Open Access
Add to BibSonomyAbstract
An interferogram conditioning procedure, for subsequent phase retrieval by Fourier demodulation, is presented here as a fast iterative approach aiming at fulfilling the classical boundary conditions imposed by Fourier transform techniques. Interference fringe patterns with typical edge discontinuities were simulated in order to reveal the edge artifacts that classically appear in traditional Fourier analysis, and were consecutively used to demonstrate the correction efficiency of the proposed conditioning technique. Optimization of the algorithm parameters is also presented and discussed. Finally, the procedure was applied to grating-based interferometric measurements performed in the hard X-ray regime. The proposed algorithm enables nearly edge-artifact-free retrieval of the phase derivatives. A similar enhancement of the retrieved absorption and fringe visibility images is also achieved.
© 2015 Optical Society of America
1. Introduction
The analysis of interference fringe patterns is a key issue in interferometric-based metrology and phase imaging applications. On fringe patterns resulting from two-dimensional gratings, it can be performed by different means, such as barycentric calculation [1–3], cross-correlation [4–6], Hilbert-Huang transformation [7], regularized phase-tracking [8], vortices localization [9] or Fourier analysis [10–12] of the modulated signal. The latter, based on the use of Fast Fourier Transform (FFT) algorithms, is a fast and convenient way to extract the information carried by a fringe pattern. However, in presence of edge discontinuities, Gibbs ringing artifacts are to be expected around the edges of the analysis window [13–15]. In some cases, such artifacts may affect the relevance of the extracted information, inducing qualitative or quantitative misleading interpretation of the collected data. Therefore, satisfying the FFT boundary conditions is a prerequisite for improving the quality of the Fourier-retrieved useful information.
C. and F. Roddier proposed in [16] a Gerchberg-type iterative approach [17] to deal with analysis pupils of arbitrary shape, and to map the complex fringe visibility of interferograms. Here, we propose to improve and extend their approach to correct for edge artifacts. The process consists in the creation of artificial fringes outside the initial support to achieve an extended interferogram ensuring signal edge continuity. For demonstration purposes, we will use simulated interference fringe patterns. After a detailed description of the interferogram conditioning method, the algorithm convergence and correction efficiency will be studied as a function of adjustable parameters. Finally, we will apply the process to experimental interferograms obtained from measurements performed in the hard X-ray regime. Indeed, X-ray phase imaging showed an increasing interest over the past decade thanks to its potentially higher contrast and sensitivity compared to classical absorption imaging [18–21]. The weak absorption properties of matter in this wavelength range can thus be overcome, providing access to unrevealed complementary information for medical diagnosis, biology or material science [22–25]. Moreover, the multimodal specificity of interferometry enables not only to retrieve the phase, but also the absorption and fringe visibility [26,27]. As it will be demonstrated, the enhancement provided by this conditioning algorithm on the phase information quality, is also reflected on the Fourier-retrieved absorption and fringe visibility images.
2. Boundary issues in Fourier analysis
Fourier analysis of interferograms is based on the use of the well-known FFT algorithms, to access the interferogram frequency-domain from which the useful information can be retrieved [10]. The relationship between the samples of the signal and their representation in the frequency domain is computed by the Discrete Fourier Transform (DFT). The latter makes use of Euler’s identity to approximate the signal by a finite series of continuous sine and cosine waves. Therefore, in presence of signal discontinuities (wherever located within the signal), FFT algorithms will create ringing artifacts also known as Gibbs phenomenon [13–15]. But this effect can also be seen as an interesting periodization property of the FFT, from which one can take advantage.
To highlight boundary issues in Fourier analysis, we chose to simulate a two-dimensional fringe pattern by considering a [0, π] checkerboard single phase grating under monochromatic parallel light illumination. The pixel size in the observation plane was adjusted to achieve a spatial sampling of 6 pixels per grating period, i.e. 3 pixels per fringe, thus ensuring best representation of the interferogram in the frequency domain for subsequent Fourier analysis. Besides, the field of view was fixed to an arbitrary size of 132 × 132 pixels, in order to fill the analysis window with an integer number of grating periods. Simulations were performed by using the following parameters: 8 µm for the grating period, 1.33 µm for the pixel size, about 4.96 cm for the propagation distance behind the grating and 17.5 keV for the photon energy. This dimensioning is close to the X-ray grating interferometer developed on the Metrology beamline at SOLEIL synchrotron and described in [28,29].
The ideal interferogram [Fig. 1(a)], ensuring signal edge continuity, was thus generated by use of an incoming plane wave composed of a flat amplitude profile and a horizontally-tilted flat wavefront. The wavefront X-tilt component was added in order to consider a constant and non-zero value X phase derivative to be retrieved. Arbitrarily fixed to 26.906 µrad, it corresponds to a phase shift of 1 pixel size at the observation distance behind the grating. Such an interferogram, referred to as the ideal case, satisfies the boundary conditions for optimal phase retrieval by Fourier demodulation. However, experimentally measured interferograms are generally discontinuous periodic functions, given that the phase carried by light leads by propagation to both intensity variations and fringe pattern deformations in the observation plane. Inhomogeneous illumination in the field of view may also be of concern. Three cases of edge discontinuities are represented in Fig. 1: by applying a smooth intensity variation to the ideal interferogram [Fig. 1(b)], by removing one pixel-column from one side of the ideal interferogram, while keeping homogeneous illumination [Fig. 1(c)], and by propagating half a period of a flat-amplitude Sine wave [Fig. 1(d)]. All these cases represent discontinuous periodic functions. Their X phase derivatives retrieved by Fourier demodulation are given in Figs. 2(b)-2(d) respectively. As expected, Gibbs ringing artifacts are observable with decreasing amplitude from the edges to the center of the supports. Moreover, one can observe on Fig. 2(b) (which represents a case of two-dimensional edge discontinuities), that transverse ringing artifacts (along Y) are also present. It demonstrates the two-dimensional nature of these effects even in the case of a one-dimensional phase derivative retrieval. In comparison, the Fourier demodulation of the ideal interferogram generates no edge artifacts [Fig. 2(a)]. The extracted X phase derivative is here estimated, within the computational error, to the theoretically expected tilt value of 26.906 µrad.
Fig. 1 (a) Continuous periodic interferogram obtained from a tilted plane wave, (b) Same as (a) with inhomogenous illumination in the field of view, (c) Same as (a) with one pixel-column removed from one side of the support, (d) Interferogram generated by half a period of an incoming flat-amplitude Sine wave.
Fig. 2 (a), (b), (c) and (d) X phase derivatives retrieved by Fourier demodulation of the interferograms given in Figs. 1(a), 1(b), 1(c) and 1(d) respectively. For comparison purposes, all Figs. (except (d)) use the same Z-scale and colorbar. The slope error is given in Peak-to-Valley (PV) value in each case.
3. Wavefront analysis by support periodization, WASP procedure
In order to overcome these edge discontinuities and the related ringing artifacts appearing through Fourier analysis, we propose to perform a prior conditioning of the fringe pattern. An obvious solution would be to spatially filter the interferogram (with help of a Hann or Blackman window for instance), but this would impinge on the collected data. Considering the need to preserve all useful information, another approach consists in creating additional signal outside the initial support, to result in obtaining an extended continuous periodic function. The procedure presented here is derived from [16] and also based on a Gerchberg-type iterative algorithm. We will name it WASP for Wavefront Analysis by Support Periodization.
For description of the algorithm, we used a simulated interferogram combining both cases of discontinuities as performed for Figs. 1(b) and 1(c). The process, illustrated in Fig. 3, is first initialized by injecting the initial interferogram into a larger support, completed with zero values in both dimensions. The positioning of the fringe pattern within this extended support has no incidence on the outcome of the WASP procedure. We aligned the initial data in one corner of the support, leading to an L-shape side-frame of zero values, i.e. the periodization area [Fig. 3]. This extended interferogram is first Fourier transformed, and filtered in the frequency domain by selecting the fringe-pattern related harmonics. Inverse Fourier transform is consecutively applied in order to recover the corresponding real-space signal, spread over the whole extended support. Doing so, the periodization area gets filled with artificial fringes. But because of the spectral filtering performed in Fourier space, the data over the base support are also affected. Therefore, the initial fringe pattern needs to be reinjected into the extended support. Finally, the resulting interferogram replaces the previous one for the second iteration, and so on. Moreover, to force the algorithm to achieve best periodization of the interferogram, the Fourier-space harmonics selection windows are progressively increased in size during convergence. Starting with a very small windowing of few pixels ensures a smooth initiation of the process on first iterations, by using only few spatial frequencies of the initial fringe pattern. As the convergence moves forward, bandwidth of the Fourier spectral filtering is progressively increased, thus allowing more spatial frequencies to let the artificial fringes evolve properly. After few tens of iterations, these latter enable a nearly-perfect joining between the edges of the interferogram [Fig. 4].
Such a conditioning enables consequently a nearly-edge-artefact-free Fourier analysis of the initial fringe pattern. Compared to [16], the implementation of an evolutionary spectral filtering allows convergence of the algorithm to the global minimum of the problem, even in the case of strongly distorted interferograms.
4. WASP parameters optimization
Convergence and performances of the WASP algorithm will now be studied as a function of adjustable parameters, such as the width of the side-frame defining the periodization area, the bandwidth evolution of the spectral filtering performed in Fourier space, and finally the correction efficiency of the algorithm over the number of iterations. The interferogram considered here is the same as the one described previously in section 3. Without prior conditioning of the fringe pattern, its X phase derivative retrieved by Fourier analysis shows edge artifacts of 76.31 µrad Peak-to-Valley (PV), and a mean value of 26.73 µrad. All PV and mean values, given in this section, were calculated over the initial support.
To fix the best settings for the algorithm, we considered ten cases of spectral filtering. These ones were defined according to the size N × N of the initial interferogram, which represents our smallest reference support. Thus, with N = 132 pixels, the spectral windowing applied in Fourier space can reach a maximum size of N/3 = 44 pixels [Fig. 3]. We set the latter to 43 pixels (~32.58% N) in order to have odd-size spectral windows to be aligned on the maximum of each of the 9 harmonics. Hence, we considered the ten following filters: - five cases of “steady-state filters” using fixed-size harmonic selection windows of 3, 11, 21, 31 and 43 pixels; - and five cases of “evolutionary filters” using a linear evolution in size with arbitrary growth rates of 20%, 40%, 80%, 160% and 308%. These latter correspond to an increase in size from 3 to 43 pixels over 200, 100, 50, 25 and 13 iterations respectively. Once one of those spectral windowings reaches the maximum size of 43 pixels, it remains at this size for the following iterations, turning from there the evolutionary filters into steady-state filters. Figure 5(a) shows the evolution of these spectral filters for N = 132 pixels.
Fig. 5 (a) Spectral windowing evolutions according to the number of iterations, for Ne = N = 132 pixels. The spectral widening is performed from odd-size to odd-size spectral windows. (b), (c), (d), (e) X phase derivative final PV and mean values, given after 100 iterations of WASP, as functions of the side-frame width, (b) and (c) with the steady-state spectral filters, (d) and (e) with the evolutionary filters. Each point is the result of a WASP convergence.
In order to generalize the algorithm parameters to any value of N and any width of the side-frame, the previous filters will be given, from now, as percentages of Ne, where Ne is the one-dimensional size of the square matrix defining the extended support (Ne corresponds to N when the width of the side-frame equals zero). This leads to fixed-size spectral windows of (2.27% Ne), (8.33% Ne), (15.91% Ne), (23.48% Ne) and (32.58% Ne) for the five steady-state filters, and to growth rates of (0.15% Ne / iter.), (0.3% Ne / iter.), (0.61% Ne / iter.), (1.21% Ne / iter.) and (2.33% Ne / iter.) for the five evolutionary filters respectively. Such a generalization ensures conservation of the spectral content, or of its evolution, whatever the size of the extended interferogram.
Finally, for the case under study, we considered 100 iterations of WASP to be enough to reach a stationary regime of correction, whatever the settings of the algorithm.
In a first step, we studied the edge artifacts amplitude reduction obtained from different sizes of the periodization area. We varied the width of the side-frame from 1 to 100 pixels (per 1 pixel step), and for each one of those we performed 100 iterations of WASP before retrieving the final PV and mean values. Thus, the influence of the side-frame width could be retrieved for each of the ten spectral filters under study. Figures 5(b) and 5(c) give the results obtained with the five steady-state filters; and Figs. 5(d) and 5(e) with the five evolutionary filters.
As one can see on Figs. 5(b) and 5(c), the (8.33% Ne) to (32.58% Ne) spectral filters enable a stable convergence of the algorithm for a width of the side-frame above 20 pixels. On the contrary, the (2.27% Ne) windowing is clearly too restrictive for the process, cutting too much of the required spatial frequencies to achieve a nearly-stationary regime of convergence. Indeed, for widths of the side-frame below 70 pixels, the algorithm behavior appears to be very erratic, depending on whether the width of the side-frame is a multiple of the grating period or not. A small region of quite stable convergence can be found between 70 and 80 pixels, but the algorithm convergence becomes unstable again for larger widths above 85 pixels.
On the other hand, the evolutionary filters [Figs. 5(d) and 5(e)] give very similar behaviors. Their correction efficiencies evolve in the same manner for widths of the side-frame above 10 pixels. However, when reaching their nearly-stationary regime of convergence, one can notice that the correction provided by the process becomes more efficient as the growth rate of the spectral filtering decreases. These results already suggest that the slowest spectral widening will be the most effective for elimination of edge artifacts.
In any case, all evolutionary filters lead to a stationary regime of correction for a width of the side-frame between 40 and 45 pixels. This is also largely true for the steady-state filters. Therefore, we set the optimal width of the side-frame to 48 pixels, as a multiple number of the grating period. It corresponds to 8 grating periods (i.e. 16 fringe periods).
In a second step, we studied the convergence of the algorithm for each of the ten filters, by considering the previously-fixed side-frame. Figures 6(a) and 6(b) give the results obtained with the steady-state filters; and Figs. 6(c) and 6(d) with the evolutionary ones.
Fig. 6 X phase derivative PV and mean values as functions of the number of iterations, (a) and (b) with the steady-state spectral filters, (c) and (d) with the evolutionary filters. Here, the width of the side-frame is fixed to 48 pixels, leading to Ne = 180 pixels.
On Fig. 6(a), we can observe that the (2.27% Ne) spectral filter is the less efficient. More precisely, in this case, the algorithm converges on the first few iterations, but then rapidly diverges to get finally stuck above the initial slope errors. Although this filter does not achieve any reduction of the edge artifacts amplitude, it is nevertheless interesting to note that it enables improvement of the mean value estimation [Fig. 6(b)]. The four other filters allow, for their part, reduction of the slope errors, with performances in correction that appear conditioned by the spectral windowing under consideration. The (8.33% Ne) spectral filter leads to the best speed of convergence on first iterations, and to the best correction efficiency after 100 iterations of WASP. The latter are both reduced progressively as the spectral windowing becomes larger. This behavior of the algorithm is reflected on the residual slope errors, as well as on the mean values. Thus, the (32.58% Ne) spectral filter, which authorizes the highest spatial frequencies, leads also to the worst correction efficiency. The slope errors are here reduced to about 16 µrad PV in a single iteration, without any further significant reduction through the following iterations. The mean value estimation remains unimproved throughout the whole process. These results demonstrate how the contribution of the high spatial frequencies in early iterations penalizes the process irremediably. Convergence to a global minimum can only be achieved through a smooth initiation of the periodization procedure. Thus, the steady-state nature of filtering appears to be a strong constraint for the algorithm, limiting the convergence to local minima only. It also highlights the complexity in finding out the best spectral windowing for a given problem.
As shown by Figs. 6(c) and 6(d), all evolutionary filters enable convergence of the WASP procedure. The filter with a growth rate of (0.15% Ne / iter.) gives, on first iterations, a convergence similar to that obtained with the (2.27% Ne) steady-state filter. Afterwards, the increase in size of the spectral windowing prevents the algorithm to diverge. Although this spectral widening leads to the slowest convergence, it offers finally the best correction efficiency, with a slope errors reduction to about 0.71 µrad PV and a mean value estimation of 26.9067 µrad after 100 iterations of WASP. As the growth rate of the spectral windowing gets higher, the speed of convergence on first iterations increases, but the final correction provided by the algorithm becomes less efficient. This behavior is in agreement with the assumption made above from Figs. 5(d) and 5(e). Thus, a compromise has to be found between the speed of convergence and the final correction efficiency, keeping in mind that only the slowest spectral widening will enable convergence to the best periodization.
In conclusion, these results demonstrate the need to increase progressively the bandwidth of the Fourier spectral filtering during the process, in order to achieve a smooth extrapolation of the interferogram. For best correction of edge artifacts, the spatial frequencies have to be taken into account, from the lowest to the highest, as slowly as possible to let the artificial fringes evolve properly within the periodization area. Of course, the final correction efficiency is also linked to the number of iterations, given that the highest spatial frequencies will only contribute at last. Thus, the number of iterations to perform depends on the complexity of the function to be periodized and on the final correction one want to achieve. This applies in particular to strongly distorted interferograms presenting a broadband spectral content.
Here, we can see that the filter with a growth rate of (0.15% Ne / iter.) ensures the most efficient convergence for the algorithm. Table 1 summarizes the WASP correction performances obtained with this filter, showing an attenuation factor of about 94% after 20 iterations and about 99% after 100 iterations. The evolution of the artificial fringes within the periodization area is the one given in Fig. 7(a). The corresponding evolution of the extended interferogram frequency-domain is given in Fig. 7(b). The latter shows how the Fourier-space signal gets progressively concentrated around the fringe-pattern related harmonics, as the function boundary conditions get fulfilled. Finally, the extracted X phase derivatives, during WASP convergence, are given in Fig. 7(c). The slope errors are here corrected to 0.71 µrad PV after 100 iterations.
Table 1. WASP correction performances obtained with the progressive spectral widening of (0.15% Ne / iter.). The values given at the 0th iteration correspond to the ones obtained without WASP.
Fig. 7 Evolutions of (a) the artificial fringes within the periodization area, (b) the extended interferogram frequency-domain, and (c) the retrieved X phase derivative, during WASP convergence with the progressive spectral widening of (0.15% Ne / iter.).
Finally, we applied the WASP algorithm (with the same parameters) to the three interferograms of Figs. 1(b), 1(c) and 1(d). For the present cases, we performed 100 iterations of WASP to achieve best correction efficiencies. The corresponding X phase derivatives are given in Fig. 8. In each case, the slope errors are corrected below 1 µrad PV (to be compared to Fig. 2), showing attenuation factors of 99.27%, 97.33% and 97.71% respectively. Of course, performing a higher number of iterations would have improved all these corrections.
Fig. 8 (a), (b) and (c) X phase derivatives retrieved by Fourier analysis after prior WASP conditioning of the interferograms of Figs. 1(b), 1(c) and 1(d) respectively. The algorithm parameters were fixed to: 48 pixels for the width of the side-frame, (0.15% Ne / iter.) for the growth rate of the spectral filtering and 100 for the number of iterations. The Z-scales and colorbars are here the same as in Figs. 2(b), 2(c), and 2(d).
After optimization of the algorithm parameters, the WASP procedure shows to be a fast and highly efficient approach to correct for edge artifacts. Typical attenuation factors of better than 97% can be achieved in few tens of iterations. Nevertheless, the number of iterations to perform still depends on the interferogram to be periodized and on the final correction to achieve. For indication only, and with our computing performances, the calculation time required by the conditioning process was about 20 ms / iteration for a 180 × 180 pixels image.
5. Application to x-ray phase imaging
To illustrate the enhancement in quality achieved on Fourier-retrieved images through WASP procedure, we used interferometric measurements performed in the hard X-ray regime. Experiments were conducted on the Metrology beamline at SOLEIL synchrotron, using a single phase grating X-ray interferometer [28,29] under 17.5 keV beam illumination. The setup uses a two-dimensional checkerboard phase grating of 6 µm pitch and 3 µm height. Made out of gold, the latter generates a ~0.85π phase shift at the working photon energy. The detection uses a 20 µm thick YAG:Ce crystal for conversion of the X-rays to visible light and a highly sensitive visible CCD camera (PCO 2000s) of 2048 × 2048 pixels (7.4 µm/pixel). The detection plane was relayed onto the CCD chip through a × 5.55 magnification optical system, leading to an effective pixel size of 1.33 µm and a field of view of 2.73 × 2.73 mm2.
Here, the sample under test is a sea shell, analyzed over a field of view of 1500 × 1500 pixels (~2 × 2 mm2). According to the previous optimization discussed in section 4, we fixed the width of the side-frame to 8 grating periods (36 pixels), and the spectral widening to a growth rate of (0.15% Ne / iter.), i.e. 2.304 pixels/iteration. Thus, the spectral windowing increases from 3 to 231 pixels over 100 iterations. Figure 9(a) shows the measured interferogram within its analysis window, and Fig. 9(b) its juxtaposition (including side-frame) in a 2 × 2 matrix configuration. By using the previously-optimized settings for the algorithm, the edges of the interferogram appear perfectly joined after 50 iterations only, with no significant improvement after 100 iterations [Figs. 9(e) and 9(f)]. In comparison, the boundary conditions are still not fulfilled after 100 iterations of WASP, when using a steady-state filter of 231 pixels (about 15% Ne) [Figs. 9(c) and 9(d)]. The gain in progressively increasing the spectral windowing is here clearly visible. For indication only, and with our computing performances, the calculation time required by the conditioning process was about 550 ms / iteration for a 1536 × 1536 pixels image.
Fig. 9 (a) Interferogram of a sea shell measured on the SOLEIL synchrotron Metrology beamline at 17.5 keV, (b) Juxtaposition in a 2 × 2 matrix configuration, including side-frame for subsequent periodization of the interferogram, Enlarged parts of the 4 corners after 50 and 100 iterations of WASP, (c) and (d) by using a 231-pixels steady-state filter, and (e) and (f) by using a progressive spectral widening of (0.15% Ne / iter.).
The X phase derivative was then retrieved from the periodized interferogram and compared to the classical retrieval performed without WASP [Fig. 10]. Enhancement in the quality of the image can be observed near the edges of the support, as well outside as within the sample. Of course ringing artifacts appear in both dimensions, since the initial interferogram has two-dimensional edge discontinuities.
A similar enhancement in quality can be achieved on the Fourier-retrieved absorption and fringe visibility images. The absorption signal or continuum is recovered from the zero-order of the interferogram frequency-domain. It reveals the sample absorption, but also phase information which turns into intensity modulations through propagation of light. On the absorption signal of the sea shell [Fig. 11], we can see how Gibbs ringing artifacts may interfere with the sample absorption footprint, leading to ambiguous interpretation of the sample structure.
Finally, the fringe visibility image was recovered from the first-orders of the interferogram frequency-domain. We can observe a similar enhancement in the quality of the image through WASP procedure [Fig. 12].
As shown by Figs. 10-12, Gibbs ringing artifacts are here corrected even in the case of a noisy data set. Indeed, since the noise component of the initial fringe pattern is also represented in the interferogram frequency-domain, it is naturally taken into account by the Fourier spectral filtering to be finally reproduced within the periodization area, along with the artificial fringes. Hence, the periodization and correction efficiencies of the proposed conditioning procedure are both insensitive to noise.
6. Conclusion
In this paper, we presented a conditioning algorithm for interference fringe patterns to be properly analyzed by Fourier demodulation. The algorithm enables periodization of the interferogram over an extended support, by creating artificial fringes ensuring boundary continuity throughout an iterative back and forth Fourier process. The information carried by the fringe pattern can thus be recovered without edge artifacts.
We demonstrated the performances and correction efficiency of the WASP algorithm by use of discontinuous periodic simulated interferograms. The algorithm parameters were optimized throughout a systematic study. The latter reveals in particular the need to increase as slowly as possible the bandwidth of the Fourier spectral filtering during the process, in order to achieve best correction efficiencies.
The WASP procedure was finally applied to interference fringe patterns obtained from measurements performed in the hard X-ray regime. The algorithm proved to be a highly efficient approach to improve the quality and relevance of the Fourier-extracted information, such as the phase derivatives or absorption and fringe visibility images. Although applied here to fully-illuminated interferograms, it is obviously also of interest for managing analysis pupils of arbitrary shape, as it often occurs in optical metrology. However, this conditioning process does not correct for intra-pupillary discontinuities.
Fourier analysis techniques can thus benefit from this approach to achieve a better qualitative and quantitative diagnosis in metrology and phase imaging applications. More generally, this noise-insensitive procedure can be applied to any signal that has to be spectrally analyzed by use of FFT algorithms, or represented in the frequency domain for further Fourier processing.
Acknowledgments
The authors would like to thank Fayçal Bouamrane for fabrication and provision of the X-ray phase grating and Paulo Da Silva for help during experiments on the SOLEIL Metrology beamline. This work has been supported by Triangle de la Physique (contract 2010-076T) and by a public grant from the “Laboratoire d’Excellence Physics Atoms Light Mater” (LabEx PALM) overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0039-PALM).
References and links
1. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971).
2. K. L. Baker and M. M. Moallem, “Iteratively weighted centroiding for Shack-Hartmann wave-front sensors,” Opt. Express 15(8), 5147–5159 (2007). [CrossRef] [PubMed]
3. P. Mercère, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett. 28(17), 1534–1536 (2003). [CrossRef] [PubMed]
4. J. P. Lewis, “Fast normalized cross-correlation,” Vision Interface 10(1), 120–123 (1995).
5. L. A. Poyneer, D. W. Palmer, K. N. LaFortune, and B. Bauman, “Experimental results for correlation-based wave-front sensing,” Proc. SPIE 5894, 58940N (2005). [CrossRef]
6. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011). [CrossRef] [PubMed]
7. M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013). [CrossRef] [PubMed]
8. J. Villa, J. A. Quiroga, and M. Servín, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39(4), 502–508 (2000). [CrossRef] [PubMed]
9. J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem in optical vortices interferometry,” Opt. Commun. 234(1), 23–28 (2004). [CrossRef]
10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
11. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005). [CrossRef] [PubMed]
12. H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35(12), 1932–1934 (2010). [CrossRef] [PubMed]
13. J. W. Gibbs, “Fourier’s series,” Nature 59(1522), 200 (1898). [CrossRef]
14. M. Bôcher, “Introduction to the theory of Fourier’s series,” Ann. Math. 7(3), 81–152 (1906). [CrossRef]
15. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986). [CrossRef] [PubMed]
16. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987). [CrossRef] [PubMed]
17. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta (Lond.) 21(9), 709–720 (1974). [CrossRef]
18. C. David, B. Nohammer, H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]
19. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003). [CrossRef] [PubMed]
20. A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44(9A), 6355–6367 (2005). [CrossRef]
21. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]
22. A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006). [CrossRef]
23. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]
24. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52(23), 6923–6930 (2007). [CrossRef] [PubMed]
25. A. Momose, Y. Takeda, W. Yashiro, A. Takeuchi, and Y. Suzuki, “X-ray phase tomography with a Talbot interferometer in combination with an x-ray imaging microscope,” J. Phys. Conf. Ser. 186, 012044 (2009). [CrossRef]
26. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]
27. W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23(7), 9233–9251 (2015). [CrossRef] [PubMed]
28. J. Rizzi, P. Mercère, M. Idir, P. D. Silva, G. Vincent, and J. Primot, “X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer,” Opt. Express 21(14), 17340–17351 (2013). [CrossRef] [PubMed]
29. J. Rizzi, P. Mercère, M. Idir, N. Guérineau, E. Sakat, R. Haïdar, G. Vincent, P. D. Silva, and J. Primot, “X-ray phase contrast imaging using a broadband x-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys. Conf. Ser. 425(19), 192002 (2013). [CrossRef]
