the holographic reconstruction (simple back-propagation D−z of m) as initial guess. Figures 2(d)–2(f) show the reconstructions obtained after 100 iteration steps with different object plane constraints, namely the range constraint, the support & range constraint and the shearlet & range constraint. The support considered for the support & range constraint is indicated by the dashed black frames in Figs. 2(a) and 2(b). For the shearlet & range constraint we use a shearlet system of 5 scales (81 shearlets in total not counting translations) and a fixed threshold parameter of θa = θp = 0.005 for both the thresholding of the phase and the amplitude.
The reconstructed phase and amplitude obtained with the shearlet & range constraint in Fig. 2(f) show the strongest resemblance to the original exit wave (cf. Figs. 2(a) and 2(b)), compared to the results obtained with the range constraint in Fig. 2(d) and the support & range constraint in Fig. 2(e). For instance, unlike in Figs. 2(d) and 2(e), the small lines on the fish’s pectoral fin and on the elephant’s trunk are clearly visible in Fig. 2(f). Furthermore, Fig. 2(f) shows a smaller noise level and a better agreement in the range of pixel values with the original exit wave field (cf. Figs. 2(a) and 2(b)) in comparison to Figs. 2(d) and 2(e). The reconstructions in Figs. 2(d) and 2(e) resemble each other strongly, i.e. the additional support information does not seem to be of particular importance in this setting. This may be attributed to the fact that the restriction of the exit wave to the box support is not a very strong constraint as the diffraction fringes in the measurement merely exceed the region of the box.
In Fig. 2(g) we have plotted the root mean square (RMS) error decay of the phase and the amplitude for the three different reconstructions, calculated by
For a fair comparison, only the region inside the box support is taken into account for the calculations. The shearlet & range constraint provides the smallest error for both the phase and the amplitude after 100 iterations. For the range constraint and the support & range constraint, the RMS error of the amplitude does not decrease but increases with the number of iterations, probably due to the high noise level leading to an inconsistent feasibility problem. With the shearlet & range constraint, the algorithm seems to be more noise tolerant. As unstructured noise is associated with small shearlet coefficients, the shearlet thresholding always implies a denoising of the reconstruction, as visible in Fig. 2(f).
The simulation demonstrates that the constraint proposed in  can be successfully extended to the physical situation with a complex-valued exit wave field behind a mixed object. Moreover, in this setting the shearlet & range constraint distinctly outperforms the range constraint and the support & range constraint.
4. Experimental data
Figure 3 shows the analysis of an experimental data set. X-ray propagation images were recorded for a Siemens star on a high-resolution test pattern (ATN/XRESO-50HC, NTT Advanced Technology) consisting of radial stripes in the form of isosceles trapezoids of different size, arranged in rings around a center. The trapezoidal small bases of each ring have a defined length as indicated by numbers in the pattern. They reach from 4 µm down to 50 nm in the center. The pattern is defined in a 200 nm thick layer of tantalum on a Ru/SiC/SiN membrane. Due to the small thickness, the specimen’s absorption is negligible at the given photon energy of E = 8 keV, i.e. we can treat the pattern as a pure phase shifting object. As the star is etched into the tantalum layer, the phase shifts of the structure are positive relative to the surroundings (i.e. here the range constraint reads |ψ| = 1, φ(ψ) ≥ 0).
The Siemens star was imaged at beamline | P10 of the PETRAIII storage ring (DESY, Hamburg) at instrument settings given in . For detection, a 2048 × 2048 pixel SCMOS camera with a gadolinium oxysulfide scintillator of 15 µm thickness was used. A divergent x-ray waveguide probe was used in order to reach nanoscale resolution. The effective near field parameters of the equivalent parallel beam geometry are computed from the variable transformation given by the Fresnel Scaling theorem . Two measurements with different waveguide-specimen and specimen-detector distances, i.e. with different magnification factors, were performed. Table 2 summarizes the relevant setup parameters for both experiments. A detail (1061 × 1061 pixel) of the recorded near field hologram from experiment 1 after applying the dark field and flat field correction is shown in the experimental setup in Fig. 1(a). For both experiments the initial guess for the phase retrieval iteration is composed of a wave field with amplitude equal to one and the phase of the holographic reconstruction.
Figure 3(a) shows the reconstruction results (1061 × 1061 pixel detail) for experiment 1 after 100 iterations of the RAAR algorithm (relaxation parameter α = 0.55) with different object plane constraints. Since the test pattern covers the whole field of view, the support constraint is not applicable. For constraints based on shearlet thresholding a shearlet system of 7 scales and a threshold parameter of θp = 0.001 is used. Compared to the reconstruction with the range constraint, the reconstructions based on shearlet thresholding show less noise and appear to exhibit a higher resolution. The rings corresponding to 4 µm down to 500 nm spatial half-period at the innermost radius are distinguishable in all three reconstructions. However, the stripes of the next smaller ring (200 nm half-period at innermost radius) are merely identifiable in the range constraint reconstruction due to the high noise level, see inlays in Fig. 3(a). The effective pixel size of about 120 nm impedes to resolve the stripes of the two smallest rings (with spatial half-periods of 100 nm and 50 nm) in all reconstructions. The higher performance in terms of resolution of the constraints based on shearlet thresholding compared to the simple range constraint is further reflected in the power spectral densities (PSDs) of the reconstructed phases, see Fig. 3(b). For the range constraint, structures can be distinguished up to spatial frequencies corresponding to a 350 nm half-period resolution before information is erased by noise. Contrarily, for the constraints based on shearlet thresholding the structures reach up to the 250 nm half-period resolution ring and, in parts, even beyond.
Figure 3(c) shows the histograms of phase values corresponding to the three reconstructions in Fig. 3(a). As the Siemens star consists of only two different thickness levels along the direction of illumination, we expect a histogram containing two sharp peaks. In the histogram of the range constraint we observe one very high peak at zero rad and two broader peaks corresponding to the two different object thicknesses. These peaks are smeared out due to noise and they would reach into the negative phase region. However, as the range constraint forces the phase to be nonnegative, the high peak at zero rad is formed. In the histogram of the shearlet & range constraint we have the same phenomenon, but the two relevant peaks are sharper and separated more clearly compared to those from the range constraint. The histogram corresponding to the shearlet constraint exhibits two very broad peaks that melt together rather strongly. Altogether, the shearlet & range constraint yields a histogram in closest resemblance to a binary object. This also reflects the high image contrast present in the corresponding reconstruction. Regarding the quantitativeness of the reconstructions, we observe the following phase shift differences between the relevant maxima (determined by Gaussian fitting) in the histograms of the range constraint and the shearlet & range constraint: Δφrange ≈ 0.31 rad and Δφshearlet&range ≈ 0.32 rad. These values agree very well with the value Δφ ≈ 0.33 rad expected from x-ray optical constants (www.cxro.lbl.gov), i.e. quantitative phase retrieval is provided in both cases.
The RAAR algorithm is also used for the phase retrieval in experiment 2. Here, a shearlet system with 8 scales (the whole 2048 × 2048 pixel camera field is reconstructed) and a threshold parameter of θp = 0.001 are employed. Figure 3(d) shows central details (700 × 700 pixel) of the reconstructed phases obtained with different object plane constraints. As in experiment 1, we observe very clean reconstructions, but at higher resolution (stronger magnification). Now, we can even distinguish the trapezoids of the ring with 100 nm spatial half-period. By-eye-inspection the shearlet & range constraint provides again the best result.
These experiments demonstrate that the constraints based on shearlet thresholding are robust against noise and imperfections of real data. Due to their denoising property they can improve the resolution of the reconstruction and provide results superior to those from the simple range constraint.
In this paper we have analyzed the performance of the shearlet constraint for iterative phase retrieval in x-ray propagation imaging, both for simulations and for experimental data, and have compared reconstruction quality to other commonly used constraints. The shearlet constraint as proposed in  uses the knowledge that the exit wave field can be sparsely represented in a suitable shearlet frame. We have shown that the original formulation can be extended to the physical situation of a specimen that causes both phase shifts and absorption of the illuminating probe (complex-valued exit wave), providing reasonable results for the challenging setting of reconstructing from a single distance exposure (Section 3). Next, the approach was tested for experimental data (Section 4). The reconstructions were found to be consistent with conventional constraints, and partly outperform these. In particular, the shearlet thresholding implies a denoising effect which can lead to better image quality and resolution of the reconstruction.
The shearlet constraint can be applied without further information about the specimen needed. However, the used threshold parameter crucially influences the reconstruction. The optimal value depends not only on the object to be reconstructed but also on the measurement, i.e. the level of noise and the Fresnel number. Within this paper, the selected parameters were found empirically.
In order to obtain a deeper understanding regarding the optimum choice of the threshold and the constraint’s performance in general, it is necessary to further analyze the shearlet constraint within the RAAR algorithm. Further future work would be aimed at extending the shearlet constraint to phase retrieval in three-dimensional x-ray tomography and to far field CDI. The results presented in this paper suggest that constraints based on sparsity information can be advantageously used for phase retrieval tasks.
We thank Martin Krenkel and Anna-Lena Robisch for numerical tools and related fruitful collaborations, and acknowledge financial support by SFB 755 Nanoscale Photonic Imaging.
References and links
1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]
2. 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(1), 133–146 (1996). [CrossRef]
3. 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]
4. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef]
5. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997). [CrossRef]
6. M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015). [CrossRef]
7. S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015). [CrossRef]
8. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (John Wiley & Sons, 2011). [CrossRef]
9. R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004). [CrossRef]
10. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006). [CrossRef]
11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]
12. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999). [CrossRef]
13. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007). [CrossRef] [PubMed]
14. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006). [CrossRef]
15. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). [CrossRef]
16. S. Marchesini, “Invited article: a unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007). [CrossRef]
17. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20(1), 40–55 (2003). [CrossRef]
18. T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1–3), 49–58 (2003). [CrossRef]
19. K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011). [CrossRef]
20. A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015). [CrossRef]
21. V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints,” Appl. Opt. 52(17), 3977–3986 (2013). [CrossRef] [PubMed]
22. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).
25. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20(6), 1025–1034 (2003). [CrossRef]
26. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005). [CrossRef]
27. S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014). [CrossRef]
28. D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).
29. K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.
30. G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, 2012). [CrossRef]
31. P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012). [CrossRef]
32. D. L. Donoho, “Sparse components of images and optimal atomic decomposition,” Constr. Approx. 17(3), 353–382 (2001). [CrossRef]
33. K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007). [CrossRef]
34. G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011). [CrossRef]
36. G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).
37. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41(3), 613–627 (1995). [CrossRef]
38. T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015). [CrossRef] [PubMed]