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]:

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(I_n)_{n∈{0,1,...,N−1}} are considered as a series of equidistantly sampled sinusoidal functions, which can be transformed by a DFT:

Thus the intensity image I_img, phase image φ_img, amplitude image A_img, and visibility image V_img are defined as

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 Γ_img, differential phase image Φ_img, and dark-field image Σ_img can be reconstructed by

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

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 Δx_n 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:

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

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:

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 I_img, A_img, φ_img, and V_img, 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 = VĪ. The moiré phase φ_o 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 (I_n)_{n∈{0,1,...,N−1}} can be generated for each pixel according to Eq. (1).

 

Fig. 2 The left figure shows the moiré phase φ_o 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 Ī_img,gt, A_img,gt, φ_img,gt, and V_img,gt. The right column shows the result of a reconstruction with incorrect phase-stepping positions. Hence the images Ī_img,art, A_img,art, φ_img,art, and V_img,art bear moiré artefacts.

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Defining the exact phase-stepping positions $x_n=\frac{2\pi }{N}n$ 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 $\left(x_n+\mathrm{\Delta }{x}_n\right)=\frac{2\pi }{N}(n+\mathrm{\Delta }n)$ the moiré artefact containing images (additional subscript ‘art’) can be reconstructed. The right column of Fig. 3 shows these artefact images. The phase image φ_img,gt and φ_img,art in Fig. 3 are unwrapped and the moiré phase φ_o 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.

Table 1. Shift Δn of each phase step n used for calculating the phase step position deviations $\mathrm{\Delta }{x}_n=\frac{2\pi }{N}\mathrm{\Delta }n$, which cause moiré artefacts in the reconstructed images.

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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 φ_o. This dependency of the weighting factors on φ_o 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 φ_o 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 Δx_n, 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 Δx_n.

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 $\frac{\mathrm{\Delta }\phi }{\phi }$ is usually larger than $\frac{\mathrm{\Delta }I}{\overline{I}}$, $\frac{\mathrm{\Delta }A}{A}$, and $\frac{\mathrm{\Delta }V}{V}$. 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 $\frac{1}{N}{\sum }_{n=0}^{N-1}\mathrm{\Delta }{x}_n$. 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 $\frac{\mathrm{\Delta }V}{V}$ is a superposition of the moiré artefact of the amplitude image normed by the amplitude signal $\frac{\mathrm{\Delta }A}{A}$ and the negative moiré artefact of the intensity image normed by the intensity signal $-\frac{\mathrm{\Delta }I}{\overline{I}}$:

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 (ΔI_img, ΔA_img, Δφ_img, ΔV_img) result in

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

Using Eqs. (16)–(19) depicts these conditions for all phase step position deviations Δx_n 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.