Abstract

Besides the well-known conventional X-ray attenuation image, Talbot-Lau X-ray imaging (TLXI) provides additional information about the small-angle scattering and refractive features of an object. In general, TLXI setups have to be mechanically robust, since already slight inaccuracies during the measurement process result in moiré artefacts. This work derives moiré artefacts as a result of phase-stepping inaccuracies. The dependency of these artefacts on the phase-stepping inaccuracies is mathematically derived by a Taylor series expansion and verified by a simulation. Among other things, it is shown that moiré artefacts can be calculated by a weighted mean of phase-stepping position deviations to their target positions. These weighting factors vary for each image. Moiré artefacts can even be affected by object features which are not displayed in the particular contrast. The findings of this work offer the possibility to develop advanced reconstruction algorithms which suppress moiré artefacts in the reconstructed images. This reduces the method’s susceptibility to setup component inaccuracies as well as external influences and hence facilitates TLXI for clinical practice.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Conventional X-ray imaging is a common technique in the fields of medical diagnostics and non-destructive testing. The reason for this is the advantage of investigating the internal structure of objects without preparing or modifying them extensively beforehand. However, the method is limited in soft tissue contrast. One possibility to cope with this deficiency is Talbot-Lau X-ray imaging (TLXI). This method allows not only for the examination of the object’s attenuation properties, but also its phase-shifting and small-angle scattering features [1–5]. Thus, the soft tissue contrast can be increased. Promising results have been achieved for various medical [6–11] and industrial applications [12–16].

A Talbot-Lau X-ray interferometer is based on a conventional X-ray setup, which is supplemented with absorbing and phase-shifting gratings to retrieve the phase-contrast and small-angle scattering information. External influences, vibrations, and inaccuracies of the setup components cause displacements of these gratings. Revol et al. [17] have already described noise as a result of stochastic uncertainty of the grating positions and Marschner et al. [18] have stated that these displacements can cause fringe artefacts in the reconstructed images.

This work proves, mathematically derives, and thoroughly discusses the dependency of moiré artefacts in the reconstructed images on the deviations of the grating positions to their target positions during the measurement process. This dependency is mathematically derived by a Taylor series expansion. In addition, a simulation is performed to verify and visualize the results.

The findings of this approach are the foundation to developing advanced reconstruction algorithms which suppress moiré artefacts in the reconstructed images. Hence, the image quality of TLXI improves. This enables further image post-processing methods, such as phase integration algorithms, which are often impeded by moiré artefacts. X-ray dark-field and phase-contrast computed tomography could also benefit from these findings, since moiré artefacts often cause data inconsistency. Additionally, the TLXI setups’ susceptibility to system component inaccuracies as well as external influences is reduced. This facilitates TLXI for clinical practice and reduces setup expenses since stability requirements of system components are decreased.

2. Methods

2.1. Talbot-Lau interferometer

Figure 1 shows a schematic illustration of a Talbot-Lau interferometer. A beam splitter grating (G1) is placed in the beam path between an X-ray source (S) and detector (D). Due to the fractional Talbot effect, an intensity distribution (I) revealing the periodic structure of the beam splitter grating occurs in certain distances behind the grating. If an object (O) is placed in front of the beam splitter grating, the intensity distribution changes due to the absorbing, scattering, and refractive characteristics of the object.

 

Fig. 1 Schematic illustration of a Talbot-Lau interferometer. A beam splitter grating (G1) is placed in the beam path between an X-ray source (S) and detector (D). An intensity distribution (I) equivalent to the periodic structure of the beam splitter grating occurs downstream of the grating. If an object (O) is placed in front of the beam splitter grating, the intensity distribution changes. The necessary spatial coherence can be achieved by using a source grating (G0). The interference fringes can be sampled by stepping an analyzer grating (G2) in front of the detector (D).

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The fractional Talbot effect requires spatially coherent radiation. To meet this requirement, a microfocus X-ray tube with a sufficient small focal spot can be used. Alternatively, a slit mask (G0) can be placed in front of the focal spot of a conventional X-ray tube. This mask absorbs certain parts of the X-ray beam and thereby creates spatially coherent slit sources. Each of these slit sources generates a self-image of the beam splitter grating. By exploiting the Lau effect, it is ensured that these self-images superimpose to a sharp intensity distribution. In general, these interference fringes are too small to be resolved by a conventional X-ray detector. To overcome this challenge, an absorbing analyzer grating (G2) with the same period as the interference fringes is placed at the plane of these fringes. This analyzer grating is used to sample the periodic intensity distribution by shifting it stepwise in its plain perpendicular to its grating bars [3,19].

2.2. Data acquisition

The intensity is measured pixelwise for each shift of the analyzer grating. This measurement method is commonly referred to as phase-stepping. The result corresponds to a sinusoidal function, which is called phase-stepping curve [4]:

In=I¯+Acos(xn+φ+φo),
in which In denotes the measured intensity of the n-th phase step, Ī the mean intensity, A the amplitude, xn=2πNn the phase step position, n the phase step number, N the total number of phase steps over one period, φ the phase shift of the sinusoidal function induced by the phase-shifting property of the object, and φo the phase without an object in the beam. φo is hereinafter referred to as moiré phase. It usually differs over the field of view due to slight misalignments and imperfections of the gratings. This moiré phase determines the basic shape of moiré artefacts in the reconstructed images. All quantities are measured pixelwise, however the indices of the detector pixel coordinates are dropped for clarity. Due to the restriction of the discrete Fourier transformation (DFT), which is used for reconstruction of the images, and for reasons of simplification of subsequent calculations, we specify that the fundamental period of the interference fringes (n ∈ {0, 1, ..., N − 1}) is equidistantly sampled by the phase-stepping method [2].

2.3. Image reconstruction by DFT

Several methods exist to reconstruct images from acquired phase-stepping data, such as the least-squares-fit [20] or deconvolution method [21,22]. This work depicts the relation of phase-stepping inaccuracies and the occurrence of moiré artefacts by example of the DFT reconstruction method [2,4,23]. Therefore, the phase-stepping intensity values(In)n∈{0,1,...,N−1} are considered as a series of equidistantly sampled sinusoidal functions, which can be transformed by a DFT:

Fj=1Nn=0N1Inexp(i2πNnj),
where Fj denotes the j-th Fourier coefficient and i the imaginary unit.

Thus the intensity image Iimg, phase image φimg, amplitude image Aimg, and visibility image Vimg are defined as

Iimg=F0,
Aimg=|F1|+|F1|=2|F1|=2(Re(F1))2+(Im(F1))2,
φimg=tan1(Im(F1)Re(F1)),
Vimg=AimgIimg,
in which Re(F1) denotes the real part and Im(F1) the imaginary part of F1. The reconstruction is performed pixelwise, though the pixel coordinates are dropped for clarity again. Since (In)n∈{0,1,...,N−1} is a real series |F1| = |F−1| is valid. If the phase-stepping process samples the intensity distribution perfectly without any grating position inaccuracies, it can easily be derived that
Iimg=I¯,
Aimg=A,
φimg=φ+φo,
Vimg=AI¯.

In general, not only an object scan (additional subscript ‘obj’), but also a flat-field reference scan (additional subscript ‘ref’) is performed and the attenuation image Γimg, differential phase image Φimg, and dark-field image Σimg can be reconstructed by

Γimg=ln(Iimg,objIimg,ref),
Φimg=φimg,objφimg,ref,
Σimg=Vimg,objVimg,ref.

To focus on the crux of the matter, moiré artefacts of Iimg, Aimg, φimg, and Vimg are investigated in this work. However, it is important to note that the differential phase image Φimg [see Eq. (12)] is free of the moiré phase φo unlike the phase image φimg [see Eq. (9)]. This is due to the fact that in normal case the moiré phase of the object scan φo,obj is equal to the moiré phase of the reference scan φo,ref and mutually annihilate for Φimg. In any case, phase-stepping inaccuracies will cause moiré artefacts even in Φimg.

3. Effect of incorrect phase-stepping positions on the reconstructed images

3.1. Analytical approach

A Taylor series expansion (TSE) is used to examine the effects of incorrect sampling positions on the reconstructed images. In general, for small deviations Δxn TSEs can be truncated after the first order. This results in an approximation of the effect of a shift of a quantity x on a function f:

Δf=f(x+Δx)f(x)=df(x)dxΔx.

Correspondingly, Eq. (14) can be expanded for N independent quantities to

Δf=n=0N1df(x0,,xN1)dxnΔxn.

Using this approximation, the effect of incorrect phase-stepping positions on the reconstructed images can be estimated. These calculations are briefly summarized in the appendix. The results are recapped in the following:

ΔIimg=I¯V1Nn=0N1sin(2πNn+φ+φo)Δxn,
ΔAimg=A1Nn=0N1sin(2(2πNn+φ+φo))Δxn,
Δφimg=1Nn=0N1(1cos(2(2πNn+φ+φo)))Δxn,
ΔVimg=1Nn=0N1(V2sin(2πNn+φ+φo)Vsin(2(2πNn+φ+φo)))Δxn,
whereby ΔIimg, ΔAimg, Δφimg, and ΔVimg denote the change of the intensity image, amplitude image, phase image, and visibility image caused by the deviation of each phase step position Δxn to its target position (n ∈ {0, 1, ..., N − 1}). The resulting image deviations ΔIimg, ΔAimg, Δφimg, and ΔVimg are often specified as moiré artefacts.

3.2. Simulative approach

To verify the results of the analytical approach and visualize the effect of phase-stepping position deviations on the reconstructed images Iimg, Aimg, φimg, and Vimg, an inaccurate phase-stepping (PS) is simulated. Therefore, an artificial PS data set is generated and reconstructed with correct and incorrect phase-stepping positions. The images resulting from the reconstruction with correct phase-stepping positions serve as ground truth images and can be compared to the artefact images resulting from the reconstruction with incorrect phase-stepping positions.

The PS data set is composed by images which describe the object features Ī, φ, and V according to Eq. (1). The amplitude A can thereby be derived by using A = . The moiré phase φo was arbitrarily defined and is displayed in Fig. 2. The left column of Fig. 3 shows the corresponding object images Ī, A, φ, and V. With this information an artificial phase-stepping curve (In)n∈{0,1,...,N−1} can be generated for each pixel according to Eq. (1).

 

Fig. 2 The left figure shows the moiré phase φo used for simulation and calculations. It is arbitrarily defined as a superposition of a linear and radial gradient. It determines the fundamental shape of the moiré artefacts in the reconstructed images and the moiré pattern in each phase step acquisition. The right figure exemplarily shows the first phase step acquisition of the generated artificial PS dataset. It constitutes an experimental raw image. The used object features Ī, φ, and A are displayed in Fig. 3. The periodic moiré pattern of the phase step acquisition is the result of the low frequency moiré phase.

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Fig. 3 The left column shows the intensity image Ī, amplitude image A, phase image φ, and visibility image V used to generate the artificial PS data set. They are also the result of the reconstruction with ideal phase step positions. Hence they serve as ground truth images Īimg,gt, Aimg,gt, φimg,gt, and Vimg,gt. The right column shows the result of a reconstruction with incorrect phase-stepping positions. Hence the images Īimg,art, Aimg,art, φimg,art, and Vimg,art bear moiré artefacts.

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Defining the exact phase-stepping positions xn=2πNn with n ∈ {0, ..., N − 1} the ground truth images can be reconstructed using the DFT method. These ground truth images (additional subscript ‘gt’) correspond to the input images [see Fig. 3, left column] according to Eqs. (7)(10). Defining the incorrect phase-stepping positions (xn+Δxn)=2πN(n+Δn) the moiré artefact containing images (additional subscript ‘art’) can be reconstructed. The right column of Fig. 3 shows these artefact images. The phase image φimg,gt and φimg,art in Fig. 3 are unwrapped and the moiré phase φo was subtracted to reveal the moiré artefacts of the phase image.

 

Fig. 4 The left column shows moiré artefacts calculated by a Taylor series expansion as derived in chapter 3.1. The center column shows moiré artefacts obtained by a reconstruction with incorrect phase step positions as shown in chapter 3.2. The right column shows the difference of the calculated and simulated moiré artefacts.

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Eight phase step positions are used for each reconstruction. The values of the corresponding phase step position deviations are randomly chosen from the interval Δn ∈ [−0.1, 0.1] and listed in Table 1. Besides these listed position deviations, several other sets of position deviations were used to compare the simulative and analytical approach. They all show equivalent results. Hence the set of position deviations listed in Table 1 are exemplarily chosen to evaluate and discuss the results of this work.

Tables Icon

Table 1. Shift Δn of each phase step n used for calculating the phase step position deviations Δxn=2πNΔn, which cause moiré artefacts in the reconstructed images.

The left column of Fig. 4 shows the moire artefacts which are calculated by Eqs. (16)(19). The center column of Fig. 4 shows the moire artefacts which are determined by the simulation. Therefore, the pixelwise difference of the simulated artefact images [see Fig. 3, right column] and the corresponding ground truth images [see Fig. 3, left column] were calculated. Thus the moiré artefacts of the images are educed. The right column of Fig. 4 shows the difference of the calculated and simulated moiré artefacts.

3.3. Evaluation and discussion

Figure 4 displays the calculated and simulated moiré artefacts. The artefacts are similar for each image. However, there are slight, barely visible deviations between the calculated and simulated artefacts. The reason for this is probably that the Taylor series expansion [Eq. (15)], which is used for calculation, is truncated after the first order. Hence, it is not as exact as the simulation results. However, the equivalent results verify the correctness of the analytical approach of chapter 3.1 for the calculation of moiré artefacts.

Equations (16)(19), which are derived by the analytical approach, confirm that incorrect phase step positions cause moiré artefacts in the reconstructed images. The phase step position deviations, which are used for the simulation and calculation in this work, are listed in table 1 and correspond to a misalignment of less than 60 nm if an analyzer grating with a period of 4.8 μm is considered. This misalignment corresponds to only 1.3% of the grating period and can easily be induced by vibrations and other external influences which are given in non-table-top setups and hence in clinical practice. It is difficult to increase the setup stability sufficiently so that enhanced reconstruction algorithms become necessary, which suppresses moiré artefacts in the reconstructed images.

Equations (16)(19) also reveal that the moiré artefact of each image can be calculated by a weighted mean of phase step position deviations. These weighting factors contain trigonometric functions and vary for each image, yet all are dependent on the moiré phase φo. This dependency of the weighting factors on φo as an argument of a sine or cosine function determines the basic shape of moiré artefacts. This becomes obvious by comparing the shape of the used moiré phase φo in Fig. 2 to the shape of the artefacts in Figs. 3 and 4. In the improbable case that the gratings of the interferometer were perfectly aligned and the gratings were free of any imperfections, the moiré phase would be constant over the image. Hence Eqs. (16)(19) show that phase-stepping inaccuracies would not result in moiré artefacts but in object feature dependent offsets. So, a precise sampling of the Talbot self-image is also important for perfectly aligned and imperfection-free gratings.

In addition, Eqs. (16)(19) show, that the moiré artefact of the intensity image ΔI is proportional to the intensity signal Ī and the artefact of the amplitude image ΔA is proportional to the amplitude signal A, whereas the moiré artefact of the phase image Δφ does not scale with the phase signal φ, which is reasonable due to the fact that φ is a cyclic variable. So the moiré artefact Δφ is equally distinct over the whole phase image, whereas ΔI and ΔA scale with their signal Ī and A, as Fig. 4 depicts.

The moiré artefact of the visibility image ΔV implies a term proportional to its visibility signal V and an additional term proportional to V2. The term, which is proportional to V, bears a factor 2 in the argument of its sine function. This results in frequency doubling of the periodic pattern of the moiré artefact. The artefacts Δφ and ΔA also bear a factor 2 in the argument of the sine and cosine functions and hence sustain a frequency doubling as well. The factor 2 also causes a change in weighting the phase step position deviations Δxn, such that the strength of ΔI and ΔA differ [see Fig. 4], although they have the same proportionality (ĪV = A), same number of phase steps N and same phase step position deviations Δxn.

Comparing the artefact containing images of Fig. 3, it becomes evident that the moiré artefact of the intensity image is not observable, whereas the moiré artefacts of the amplitude and visibility images are clearly visible, although ΔI, ΔA, and ΔV are similar in strength [see Fig. 4]. This is due to the difference in the underlying signal strength Ī, A, and V. The intensity signal Ī is stronger than the amplitude signal A and visibility signal V. Moreover, the intensity moiré artefact is often better masked by image features due to its lower frequency compared to the moiré artefacts of the other images. Equations (16)(19) also show that Δφφ is usually larger than ΔII¯, ΔAA, and ΔVV. This is partially due to the fact that the underlying phase signal φ is normally weaker than Ī, A, and V. Hence the phase image sustains the most impact of incorrect phase step positions, which is in agreement with experimental observations. In addition, Δφ is composed of two terms [see Eq. (18)]. The cosine term which determines the moiré artefact and the term 1Nn=0N1Δxn. The latter term displays an unweighted mean of the phase step position deviations and determines a global phase offset induced by the inaccurate phase step positions.

Equations (16)(19) also show, that all moiré artefacts depend on φ; moreover, ΔI is proportional to V. Hence moiré artefacts can be affected by object features which are not displayed in the particular images and thus can appear independent of the shown morphology. Furthermore by comparing ΔI [Eq. (16)] and ΔA [Eq. (17)] with ΔV [Eq. (19)] it can easily be derived that the moiré artefact of the visibility image normed by the visibility signal ΔVV is a superposition of the moiré artefact of the amplitude image normed by the amplitude signal ΔAA and the negative moiré artefact of the intensity image normed by the intensity signal ΔII¯:

ΔVV=ΔAAΔII¯.

Considering Eqs. (11)(13) could suggest that moiré artefacts of reference images (additional subscript ‘ref’) and object images (additional subscript ‘obj’) mutually annihilate resulting in moiré artefact free attenuation, differential phase and dark-field images.

Using Eqs. (11)(13) and separating the intensity, amplitude, phase, and visibility images in their ground truth images (Ī, A, φ, V) and moiré artefact images (ΔIimg, ΔAimg, Δφimg, ΔVimg) result in

Γimg=ln(I¯obj+ΔIimg,objI¯ref+ΔIimg,ref),
Φimg=(φobj+Δφimg,obj)(φref+Δφimg,ref),
img=Vobj+ΔVimg,objVref+ΔVimg,ref.

Consequently, the conditions for moiré artefact free attenuation, differential phase, and dark-field images are

ΔVimg,objΔVimg,ref=I¯objI¯ref,
Δφimg,obj=Δφimg,ref,
ΔVimg,objΔVimg,ref=VobjVref.

Using Eqs. (16)(19) depicts these conditions for all phase step position deviations Δxn of the object and reference scan. These conditions have to be simultaneously fulfilled that moiré artefacts of object and reference images mutually annihilate. Since phase step position deviations can be caused by external influences, vibrations, and inaccuracies of setup components, the deviations can be considered to be stochastic and in general do not fulfill these conditions. Furthermore, the weighting factors of Eqs. (16)(19) differ for object and reference images. Hence moiré artefacts of object and reference images usually do not mutually annihilate and attenuation, differential phase, and dark-field images bear moiré artefacts in the case of inaccurate phase-stepping.

Acquisition time and photon flux deviations between phase step acquisitions could be other potential causes of moiré artefacts, since they cause intensity deviations in the acquired phase stepping data set. As we experienced that X-ray tubes and detectors are in general sufficiently stable and precise, this is mentioned for the sake of completeness and is not further discussed.

4. Conclusion

The result of inaccurate phase-stepping are moiré artefacts in the reconstructed images. The crucial length scale for the inaccuracy amounts to only fractions of a micrometer. The resulting artefacts can be derived by a Taylor series expansion. This approach reveals that moiré artefacts can be calculated as weighted means of the phase step position deviations. These weighting factors vary for each image, but all depend on the moiré phase which determines the basic shape of moiré artefacts. Moreover moiré artefacts can even be affected by object features which are not displayed in the particular contrast and thus can appear independent of the shown morphology. The least impact of inaccurate phase-stepping positions sustains the intensity image, the most impact sustains the phase image. The phase image also sustains a global phase offset by phase-stepping inaccuracies. The moiré artefacts of amplitude, phase, and visibility images show a higher frequency in their periods than the one of the intensity image. The artefact of the visibility image is a superposition of the amplitude moiré artefact and negative intensity moiré artefact.

The derived dependency of moiré artefacts on phase-stepping inaccuracies could be used to develop advanced reconstruction algorithms suppressing moiré artefacts in the reconstructed images. Therefore, an image-based cost function can be defined which is sensitive on moiré artefacts. This cost function is minimized by rearranging the phase step positions of the reference and object scan. The findings of this work can be used to form constraints for the optimization method to improve the results and accelerate the method. This improvement of image quality also enables further image post-processing methods, such as phase integration algorithms, which are often impeded by moiré artefacts. Additionally, the setups’ susceptibility to vibrations, external influences, and system component inaccuracies is reduced. This is an essential requirement to facilitate TLXI for clinical practice. Moreover, it reduces setup expenses, since accuracies and stability requirements of the system components can be loosened. Hence, this work provides the foundation to develop an advanced reconstruction algorithm, which constitutes a substantial and necessary enhancement for TLXI.

Appendix

The appendix contains a brief summary of the calculations of the moiré artefacts caused by inaccurate phase-stepping for the intensity, amplitude, phase, and visibility images.

Intensity image

According to Eq. (15) the moiré artefact ΔIimg of the intensity image Iimg can be derived by

ΔIimg=n=0N1dIimgdxnΔxn,
where Δxn denotes the phase-stepping position deviations to their target positions, N the number of phase steps, and ddxn the derivative with respect to the n-th phase step position xn. Using Eqs. (1)(3) the intensity image can be derived by
Iimg=F0=1Nn=0N1In=1Nn=0N1(I¯+Acos(xn+φ+φo)).
Hence, the derivative of the intensity image with respect to xn can be calculated:
dIimgdxn=A1Nsin(xn+φ+φo).
The final expression of ΔIimg can be achieved by evaluating xn of Eq. (29) at its target positions 2πNn and substituting the resulting expression in Eq. (27)
ΔIimg=A1Nn=0N1sin(2πNn+φ+φo)Δxn.

Amplitude image

The moiré artefact ΔAimg of the amplitude image Aimg can also be derived by Eq. (15):

ΔAimg=n=0N1dAimgdxnΔxn.
The amplitude image Aimg can be obtained by Eq. (4):
Aimg=2|F1|=2(Re(F1))2+(Im(F1))2.
The first order Fourier coefficient F1 is defined by Eq. (2):
F1=1Nn=0N1Inexp(i2πNn)=1Nn=0N1(I¯+Acos(xn+φ+φo))exp(i2πNn),
where Eq. (1) was used to substitute In. Hence, the derivative of the amplitude image with respect to xn can be calculated:
dAimgdxn=ANsin(2(2πNn+φ+φo)),
where xn was evaluated at the target positions 2πNn. By substituting Eq. (34) in Eq. (31) the moiré artefact of the amplitude image can be derived:
Aimg=A1Nn=0N1sin(2(2πNn+φ+φo))Δxn.

Phase image

Equation (15) is also used to calculate the moiré artefact Δφimg of the phase image φimg:

Δφimg=n=0N1dφimgdxnΔxn.
The phase image φimg is described by Eq. (5):
φimg=tan1(Im(F1)Re(F1)).
Accordingly, the derivative of the phase image with respect to the phase-stepping positions can be calculated:
dφimgdxn=1N(1cos(2(2πNn+φ+φo))),
where xn was evaluated at its target positions 2πNn. Substituting Eq. (38) into Eq. (36) results in
Δφimg=1Nn=0N1(1cos(2(2πNn+φ+φo)))Δxn.

Visibility image

The moiré artefact ΔVimg of the visibility image Vimg is derived by Eq. (15) as well:

ΔVimg=n=0N1dVimgdxnΔxn,
whereby the visibility image is defined according to Eq. (6):
Vimg=AimgIimg.
The derivative of the visibility image with respect to the n-th phase step positions can be calculated by
dVimgdxn=ddxnAimgIimg1N(V2sin(2πNn+φ+φo)Vsin(2(2πNn+φ+φo))),
where xn was evaluated at its target positions 2πNn. Substituting Eq. (42) in Eq. (40) derives the moiré artefact of the visibility image:
ΔVimg=1Nn=0N1(V2sin(2πNn+φ+φo)Vsin(2(2πNn+φ+φo)))Δxn.

Acknowledgment

The authors would like to thank the Medical Physics group of the Friedrich-Alexander-University Erlangen-Nuremberg for their scientific input, proliferous discussions, and constructive feedback.

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12. T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009). [CrossRef]   [PubMed]  

13. V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011). [CrossRef]  

14. F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013). [CrossRef]   [PubMed]  

15. M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013). [CrossRef]  

16. S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

17. V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010). [CrossRef]   [PubMed]  

18. M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016). [CrossRef]   [PubMed]  

19. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008). [CrossRef]  

20. C. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef]   [PubMed]  

21. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009). [CrossRef]  

22. P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012). [CrossRef]   [PubMed]  

23. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009). [CrossRef]  

References

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  1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
    [Crossref]
  2. 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, 6296–6304 (2005).
    [Crossref] [PubMed]
  3. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
    [Crossref]
  4. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
    [Crossref] [PubMed]
  5. A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355 (2005).
    [Crossref]
  6. D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
    [Crossref] [PubMed]
  7. M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
    [Crossref] [PubMed]
  8. J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
    [Crossref]
  9. M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
    [Crossref] [PubMed]
  10. T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
    [Crossref] [PubMed]
  11. G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
    [Crossref]
  12. T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
    [Crossref] [PubMed]
  13. V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
    [Crossref]
  14. F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
    [Crossref] [PubMed]
  15. M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
    [Crossref]
  16. S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.
  17. V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
    [Crossref] [PubMed]
  18. M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
    [Crossref] [PubMed]
  19. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
    [Crossref]
  20. C. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [Crossref] [PubMed]
  21. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
    [Crossref]
  22. P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
    [Crossref] [PubMed]
  23. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
    [Crossref]

2016 (1)

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

2013 (6)

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

2012 (1)

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

2011 (3)

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

2010 (1)

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

2009 (3)

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

2008 (2)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

2005 (2)

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

1982 (1)

Anton, G.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

Baumann, J.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Bayer, F.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Bech, M.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Beck, T. J.

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

Beckmann, M.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Bednarzik, M.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Bellotti, R.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Bingham, C. O.

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

Brendel, C.

Brönnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Bruyndonckx, P.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Bunk, O.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

Carrino, J. A.

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

Chen, Z.-Q.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

Christensen, L. B.

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

Cloetens, P.

David, C.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

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, 6296–6304 (2005).
[Crossref] [PubMed]

Diaz, A.

Dobes, K.

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Donath, T.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

Durst, J.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Engelhardt, M.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Fasching, P.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Fehringer, A.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Feidenhans, R.

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

Groot, W.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Grünzweig, C.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Haas, W.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Hartmann, A.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Hauser, N.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Hempel, E.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Henrich, B.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

Herzen, J.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Hoheisel, M.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Hohl, M. K.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Hoshino, Y.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Hostens, J.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Huang, Z.-F.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

Jerjen, I.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Kang, K.-J.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

Kastner, J.

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Kaufmann, R.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Kido, K.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Kiyohara, J.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Kottler, C.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Kubik-Huch, R. A.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Lauridsen, T.

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

Lüthi, T.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Makifuchi, C.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Marschner, M.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Michel, T.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Modregger, P.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Momose, A.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355 (2005).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Morgan, C.

Nagashima, M.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Nagatsuka, S.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Nielsen, M. S.

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

Nishino, S.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Noël, P.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Pauwels, B.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Pelzer, G.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Pfeiffer, F.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

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, 6296–6304 (2005).
[Crossref] [PubMed]

Pinzer, B.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Plank, B.

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Popescu, S.

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Potdevin, G.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Radicke, M.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Rauh, C.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Revol, V.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Rieger, J.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Ritter, A.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Roessl, E.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Sasov, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Scattarella, F.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Scheerer, M.

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Schroer, C.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Schulz-Wendtland, R.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Schuster, M.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Schütz, P.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Senck, S.

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

Sennhauser, U.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Sievers, P.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Singer, G.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Stampanoni, M.

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

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, 6296–6304 (2005).
[Crossref] [PubMed]

Straumann, U.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Stutman, D.

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Tanaka, J.

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Tapfer, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Thüring, T.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Trippel, M.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Uder, M.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Urban, C.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Velroyen, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Wachter, D.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Wang, Z.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

Wang, Z.-T.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

Weber, T.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based x-ray dark-field imaging of ordered structures,” Opt. Express 21, 19922–19933 (2013).
[Crossref] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

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, 6296–6304 (2005).
[Crossref] [PubMed]

Wenkel, E.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Willner, M.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Wucherer, L.

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

Yaroshenko, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Zabler, S.

Zang, A.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

Ziegler, E.

Appl. Phys. Lett. (1)

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

Food Control (1)

M. S. Nielsen, T. Lauridsen, L. B. Christensen, and R. Feidenhans, “X-ray dark-field imaging for detection of foreign bodies in food,” Food Control 30, 531–535 (2013).
[Crossref]

Investigative Radiology (1)

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Investigative Radiology 46, 801–806 (2011).
[Crossref] [PubMed]

J. Appl. Phys. (2)

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105, 102006 (2009).
[Crossref]

J. Microscopy (1)

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microscopy 232, 145–157 (2008).
[Crossref]

Jpn. J. Appl. Phys. (2)

A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355 (2005).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866 (2003).
[Crossref]

Nat. Materials (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Materials 7, 134–137 (2008).
[Crossref] [PubMed]

Nat. Physics (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Physics 2, 258–261 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (2)

T. Michel, J. Rieger, G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713 (2013).
[Crossref] [PubMed]

D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56, 5697 (2011).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel, and C. David, “Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source,” Rev. Sci. Instrum. 80, 053701 (2009).
[Crossref] [PubMed]

Sci. Rep. (2)

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Zeitschrift für Medizinische Physik (1)

G. Anton, F. Bayer, M. Beckmann, J. Durst, P. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, M. Radicke, C. Rauh, J. Rieger, A. Ritter, R. Schulz-Wendtland, M. Uder, D. Wachter, T. Weber, E. Wenkel, and L. Wucherer, “Grating-based darkfield imaging of human breast tissue,” Zeitschrift für Medizinische Physik 23, 228–235 (2013).
[Crossref]

J. Tanaka, M. Nagashima, K. Kido, Y. Hoshino, J. Kiyohara, C. Makifuchi, S. Nishino, S. Nagatsuka, and A. Momose, “Cadaveric and in vivo human joint imaging based on differential phase contrast by x-ray Talbot-Lau interferometry,” Zeitschrift für medizinische Physik 23, 222–227 (2013).
[Crossref]

Other (1)

S. Senck, M. Scheerer, V. Revol, K. Dobes, B. Plank, and J. Kastner, “Non-destructive evaluation of defects in polymer matrix composites for aerospace applications using x-ray Talbot-Lau interferometry and micro CT,” in “58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,” (2017), p. 0355.

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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of a Talbot-Lau interferometer. A beam splitter grating (G1) is placed in the beam path between an X-ray source (S) and detector (D). An intensity distribution (I) equivalent to the periodic structure of the beam splitter grating occurs downstream of the grating. If an object (O) is placed in front of the beam splitter grating, the intensity distribution changes. The necessary spatial coherence can be achieved by using a source grating (G0). The interference fringes can be sampled by stepping an analyzer grating (G2) in front of the detector (D).
Fig. 2
Fig. 2 The left figure shows the moiré phase φo used for simulation and calculations. It is arbitrarily defined as a superposition of a linear and radial gradient. It determines the fundamental shape of the moiré artefacts in the reconstructed images and the moiré pattern in each phase step acquisition. The right figure exemplarily shows the first phase step acquisition of the generated artificial PS dataset. It constitutes an experimental raw image. The used object features Ī, φ, and A are displayed in Fig. 3. The periodic moiré pattern of the phase step acquisition is the result of the low frequency moiré phase.
Fig. 3
Fig. 3 The left column shows the intensity image Ī, amplitude image A, phase image φ, and visibility image V used to generate the artificial PS data set. They are also the result of the reconstruction with ideal phase step positions. Hence they serve as ground truth images Īimg,gt, Aimg,gt, φimg,gt, and Vimg,gt. The right column shows the result of a reconstruction with incorrect phase-stepping positions. Hence the images Īimg,art, Aimg,art, φimg,art, and Vimg,art bear moiré artefacts.
Fig. 4
Fig. 4 The left column shows moiré artefacts calculated by a Taylor series expansion as derived in chapter 3.1. The center column shows moiré artefacts obtained by a reconstruction with incorrect phase step positions as shown in chapter 3.2. The right column shows the difference of the calculated and simulated moiré artefacts.

Tables (1)

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Table 1 Shift Δn of each phase step n used for calculating the phase step position deviations Δ x n = 2 π N Δ n, which cause moiré artefacts in the reconstructed images.

Equations (43)

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I n = I ¯ + A cos ( x n + φ + φ o ) ,
F j = 1 N n = 0 N 1 I n exp ( i 2 π N n j ) ,
I img = F 0 ,
A img = | F 1 | + | F 1 | = 2 | F 1 | = 2 ( Re ( F 1 ) ) 2 + ( Im ( F 1 ) ) 2 ,
φ img = tan 1 ( Im ( F 1 ) Re ( F 1 ) ) ,
V img = A img I img ,
I img = I ¯ ,
A img = A ,
φ img = φ + φ o ,
V img = A I ¯ .
Γ img = ln ( I img , obj I img , ref ) ,
Φ img = φ img , obj φ img , ref ,
Σ img = V img , obj V img , ref .
Δ f = f ( x + Δ x ) f ( x ) = d f ( x ) d x Δ x .
Δ f = n = 0 N 1 d f ( x 0 , , x N 1 ) d x n Δ x n .
Δ I img = I ¯ V 1 N n = 0 N 1 sin ( 2 π N n + φ + φ o ) Δ x n ,
Δ A img = A 1 N n = 0 N 1 sin ( 2 ( 2 π N n + φ + φ o ) ) Δ x n ,
Δ φ img = 1 N n = 0 N 1 ( 1 cos ( 2 ( 2 π N n + φ + φ o ) ) ) Δ x n ,
Δ V img = 1 N n = 0 N 1 ( V 2 sin ( 2 π N n + φ + φ o ) V sin ( 2 ( 2 π N n + φ + φ o ) ) ) Δ x n ,
Δ V V = Δ A A Δ I I ¯ .
Γ img = ln ( I ¯ obj + Δ I img , obj I ¯ ref + Δ I img , ref ) ,
Φ img = ( φ obj + Δ φ img , obj ) ( φ ref + Δ φ img , ref ) ,
img = V obj + Δ V img , obj V ref + Δ V img , ref .
Δ V img , obj Δ V img , ref = I ¯ obj I ¯ ref ,
Δ φ img , obj = Δ φ img , ref ,
Δ V img , obj Δ V img , ref = V obj V ref .
Δ I img = n = 0 N 1 d I img d x n Δ x n ,
I img = F 0 = 1 N n = 0 N 1 I n = 1 N n = 0 N 1 ( I ¯ + A cos ( x n + φ + φ o ) ) .
d I img d x n = A 1 N sin ( x n + φ + φ o ) .
Δ I img = A 1 N n = 0 N 1 sin ( 2 π N n + φ + φ o ) Δ x n .
Δ A img = n = 0 N 1 d A img d x n Δ x n .
A img = 2 | F 1 | = 2 ( Re ( F 1 ) ) 2 + ( Im ( F 1 ) ) 2 .
F 1 = 1 N n = 0 N 1 I n exp ( i 2 π N n ) = 1 N n = 0 N 1 ( I ¯ + A cos ( x n + φ + φ o ) ) exp ( i 2 π N n ) ,
d A img d x n = A N sin ( 2 ( 2 π N n + φ + φ o ) ) ,
A img = A 1 N n = 0 N 1 sin ( 2 ( 2 π N n + φ + φ o ) ) Δ x n .
Δ φ img = n = 0 N 1 d φ img d x n Δ x n .
φ img = tan 1 ( Im ( F 1 ) Re ( F 1 ) ) .
d φ img d x n = 1 N ( 1 cos ( 2 ( 2 π N n + φ + φ o ) ) ) ,
Δ φ img = 1 N n = 0 N 1 ( 1 cos ( 2 ( 2 π N n + φ + φ o ) ) ) Δ x n .
Δ V img = n = 0 N 1 d V img d x n Δ x n ,
V img = A img I img .
d V img d x n = d d x n A img I img 1 N ( V 2 sin ( 2 π N n + φ + φ o ) V sin ( 2 ( 2 π N n + φ + φ o ) ) ) ,
Δ V img = 1 N n = 0 N 1 ( V 2 sin ( 2 π N n + φ + φ o ) V sin ( 2 ( 2 π N n + φ + φ o ) ) ) Δ x n .

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