# Evaluation of the Weighted Mean X-ray Energy for an Imaging System Via Propagation-Based Phase-Contrast Imaging

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement Setup at GSI

#### 2.2. Measurement Setup at DLS

#### 2.3. Propagation-Based Phase-Contrast

#### 2.4. Computer Simulation

#### 2.5. Energy Evaluation for GSI Data

## 3. Results

#### 3.1. Evaluation of the Dominant X-ray Energy

#### 3.2. Validation of the Evaluated Dominant X-ray Energy

#### 3.3. Validation of the Evaluation Method with Monochromatic Images

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Le Pape, S.; Neumayer, P.; Fortmann, C.; Döppner, T.; Davis, P.; Kritcher, A.; Landen, O.; Glenzer, S. X-ray radiography and scattering diagnosis of dense shock-compressed matter. Phys. Plasmas
**2010**, 17, 056309. [Google Scholar] [CrossRef][Green Version] - Hochhaus, D.C.; Aurand, B.; Basko, M.; Ecker, B.; Kühl, T.; Ma, T.; Rosmej, F.; Zielbauer, B.; Neumayer, P. X-ray radiographic expansion measurements of isochorically heated thin wire targets. Phys. Plasmas
**2013**, 20, 062703. [Google Scholar] [CrossRef][Green Version] - Schwoerer, H.; Gibbon, P.; Düsterer, S.; Behrens, R.; Ziener, C.; Reich, C.; Sauerbrey, R. MeV X Rays and Photoneutrons from Femtosecond Laser-Produced Plasmas. Phys. Rev. Lett.
**2001**, 86, 2317–2320. [Google Scholar] [CrossRef] [PubMed][Green Version] - Momose, A. Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer. Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrom. Detect. Assoc. Equip.
**1995**, 352, 622–628. [Google Scholar] [CrossRef] - Momose, A. Recent Advances in X-ray Phase Imaging. Jpn. J. Appl. Phys.
**2005**, 44, 6355. [Google Scholar] [CrossRef] - Pfeiffer, F.; Weitkamp, T.; Bunk, O.; David, C. Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources. Nat. Phys.
**2006**, 2, 258–261. [Google Scholar] [CrossRef] - Willner, M.; Herzen, J.; Grandl, S.; Auweter, S.; Mayr, D.; Hipp, A.; Chabior, M.; Sarapata, A.; Achterhold, K.; Zanette, I.; et al. Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging. Phys. Med. Biol.
**2014**, 59, 1557–1571. [Google Scholar] [CrossRef] - Myers, G.R.; Mayo, S.C.; Gureyev, T.E.; Paganin, D.M.; Wilkins, S.W. Polychromatic cone-beam phase-contrast tomography. Phys. Rev. A
**2007**, 76, 045804. [Google Scholar] [CrossRef] - Wilkins, S.; Gureyev, T.E.; Gao, D.; Pogany, A.; Stevenson, A. Phase-contrast imaging using polychromatic hard X-rays. Nature
**1996**, 384, 335. [Google Scholar] [CrossRef] - Snigirev, A.; Snigireva, I.; Kohn, V.; Kuznetsov, S.; Schelokov, I. On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation. Rev. Sci. Instrum.
**1995**, 66, 5486–5492. [Google Scholar] [CrossRef] - Meadowcroft, A.L.; Bentley, C.D.; Stott, E.N. Evaluation of the sensitivity and fading characteristics of an image plate system for X-ray diagnostics. Rev. Sci. Instrum.
**2008**, 79, 113102. [Google Scholar] [CrossRef] - Clark, J.N.; Putkunz, C.T.; Pfeifer, M.A.; Peele, A.G.; Williams, G.J.; Chen, B.; Nugent, K.A.; Hall, C.; Fullagar, W.; Kim, S.; et al. Use of a complex constraint in coherent diffractive imaging. Opt. Express
**2010**, 18, 1981. [Google Scholar] [CrossRef] [PubMed] - Bagnoud, V.; Aurand, B.; Blazevic, A.; Borneis, S.; Bruske, C.; Ecker, B.; Eisenbarth, U.; Fils, J.; Frank, A.; Gaul, E.; et al. Commissioning and early experiments of the PHELIX facility. Appl. Phys. B
**2010**, 100, 137–150. [Google Scholar] [CrossRef] - Antonelli, L.; Barbato, F.; Mancelli, D.; Trela, J.; Zeraouli, G.; Boutoux, G.; Neumayer, P.; Atzeni, S.; Schiavi, A.; Volpe, L.; et al. X-ray phase-contrast imaging for laser-induced shock waves. EPL (Europhys. Lett.)
**2019**, 125, 35002. [Google Scholar] [CrossRef][Green Version] - Schropp, A.; Hoppe, R.; Meier, V.; Patommel, J.; Seiboth, F.; Ping, Y.; Hicks, D.G.; Beckwith, M.A.; Collins, G.W.; Higginbotham, A.; et al. Imaging Shock Waves in Diamond with Both High Temporal and Spatial Resolution at an XFEL. Sci. Rep.
**2015**, 5. [Google Scholar] [CrossRef][Green Version] - Fiksel, G.; Marshall, F.J.; Mileham, C.; Stoeckl, C. Note: Spatial resolution of Fuji BAS-TR and BAS-SR imaging plates. Rev. Sci. Instrum.
**2012**, 83, 086103. [Google Scholar] [CrossRef] - Weitkamp, T.; Haas, D.; Wegrzynek, D.; Rack, A. ANKAphase: Software for single-distance phase retrieval from inline X-ray phase-contrast radiographs. J. Synchrotron Radiat.
**2011**, 18, 617–629. [Google Scholar] [CrossRef] [PubMed] - Burvall, A.; Lundström, U.; Takman, P.A.C.; Larsson, D.H.; Hertz, H.M. Phase retrieval in X-ray phase-contrast imaging suitable for tomography. Opt. Express
**2011**, 19, 10359–10376. [Google Scholar] [CrossRef] [PubMed] - Souvorov, A.; Ishikawa, T.; Kuyumchyan, A. Multiresolution phase retrieval in the Fresnel region by use of wavelet transform. J. Opt. Soc. Am. Opt. Image Sci.
**2006**, 23, 279–287. [Google Scholar] [CrossRef] - Voelz, D.G. Computational Fourier Optics: A MATLAB Tutorial; SPIE Press: Bellingham, WA, USA, 2011. [Google Scholar]
- Giewekemeyer, K.; Krüger, S.P.; Kalbfleisch, S.; Bartels, M.; Beta, C.; Salditt, T. X-ray propagation microscopy of biological cells using waveguides as a quasipoint source. Phys. Rev. A
**2011**, 83. [Google Scholar] [CrossRef] - Hagemann, J.; Salditt, T. Divide and update: Towards single-shot object and probe retrieval for near-field holography. Opt. Express
**2017**, 25, 20953–20968. [Google Scholar] [CrossRef] [PubMed][Green Version] - Paganin, D.M. Coherent X-ray Optics; Oxford University Press (OUP): Oxford, UK, 2006. [Google Scholar] [CrossRef]
- Paganin, D.; Mayo, S.C.; Gureyev, T.E.; Miller, P.R.; Wilkins, S.W. Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J. Microsc.
**2002**, 206, 33–40. [Google Scholar] [CrossRef] [PubMed] - Bronnikov, A.V. Reconstruction formulas in phase-contrast tomography. Opt. Commun.
**1999**, 171, 239–244. [Google Scholar] [CrossRef] - Cloetens, P.; Ludwig, W.; Baruchel, J.; Van Dyck, D.; Van Landuyt, J.; Guigay, J.P.; Schlenker, M. Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays. Appl. Phys. Lett.
**1999**, 75, 2912–2914. [Google Scholar] [CrossRef][Green Version] - Guigay, J.P.; Langer, M.; Boistel, R.; Cloetens, P. Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region. Opt. Lett.
**2007**, 32, 1617–1619. [Google Scholar] [CrossRef] - Henke, B.; Gullikson, E.; Davis, J. X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E = 50–30,000 eV, Z = 1–92. At. Data Nucl. Data Tables
**1993**, 54, 181–342. [Google Scholar] [CrossRef][Green Version] - Morgan, K.S.; Siu, K.K.W.; Paganin, D.M. The projection approximation and edge contrast for X-ray propagation-based phase contrast imaging of a cylindrical edge. Opt. Express
**2010**, 18, 9865–9878. [Google Scholar] [CrossRef] - Matsushima, K.; Shimobaba, T. Band-Limited Angular Spectrum Method for Numerical Simulation of Free-Space Propagation in Far and Near Fields. Opt. Express
**2009**, 17, 19662–19673. [Google Scholar] [CrossRef][Green Version] - Bearden, J.A. X-ray Wavelengths. Rev. Mod. Phys.
**1967**, 39, 78–124. [Google Scholar] [CrossRef]

**Figure 1.**Measurement setup at Gesellschaft für Schwerionenforschung (GSI). The laser beam is split into backlighter and heater. The backlighter is shot at a tungsten wire for generating X-rays. The heater can optionally be shot at an object for generating shock waves or explosions. The inset shows a zoom-in on the tungsten backlighter wire (B) and the titanium wire (T) positions, depicted as green circles. In the presented measurement the heater is not shot at the titanium wire and the cold titanium wire is imaged. The images are obtained with imaging plates. The magnet is used to divert electrons, which occur due to the explosion of the wire and cause noise in the acquired images.

**Figure 2.**Measurement of a titanium wire with regions of interest (ROI).

**Green Rectangle**: ROI that is used to reconstruct the wire.

**Red Rectangle**: background.

**Figure 3.**

**Left**: phase reconstructions for the green marked region of interest in Figure 2 retrieved with the following energies (from top to bottom): $E=[2.7;4.7;11.8;22.0]\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$.

**Right**: lineplots (blue) of the retrieved phase, which is shown in the

**Left**images, and theoretical phase at the centre of the wire (red).

**Figure 4.**Evaluation of the dominant energy. (

**a**) Absolute phase difference between theoretical phase and retrieved mean phase in dependence of the energy. The calculated energy range is between $2\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $22\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$. (

**b**) Detailed view on the minimum with a finer step-size (purple). For the finer step-size the evaluated energy range is between $11.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $12.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$. In blue a section of the plot shown in (

**a**) can be seen. The error is in the regime of $0.08\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}$.

**Figure 5.**Absolute difference between the mean phase-shift of the green region of interest and the theoretical value depending on the energy. (

**a**) The grey curve results from the reconstruction process with a uniform reference and the orange curve from a noisy reference with $\mathrm{NL}=31.81\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$. (

**b**) Results for different assumed diameters of the titanium wire. The blue curve presents the result shown in Figure 4 for the given value of the diameter of $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. In green, red and yellow the absolute phase difference for a theoretical phase-shift of a $45\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$, a $55\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ and a $60\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ wire is shown, respectively. The error is in the regime of $0.08\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}$.

**Figure 6.**Monochromatic simulations of the propagation signature of a $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ titanium wire for different energies. Source-blurring is neglected in the simulations. A lineplot of the measurement shown in Figure 2 is plotted in blue for comparative reasons. In yellow the simulation for $10\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$, in orange the simulation for $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and in red the simulation for $12\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ is shown. The simulation in orange for $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ yields the best fit for the measured values at the centre of the wire.

**Figure 7.**Monochromatic simulation of the propagation signature of titanium wires with varying diameter of $\pm 10\%$ of the supposed diameter of $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. The results for the best fitting energies are shown for each diameter. Source-blurring is included in the simulations. A lineplot of the measurement shown in Figure 2 is plotted in blue for comparative reasons. In yellow the simulation of a wire with $45\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ diameter for $10.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$, in orange the simulation of a wire with $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ diameter for $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and in red the simulation of a wire with $55\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ diameter for $11.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ is shown.

**Figure 8.**Simulation of the propagation signature of a titanium wire of $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ diameter. On the left (

**a**,

**c**,

**e**), three toy spectra with different weighted mean energies (red line) and with different energy distributions are assumed. Three monochromatic lines (green) at $5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$, $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $20\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ ((

**b**)) or $5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$, $15\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $20\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (c/d and e/f), respectively, are used to illustrate the spectra. On the right (

**b**,

**d**,

**f**), the corresponding simulated propagation signatures of a $50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ titanium wire are shown in orange. Source-blurring is neglected in the simulations. The blue line shows a lineplot of the measurement of the wire. From top to bottom the weighted mean energy of the spectra is $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (a/b), $13\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (c/d) and $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (e/f). The maximal contributing energy bin of the spectra is $11\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (a/b), $15\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (c/d) and $15\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ (e/f).

**Figure 9.**(

**a**) Detector read-out of the propagation signatures of three carbon wires acquired at Diamond Light Source (DLS). (

**b**) Reconstructed thickness of the carbon wires at the correct energy of $10\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$. (

**c**,

**d**) Absolute difference between mean phase-shift of the region marked with the red line in (

**a**) and theoretical value depending on the energy. (

**c**) Whole energy range between $2\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $22\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$. (

**d**) Energy range around the minimum between $9.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ and $10.5\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}$ in finer steps (purple). In blue the corresponding section of the plot in (

**c**) is shown. The error is in the range of $0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}$.

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**MDPI and ACS Style**

Seifert, M.; Weule, M.; Cipiccia, S.; Flenner, S.; Hagemann, J.; Ludwig, V.; Michel, T.; Neumayer, P.; Schuster, M.; Wolf, A.;
et al. Evaluation of the Weighted Mean X-ray Energy for an Imaging System Via Propagation-Based Phase-Contrast Imaging. *J. Imaging* **2020**, *6*, 63.
https://doi.org/10.3390/jimaging6070063

**AMA Style**

Seifert M, Weule M, Cipiccia S, Flenner S, Hagemann J, Ludwig V, Michel T, Neumayer P, Schuster M, Wolf A,
et al. Evaluation of the Weighted Mean X-ray Energy for an Imaging System Via Propagation-Based Phase-Contrast Imaging. *Journal of Imaging*. 2020; 6(7):63.
https://doi.org/10.3390/jimaging6070063

**Chicago/Turabian Style**

Seifert, Maria, Mareike Weule, Silvia Cipiccia, Silja Flenner, Johannes Hagemann, Veronika Ludwig, Thilo Michel, Paul Neumayer, Max Schuster, Andreas Wolf,
and et al. 2020. "Evaluation of the Weighted Mean X-ray Energy for an Imaging System Via Propagation-Based Phase-Contrast Imaging" *Journal of Imaging* 6, no. 7: 63.
https://doi.org/10.3390/jimaging6070063