Abstract

X-ray phase contrast imaging (PCI) has shown great potential for clinical investigation of soft tissues. However, most of the existing X-ray PCI modalities require either a partially coherent source such as a synchrotron or complex setups that are barely compatible with low-dose and patient tomography. This work demonstrates the possibility to efficiently achieve PCI on a low coherence system with a conventional X-ray tube and a detector compatible with a clinical routine. This was accomplished by adapting the speckle-based imaging setup and the numerical phase retrieval processing methods to the low coherence and the low resolution of the experiment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The past decade saw a growing interest in preclinical and clinical applications of X-ray phase contrast imaging (PCI), which generated a significant increase in resources invested into the field [1,2]. The imaging method’s attractiveness comes from its demonstrated ability to provide better contrast for soft tissue than conventional radiography at lower deposited radiation costs. Although PCI with visible light dates back to the 1930s, the idea was only attempted in the X-ray regime in 1965 and took off as a research field in the mid 1990s. This was due to the advent of third generation synchrotrons that could offer temporally and spatially coherent X-ray photons, hence enabling the development of new and more robust phase sensitive methods. Coherent light is indeed an essential requirement for pioneer X-ray PCI techniques such as the popular propagation-based approach. Although some imaging clinical trials were carried at synchrotron facilities in the past [3] and others are still ongoing or planned for the near future with similar sources [4], routine clinical applications using large scale synchrotrons are impossible due to their very high cost and limited access. While compact coherent sources have been developed to alleviate this issue, PCI techniques able to work with low coherence sources are still highly desirable since they can be adapted to existing laboratory sources and make PCI a widespread technique.

In this regard, techniques were successfully adapted to conventional, noncoherent sources [59] with the first results of clinical application recently published [10]. However, even if some of techniques can be adapted for having a 2D sensitivity (see, for instance [11,12]), the majority of the published works deals with sample refraction recovery in only one direction. Moreover, the main drawbacks of those approaches for tomographic clinical implementation are (1) the complex optical setups that are difficult to install in a rotation gantry and (2) their nonoptimized radiation dose deposition due to the additional optical elements that reduce the number of photons reaching the detector after crossing the sample.

More recently, speckle-based imaging (SBI) was developed [13,14] that uses random pattern masks to modulate the X-ray beam rather than periodic structures. This technique, which allows the retrieval of the phase gradient in two directions was later adapted to a laboratory experimental installation using a liquid metal jet source and a high-resolution detector with pixels smaller than 10 µm [15]. It is far beyond what is employed today for imaging a dose-sensitive sample.

Here, we present the successful implementation of a PCI method that allows the retrieval of phase information using a detector with much larger pixels (48 µm) combined with a standard X-ray tube. This marks an important first step toward the clinical implementation of PCI.

The first works on SBI used pieces of sandpaper to randomly modulate the beam wavefront, thanks to the speckle effect arising from the coherent interference of the beam photons. Unfortunately, this phenomenon is hardly achievable with conventional sources due to their weak coherence properties. Thus, when using conventional X-ray systems, random structuring masks achieve the X-ray beam modulation thanks to local differences in attenuation of granular membranes. Despite this difference, the principle of tracking modulations to retrieve the refraction of the sample remains the same.

In practice, a randomly structured membrane is introduced in the path of the beam to generate a random pattern on the detector and a first image ${I_r}$ is acquired as a reference. A second image ${I_s}$ is then obtained with the sample inserted into the beam (see Fig. 1). The interactions of the photons with the sample generate local distortions of the reference pattern. By numerical comparison of the images ${I_s}$ and ${I_r}$, one can retrieve the refraction induced by the sample because it will translate in small local displacement of the random intensity pattern. To achieve better results, several pairs of images can be collected while moving the mask to different positions.

 figure: Fig. 1.

Fig. 1. Random mask phase contrast imaging experimental setup: A randomly structured membrane is placed in the X-ray beam to generate a random intensity modulation on the images. A first image ${I_r}$ is acquired as a reference. Without moving the membrane, a second image ${I_s}$ is captured with the sample inserted into the beam. By analyzing the modulation intensity variations in the two images one can retrieve the refraction caused by the sample.

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A few algorithms are already available in the literature for tracking the pattern movements between the reference and sample images. The first set of methods used an explicit tracking of the modulations by locally comparing, pixel by pixel, the individual displacement of each modulation, using cross-correlation maximization [13,14,16] or functional minimization [17]. These algorithms are not limited by any specific hypothesis and can be applied to a broad range of samples but are computationally expensive. In contrast, the second kind of algorithm employs an implicit tracking where the method does not seek the modulation displacement locally, but rather assumes total photon flux conservation [1820]. The starting point of these algorithms is the transport of intensity equation (TIE) that can be used in the Fresnel regime to describe the evolution of intensity due to the insertion of a nonabsorbing sample into the beam. This equation can be written as

$${I_r}(x,y) - {I_s}(x,y) \approx {\nabla _ \bot} \cdot [{I_r}(x,y){D_ \bot}(x,y)],$$
where, ${D_ \bot} = ({D_x},{D_y})$ is the transverse displacement field, and ${\nabla _ \bot} = \partial /\partial x + \partial /\partial y$ is the 2D transverse gradient operator. The displacement maps are linked to the phase derivatives assuming small angles:
$$\left({\begin{array}{*{20}{c}}{{\nabla _x}}\\{{\nabla _y}}\end{array}} \right)\phi (x,y) = \frac{{2\pi}}{{\lambda \Delta}}\left({\begin{array}{*{20}{c}}{\textbf{x}}\\{\textbf{y}}\end{array}} \right){D_ \bot}(x,y),$$
where $\Delta$ is the sample-to-detector distance and $\lambda$ is the beam photons wavelength. An efficient way to solve this equation for pure phase objects is to use the Fourier derivative theorem as developed by Paganin et al. in [18]. This approach, built on the restricting hypothesis of nonadsorbing sample and the partial use of the beam coherence, is not well-suited for a transfer to low coherence setups and human tissues.

Here, we derive a modified phase retrieval approach, alleviating the above assumptions, that we call a low coherence system (LCS). Adjusting the implicit tracking approach starting from the TIE by accounting for absorption and expanding the right-hand term, Eq. (1) becomes

$$\begin{split}&{{I_r}(x,y) - \frac{{{I_s}(x,y)}}{{{I_{{\rm{obj}}}}(x,y)}} \approx {I_r}(x,y)\nabla {D_ \bot}(x,y)}\\&\quad + {D_ \bot}(x,y){\nabla _ \bot}{I_r}(x,y),\end{split}$$
where ${I_{{\rm{obj}}}}$ is the attenuation image of the object. In Eq. (3), the first right-hand side term corresponds to the variations of intensity due to interference fringes appearing with propagation [21]. The second term mirrors the geometric displacement of the variations of ${I_r}$ due to refraction. When using a low coherence system, the interference fringes will not be resolved and the first term can be neglected, leading to
$$\begin{split}&{{I_r}(x,y) - \frac{{{I_s}(x,y)}}{{{I_{{\rm{obj}}}}(x,y)}} \approx {D_ \bot}(x,y){\nabla _ \bot}{I_r}(x,y)}\\&\quad \approx {D_x}(x,y)\frac{{\partial {I_r}(x,y)}}{{\partial x}} + {D_y}(x,y)\frac{{\partial {I_r}(x,y)}}{{\partial y}}\end{split}.$$

Following the resolution approach of [20], with three (or more) pairs of images acquired at different membrane positions, the variables ${D_x}$, ${D_y}$, and $1/{I_{{\rm{obj}}}}$ are extracted from the resolution of the system of N equations:

$$\begin{split}{I_r^{(k)}(x,y)}&= \frac{1}{{{I_{{\rm{obj}}}}(x,y)}}I_s^{(k)}(x,y) + {D_x}(x,y)\frac{{\partial I_r^{(k)}(x,y)}}{{\partial x}}\\&\quad + {D_y}(x,y)\frac{{\partial I_r^{(k)}(x,y)}}{{\partial y}},\end{split}$$
where $k \in \{1\ldots N\} ,N \ge 3$. In the case of an overdetermined system, we use a least squares resolution [20].

The algorithm is able to retrieve displacements of the modulation pattern well below the pixel size. The main limitations arise from a reduction in the visibility of the modulations due to a blurring caused by the source size and the detector point spread function (PSF), and the shot noise. To limit this issue, we use membranes made of high-Z element powders, thus providing very strong modulations [22]. Additionally, a deconvolution Wiener filter is applied to the acquisitions considering the PSF of the detector and the projected source size. More precisely, the kernel was approximated by a Gaussian shape based on the modulation transfer function measured. Those allowed increase the visibility of the modulations and better track their subpixelic displacements. Finally, as the calculation is done individually for each pixel, the shot noise present in ${D_ \bot}(x,y)$ is removed by a median filter with a radius of three pixels.

The imaging of a domestic fly shows the efficiency of the method with a commercially available X-ray imaging device on a small and complex sample.

This biological sample was analyzed with the LCS on an adapted classic X-ray imaging setup: a conventional microfocus source on an EasyTom XL tomographic setup (RX Solutions, Chavanod, France). The source size was 4 µm FWHM and was operating at 40 kVp. The detector was a Quad-RO CCD detector (Teledyne Princeton Instruments, Trenton, NJ, USA) with a pixel size of $48 \times 48\;{{\unicode{x00B5}}}{{\rm{m}}^2}$. The acquisition time for one projection was 30 s. The standard deviation of the PSF of this detector was measured to be 1.2 pixels. The sample was mounted 12 cm away from the source and a membrane placed at 18.5 cm of the source. The distance from the source to the detector was 53 cm. The membrane was made of TiC grains of average size 100 µm (given by the manufacturer). The random modulation obtained on the detector has an average size of 8 pixels in diameter due to the oval shape. The visibility ($\frac{\sigma}{\mu}$) of 20% where $\sigma$ is the standard deviation, $\mu$ the average measured in a homogeneous zone. Twenty pairs of images were acquired.

Figure 2 displays the displacement maps ${D_x}$ and ${D_y}$ calculated with the LCS approach from 20 pairs of images. It demonstrates a high visual quality, thanks to the high resolution offered by LCS. Counterintuitively, the deconvolution of the acquisition step removes the high frequency noise present in the displacement maps.

 figure: Fig. 2.

Fig. 2. Influence of the deconvolution: Displacement maps ${D_x}$ and ${D_y}$ of the fly sample retrieved with the low coherence system approach without (left) and with (right) deconvolution with a Wiener filter algorithm applied onto the original data (${I_s}$, ${I_r}$).

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 figure: Fig. 3.

Fig. 3. Sensitivity study: Sensitivity of the retrieved refraction angle image versus the number of image pairs $(Is,Ir)$ used in the low coherence system algorithm. Each pair of images were deconvoluted with a Wiener filter. The grayscale values were converted to refraction angles by calculating the angle according to $\alpha = \frac{{{D_ \bot}}}{\Delta}$.

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Figure 3 illustrates the method’s angular sensitivity with respect to the number of measurements. Displacement maps were computed with the LCS with an increasing number of deconvoluted acquisitions (from three to 20). The sensitivity plotted in Fig. 3(b) is computed using the standard deviation within $300 \times 300$ pixels windows in the displacement images outside of the sample. It comes as no surprise that the sensitivity converges asymptotically with the number of measurements reaching a sensitivity of a few microradians (corresponding to a few hundredths of a pixel). Note that by using only a six equations overdetermined system, the results reach an inflection point. Such nonlinear sensitivity evolution will play an important role in choosing the experimental parameter to trade between the accuracy and the dose delivered to the patient.

Finally, we compare the implicit LCS algorithm to an explicit one from the literature: the unified modulated pattern analysis (UMPA) [17].

The algorithms employed for the data processing were implemented in Python3. The UMPA method [17] is accessible at [23], it does not require hypothesis on the sample, and was the one used in the most recent SBI studies on conventional sources [15]. The UMPA parameters used for the fly phase retrieval were a window size of 11x11 pixels and a maximum displacement limited to one pixel. These parameters were obtained iteratively by trial and error until the best results in terms of visual quality were achieved. The implementation of LCS can be found at [24]. The original data can be found at [25].

The integration of the phase gradient images to recover the full phase map was done using a Fourier-based approach derived by Arnison et al. [26].

 figure: Fig. 4.

Fig. 4. Comparison of the implicit and an explicit tracking approaches: (a) Phase images of a fly integrated from the LCS and the UMPA displacement maps. The two algorithms use the same 10 pairs of measurements deconvoluted with a Wiener filter. (b) Extracted profiles from the blue and orange lines in the inset images.

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Figure 4 displays the fly sample phase images integrated from displacement maps obtained with those two algorithms. Visually, LCS renders sharper and straighter edges than UMPA, avoiding staircase artifacts. The resolution obtained with LCS is also improved, as shown in the profiles of Fig. 4(b). LCS successfully renders close lines separated by only 36 µm while UMPA fails to do so. Such improvements in performance can be explained by the pixel-wise analysis approach of LCS, where UMPA uses probing windows. Data presented in Supplement 1 show the results of numerical simulations using a lower coherence source and a geometry compatible with a clinical environment. These data show that the method would be efficient to distinguish muscle from cartilage, which is usually difficult to assess in conventional absorption-based radiography. Although the computational efficiency of the algorithms is difficult to evaluate, the processing of the same images with the same machine and programming language showed that the LCS processing time is reduced by a factor of 30 (2 min per image with the current nonparallel implementation) compared to explicit algorithms. This will be a crucial parameter when treating large volumes of clinical data such as tomography.

The development of an implicit tracking algorithm, specific to low coherence systems, combined with the use of strongly absorbing random masks, causing high gradient modulations, demonstrates the possibility to perform X-ray phase contrast imaging using a conventional source and a low-resolution detector. We presented in this work results obtained on a small sample at a high resolution obtained through geometrical magnification. However, the phase sensitivity of the system is limited by the sample-to-detector distance and the pixel size of the detector, and since these parameters can readily be chosen to be compatible with a clinical system, one can already envisage achieving a similar sensitivity at the clinics (few microradians).

The experimental practicality of the setup combined with the simplicity of the numerical approach paves the way for phase contrast tomography at the clinics with complex absorbing tissues. Moreover, this algorithm may be used for wavelengths other than X-rays as various studies employ similar approaches with visible light beams scrambled by random masks [2729]. Concerning X-rays, the method used here is not restricted to this particular range of energy and might be applicable to higher ones if needed. Finally, the method presented here does not take into account the scattering signal because it is considered to have a very minor influence on the calculated phase based on previous experiments performed in synchrotrons. This will be investigated in future work.

Acknowledgment

The authors acknowledge Luc Salvo and Pierre Lhuissier for their help during the experiment. The authors also thank Simap laboratory for granting beam time on the microCT machine. We acknowledge useful discussions with Kaye Morgan, David Paganin, and Konstantin Pavlov.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this Letter are available at [25].

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012). [CrossRef]  

2. H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020). [CrossRef]  

3. E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011). [CrossRef]  

4. Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015). [CrossRef]  

5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006). [CrossRef]  

6. M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009). [CrossRef]  

7. A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]  

8. C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014). [CrossRef]  

9. F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015). [CrossRef]  

10. C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020). [CrossRef]  

11. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007). [CrossRef]  

12. G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015). [CrossRef]  

13. S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012). [CrossRef]  

14. K. S. Morgan, D. M. Paganin, and K. K. Siu, Appl. Phys. Lett. 100, 124102 (2012). [CrossRef]  

15. M.-C. Zdora, I. Zanette, T. Walker, N. Phillips, R. Smith, H. Deyhle, S. Ahmed, and P. Thibault, Appl. Opt. 59, 2270 (2020). [CrossRef]  

16. S. Berujon and E. Ziegler, Phys. Rev. A 92, 013837 (2015). [CrossRef]  

17. M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017). [CrossRef]  

18. D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018). [CrossRef]  

19. K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020). [CrossRef]  

20. K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020). [CrossRef]  

21. T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, J. Opt. Soc. Am. A 23, 34 (2006). [CrossRef]  

22. H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patent WO2021005283 (14 January 2020).

23. P. Thibault, “Github repository of unified modulated pattern analysis,” Github (2017 ), https://github.com/pierrethibault/UMPA.

24. E. Brown, “Popcorn,” Github (2021), https://github.com/DoctorEmmetBrown/popcorn.

25. E. Brun, “DomesticFlyPCIConventionalSource,” OSF (2021), https://osf.io/4ruq2/.

26. M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004). [CrossRef]  

27. L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019). [CrossRef]  

28. Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021). [CrossRef]  

29. P. Berto, H. Rigneault, and M. Guillon, Opt. Lett. 42, 5117 (2017). [CrossRef]  

References

  • View by:

  1. A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012).
    [Crossref]
  2. H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
    [Crossref]
  3. E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
    [Crossref]
  4. Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
    [Crossref]
  5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
    [Crossref]
  6. M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
    [Crossref]
  7. A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007).
    [Crossref]
  8. C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
    [Crossref]
  9. F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
    [Crossref]
  10. C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
    [Crossref]
  11. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
    [Crossref]
  12. G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
    [Crossref]
  13. S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012).
    [Crossref]
  14. K. S. Morgan, D. M. Paganin, and K. K. Siu, Appl. Phys. Lett. 100, 124102 (2012).
    [Crossref]
  15. M.-C. Zdora, I. Zanette, T. Walker, N. Phillips, R. Smith, H. Deyhle, S. Ahmed, and P. Thibault, Appl. Opt. 59, 2270 (2020).
    [Crossref]
  16. S. Berujon and E. Ziegler, Phys. Rev. A 92, 013837 (2015).
    [Crossref]
  17. M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
    [Crossref]
  18. D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
    [Crossref]
  19. K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
    [Crossref]
  20. K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
    [Crossref]
  21. T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, J. Opt. Soc. Am. A 23, 34 (2006).
    [Crossref]
  22. H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patentWO2021005283 (14January2020).
  23. P. Thibault, “Github repository of unified modulated pattern analysis,” Github (2017 ), https://github.com/pierrethibault/UMPA .
  24. E. Brown, “Popcorn,” Github (2021), https://github.com/DoctorEmmetBrown/popcorn .
  25. E. Brun, “DomesticFlyPCIConventionalSource,” OSF (2021), https://osf.io/4ruq2/ .
  26. M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
    [Crossref]
  27. L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
    [Crossref]
  28. Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
    [Crossref]
  29. P. Berto, H. Rigneault, and M. Guillon, Opt. Lett. 42, 5117 (2017).
    [Crossref]

2021 (1)

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

2020 (5)

K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
[Crossref]

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
[Crossref]

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

M.-C. Zdora, I. Zanette, T. Walker, N. Phillips, R. Smith, H. Deyhle, S. Ahmed, and P. Thibault, Appl. Opt. 59, 2270 (2020).
[Crossref]

2019 (1)

L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
[Crossref]

2018 (1)

D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
[Crossref]

2017 (2)

P. Berto, H. Rigneault, and M. Guillon, Opt. Lett. 42, 5117 (2017).
[Crossref]

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
[Crossref]

2015 (4)

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

S. Berujon and E. Ziegler, Phys. Rev. A 92, 013837 (2015).
[Crossref]

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

2014 (1)

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

2012 (3)

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, Appl. Phys. Lett. 100, 124102 (2012).
[Crossref]

2011 (1)

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

2009 (1)

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

2007 (2)

2006 (2)

2004 (1)

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[Crossref]

Abrami, A.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Ahmed, S.

Arboleda, C.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Arfelli, F.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[Crossref]

Basta, D.

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

Bech, M.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

Berto, P.

Berujon, S.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
[Crossref]

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
[Crossref]

D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
[Crossref]

S. Berujon and E. Ziegler, Phys. Rev. A 92, 013837 (2015).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012).
[Crossref]

H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patentWO2021005283 (14January2020).

Bohic, S.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

Boss, A.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012).
[Crossref]

Bregant, P.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Brown, J. M.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

Brun, E.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
[Crossref]

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
[Crossref]

D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
[Crossref]

H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patentWO2021005283 (14January2020).

Brun, F.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

Bunk, O.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[Crossref]

Burns, Z.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Casarin, K.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Castelli, E.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Chen, Q.

L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
[Crossref]

Chenda, V.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Coan, P.

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012).
[Crossref]

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[Crossref]

Cova, M. A.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

David, B.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

David, C.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[Crossref]

Deyhle, H.

Diemoz, P.

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

Diemoz, P. C.

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

Dreossi, D.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Endrizzi, M.

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

Fan, Y.

L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
[Crossref]

Fayard, B.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

Feidenhans, R.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

Forte, S.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Gaudin, P.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

Guillon, M.

Gureyev, T. E.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, J. Opt. Soc. Am. A 23, 34 (2006).
[Crossref]

Hagen, C.

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

Jefimovs, K.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Jensen, T. H.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

Kallon, G. K.

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

Khorashad, L. K.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Kitchen, M. J.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

Koch, F. J.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
[Crossref]

Koehler, T.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Kottler, C.

Kubik-Huch, R. A.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Kuhn, N.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Labriet, H.

D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
[Crossref]

H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patentWO2021005283 (14January2020).

Lång, K.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[Crossref]

Last, A.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
[Crossref]

Lee, Y. U.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Leo, C.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Li, G.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Li, H. T.

K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
[Crossref]

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
[Crossref]

Liu, Z.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Lockie, D.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

Longo, R.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Lu, L.

L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
[Crossref]

Ma, Q.

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
[Crossref]

Marcon, M.

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
[Crossref]

Mathieu, H.

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

Mayo, S. C.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

Menk, R. H.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Millard, T. P.

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

Morgan, K. S.

K. S. Morgan, D. M. Paganin, and K. K. Siu, Appl. Phys. Lett. 100, 124102 (2012).
[Crossref]

Munro, P.

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

Nesterets, Y. I.

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, J. Opt. Soc. Am. A 23, 34 (2006).
[Crossref]

Olivo, A.

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
[Crossref]

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
[Crossref]

A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

Paganin, D. M.

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
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D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
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K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
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K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
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Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
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Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
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C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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Rigon, L.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
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H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
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Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
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Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
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M.-C. Zdora, I. Zanette, T. Walker, N. Phillips, R. Smith, H. Deyhle, S. Ahmed, and P. Thibault, Appl. Opt. 59, 2270 (2020).
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Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
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L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
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Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
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M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
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L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
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Appl. Opt. (1)

Appl. Phys. Lett. (4)

G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, Appl. Phys. Lett. 107, 204105 (2015).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, Appl. Phys. Lett. 100, 124102 (2012).
[Crossref]

A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

F. A. Vittoria, G. K. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, Appl. Phys. Lett. 106, 224102 (2015).
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Eur. Radiol. (1)

C. Arboleda, Z. Wang, K. Jefimovs, T. Koehler, U. Van Stevendaal, N. Kuhn, B. David, S. Prevrhal, K. Lång, S. Forte, R. A. Kubik-Huch, C. Leo, G. Singer, M. Marcon, A. Boss, E. Roessl, and M. Stampanoni, Eur. Radiol. 30, 1419 (2020).
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Front. Phys. (1)

L. Lu, J. Sun, J. Zhang, Y. Fan, Q. Chen, and C. Zuo, Front. Phys. 7, 77 (2019).
[Crossref]

J. Microsc. (1)

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
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J. Opt. (1)

K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, J. Opt. 22, 125604 (2020).
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J. Opt. Soc. Am. A (1)

J. Synchrotron Radiat. (1)

Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, D. Thompson, J. M. Brown, M. J. Kitchen, K. M. Pavlov, D. Lockie, F. Brun, and G. Tromba, J. Synchrotron Radiat. 22, 1509 (2015).
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Med. Phys. (1)

C. Hagen, P. Munro, M. Endrizzi, P. Diemoz, and A. Olivo, Med. Phys. 41, 070701 (2014).
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Nat. Commun. (1)

Y. U. Lee, J. Zhao, Q. Ma, L. K. Khorashad, C. Posner, G. Li, G. B. M. Wisna, Z. Burns, J. Zhang, and Z. Liu, Nat. Commun. 12, 1 (2021).
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Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
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Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, Phys. Med. Biol. 54, 2747 (2009).
[Crossref]

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2012).
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Phys. Rev. A (3)

S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012).
[Crossref]

S. Berujon and E. Ziegler, Phys. Rev. A 92, 013837 (2015).
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D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, Phys. Rev. A 98, 053813 (2018).
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Phys. Rev. Appl. (1)

K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, Phys. Rev. Appl. 13, 054023 (2020).
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Phys. Rev. Lett. (1)

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, Phys. Rev. Lett. 118, 203903 (2017).
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Radiology (1)

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, Radiology 259, 684 (2011).
[Crossref]

Sci. Rep. (1)

H. Rougé-Labriet, S. Berujon, H. Mathieu, S. Bohic, B. Fayard, J.-N. Ravey, Y. Robert, P. Gaudin, and E. Brun, Sci. Rep. 10, 1 (2020).
[Crossref]

Other (4)

H. Labriet, S. Berujon, and E. Brun, “X-ray imaging device and associated imaging method,” France patentWO2021005283 (14January2020).

P. Thibault, “Github repository of unified modulated pattern analysis,” Github (2017 ), https://github.com/pierrethibault/UMPA .

E. Brown, “Popcorn,” Github (2021), https://github.com/DoctorEmmetBrown/popcorn .

E. Brun, “DomesticFlyPCIConventionalSource,” OSF (2021), https://osf.io/4ruq2/ .

Supplementary Material (1)

NameDescription
Supplement 1       Results of a numerical simulation on a biological phantom using a clinical compatible experimental set-up with a large spot (70 µm) X-ray source, a pixel size of 70µm and a source to detector distance of 1m

Data availability

Data underlying the results presented in this Letter are available at [25].

25. E. Brun, “DomesticFlyPCIConventionalSource,” OSF (2021), https://osf.io/4ruq2/.

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Figures (4)

Fig. 1.
Fig. 1. Random mask phase contrast imaging experimental setup: A randomly structured membrane is placed in the X-ray beam to generate a random intensity modulation on the images. A first image ${I_r}$ is acquired as a reference. Without moving the membrane, a second image ${I_s}$ is captured with the sample inserted into the beam. By analyzing the modulation intensity variations in the two images one can retrieve the refraction caused by the sample.
Fig. 2.
Fig. 2. Influence of the deconvolution: Displacement maps ${D_x}$ and ${D_y}$ of the fly sample retrieved with the low coherence system approach without (left) and with (right) deconvolution with a Wiener filter algorithm applied onto the original data (${I_s}$, ${I_r}$).
Fig. 3.
Fig. 3. Sensitivity study: Sensitivity of the retrieved refraction angle image versus the number of image pairs $(Is,Ir)$ used in the low coherence system algorithm. Each pair of images were deconvoluted with a Wiener filter. The grayscale values were converted to refraction angles by calculating the angle according to $\alpha = \frac{{{D_ \bot}}}{\Delta}$.
Fig. 4.
Fig. 4. Comparison of the implicit and an explicit tracking approaches: (a) Phase images of a fly integrated from the LCS and the UMPA displacement maps. The two algorithms use the same 10 pairs of measurements deconvoluted with a Wiener filter. (b) Extracted profiles from the blue and orange lines in the inset images.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

I r ( x , y ) I s ( x , y ) [ I r ( x , y ) D ( x , y ) ] ,
( x y ) ϕ ( x , y ) = 2 π λ Δ ( x y ) D ( x , y ) ,
I r ( x , y ) I s ( x , y ) I o b j ( x , y ) I r ( x , y ) D ( x , y ) + D ( x , y ) I r ( x , y ) ,
I r ( x , y ) I s ( x , y ) I o b j ( x , y ) D ( x , y ) I r ( x , y ) D x ( x , y ) I r ( x , y ) x + D y ( x , y ) I r ( x , y ) y .
I r ( k ) ( x , y ) = 1 I o b j ( x , y ) I s ( k ) ( x , y ) + D x ( x , y ) I r ( k ) ( x , y ) x + D y ( x , y ) I r ( k ) ( x , y ) y ,