Advanced Phase-Retrieval for Stepping-Free X-Ray Dark-Field Computed Tomography

Grating-based phase- and dark-field-contrast X-ray imaging is a novel technology that aims to extend conventional attenuation-based X-ray imaging by unlocking two additional contrast modalities. The so called phase-contrast and dark-field channels provide enhanced soft tissue contrast and additional microstructural information. Accessing this additional information comes at the expense of a more intricate measurement setup and necessitates sophisticated data processing. A big challenge for translating grating-based dark-field computed tomography to medical applications lies in minimizing the data acquisition time. While a continuously moving detector is ideal, it prohibits conventional phase stepping techniques that require multiple projections under the same angle with different grating positions. One solution to this problem is the so-called sliding window processing approach that is compatible with continuous data acquisition. However, conventional sliding window techniques lead to crosstalk-artifacts between the three image channels, if the projection of the sample moves too fast on the detector within a processing window. In this work we introduce a new interpretation of the phase retrieval problem for continuous acquisitions as a demodulation problem. In this interpretation, we identify the origin of the crosstalk-artifacts as partially overlapping modulation side bands. Furthermore, we present three algorithmic extensions that improve the conventional sliding-window-based phase retrieval and mitigate crosstalk-artifacts. The presented algorithms are tested in a simulation study and on experimental data from a human-scale dark-field CT prototype. In both cases they achieve a substantial reduction of the occurring crosstalk-artifacts.


I. INTRODUCTION
X -RAY computed tomography (CT) is an indispensable technique in medical imaging that yields unobstructed 3D-views of a patient's anatomy and enables fast quantitative measurements.However, in its conventional form CT imaging is limited to attenuation-based contrast and cannot exploit waveproperties of the incident X-rays.Grating-based phase-contrast and dark-field X-ray imaging is a promising technique to unlock additional contrast channels based on the radiation's waveproperties for medical applications.Accessing this additional information can be achieved, for example, by introducing a so called Talbot-Lau interferometer in the beam path [1]- [3] or other approaches such as speckle or edge-illumination-based imaging [4], [5].
The added diffraction gratings modulate the X-ray wavefront and create a reference interference pattern on the detector.Once a sample is introduced to the X-ray beam, its interactions with the incident wavefront distort the observed pattern in three major ways.First, the sample's attenuation decreases the overall intensity.Second, porous materials induce small-angle scattering that smears out the observed interference pattern and lowers its visibility.Third, the sample induces a phase shift that leads to a shift of the observed pattern.A proper analysis of the interferometer's interference pattern thus yields two additional imaging contrasts, the so called dark-field and phase contrast channels.Conventionally, the process to separate these three contrasts is called phase retrieval.In the case of continuous data acquisition, we will also refer to it as signal demodulation.
While the phase contrast modality improves soft tissue contrast [2], [6], [7], the dark-field signal has proven a promising new tool in the diagnosis of lung diseases, as it can show microstructural changes in the lung parenchyma [8]- [14].
Recently, the potential of dark-field imaging for the diagnosis of pulmonary diseases was demonstrated in first clinical studies with a prototype radiography system [15]- [17].However, this system is only capable of measuring chest radiographs, thus lacking 3D information.In a first step to combine dark-field imaging and CT for clinical use, a human sized dark-field CT prototype was presented in [18].
Previous dark-field CT systems were limited to small animal fields-of-view and comparably slow acquisition, with measurement times of several minutes and without continuous rotation of the gantry [19].In contrast, this prototype achieves scan times comparable to an existing clinical CT scanner.Here, the continuous rotation of the gantry prohibits the use of conventional stepping-based data acquisition since the detector is positioned differently in each shot.
There exist three conceptual approaches to perform phase retrieval for a system with a continuously rotating gantry.The first and most involved one is to combine the problems of phase retrieval and tomographic reconstruction: A single statistical iterative reconstruction considers the full interferometric image formation process and reconstructs all three imaging channels concurrently [20], [21].Unfortunately, this approach suffers greatly from the fact that the underlying optimization problem is non-convex, highly ill-posed, and has problematic convergence behavior.
The second approach is the so-called sliding window approach introduced by [22].Here, adjacent shots are grouped together and processed jointly with conventional steppingbased phase retrieval algorithms in the projection domain.Consequently, the sample's projection is approximated to be stationary within one processing window.Next, the obtained projections can be reconstructed, for example, by filtered back projection (FBP).Naturally, the so obtained reconstructions can be used as is or to initialize the more sophisticated iterative algorithms.
The third approach uses spatially high-frequent carrier fringes and derives the fringe phase by Fourier analysis [23] or by processing local patches [24].Similar to the sliding window approach, the assumption here is that the measured differences in intensity within the patch are mostly due to the different fringe phase and not due to changes in the underlying sample features.
While the sliding window and the carrier fringe approach yield faster reconstruction results, they are prone to artifacts, which appear if the projection of samples moves too quickly on the detector (in the sliding window approach) or if the sample's projection changes spatially too rapidly (in the carrier fringe approach).These artifacts can be traced back to crosstalk between the three imaging domains; hence, we refer to them as crosstalk-artifacts.
In this work, we present three algorithmic extensions of the conventional sliding window demodulation approach, including a combination with the carrier fringe method, which mitigate the occurring crosstalk-artifacts substantially and enhance the image quality.

II. MODEL ANALYSIS
The basic forward model of grating-based phase-contrast and dark-field X-ray imaging is given by [2] where the state of the interferometer is described by the parameters I, V , and ϕ which denote the mean intensity, the fringe amplitude (visibility), and the interferometer phase, respectively.Once a sample is introduced it attenuates, smallangle scatters, and phase-shifts the incident wave, which alters the measured fringe pattern.These changes are modeled by three sample parameters: the sample transmission T , the darkfield D, and the (differential) phase Φ.For a data acquisition scheme utilizing a continuously moving detector and in the presence of vibrations and other fluctuations, all the introduced parameters generally depend on the individual detector pixel ⃗ x and time t.
It is, however, instructive to first consider the case of a constant intensity I 0 and visibility V 0 , and focus on one pixel.In this case, equation ( 1) can be re-written as: where we defined: While the time-dependence of the interferometer phase ϕ stems from the motion of the gratings, there is also a time-dependence of the sample parameters M , C and S due to the continuous gantry motion.
The measured signal can be thus thought of as a superposition of three amplitude modulated carrier signals.The first carrier is the constant (DC) function that is modulated by M (t).The second and third carriers are the orthogonal phase stepping carrier signals sin(ϕ(t)) and cos(ϕ(t)), which depend on the grating movement.Their respective modulation encodes the signals S(t) and C(t) in a shared frequency band.The Fourier spectrum of a qualitative example signal y is illustrated in Fig. 1A.For this example, a linear change of the interferometer phase in time was simulated: Such a phase movement corresponds to a shift of the analyzer grating with a constant velocity.While the information of M (t) is decoded in the low frequency band around DC, the signals C(t) and S(t) are decoded in the side bands around the phase stepping carrier frequency f ϕ .Sliding window phase retrieval consequently consists of three steps: (a) First, the amplitude modulation of the DC carrier needs to be separated from the modulated phase stepping carriers.In the case of a linear phase stepping, this can be accomplished by low-pass filtering the signal, which directly yields M (t).(b) Next, demodulation has to be performed on the remaining high-pass filtered signal to extract C(t) and S(t).(c) Third, the parameters M , C and S have to be transformed back into T , D and Φ using: Achieving step (a) depends on a sufficient spectral separation of the modulated DC carrier and the phase stepping carriers, i.e. f ϕ needs to be sufficiently high to separate the modulation side bands around f ϕ from the low frequency band around DC. Otherwise, the bands overlap and crosstalk-artifacts emerge in the demodulated channels because an unambiguous separation of the channels is no longer possible.Typically, high-frequency information from the DC modulation blends into the modulation channels C and S as the attenuation signal decoded in M is the strongest signal in setups suitable for imaging humans, due to constrained geometry and limited sensitivity.This leads to the emergence of movement-induced crosstalk-artifacts in the dark-field and phase channels at positions of moving sharp edges in the transmission.
In general, the design of the interferometer phase ϕ is thus vital for achieving a good demodulation as it governs the properties of the carrier signal.While linear phase stepping is favorable due to its simplicity, in some settings fast data acquisition cannot be performed easily with a controlled stepping procedure.In this case, the current grating position has to be estimated for each shot in a pre-processing step and ϕ(t) can have a more complex shape.The dark-field CT prototype discussed in [18], for example, achieves a fast phase movement by exploiting system vibrations, which lead to a sinusoidal phase movement [25].An example Fourier spectrum of such an interferometric signal y is shown in Fig. 1B.Optimizing the phase stepping for sliding window phase retrieval to maximize the spectral separation of the different carrier terms in this general setting is not trivial and is not discussed here.In this work, we focus on improving the demodulation algorithm itself, even for signals with non-optimal phase stepping.Nevertheless, we want to highlight the importance of proper phase sampling as a major factor in avoiding crosstalk-artifacts.

III. SLIDING-WINDOW-BASED PHASE RETRIEVAL
For an arbitrary phase stepping scheme, phase retrieval can be performed by solving a least-squares optimization problem that fits the general model (1) to the measured data y.Since statistical information from the measured data can be included into optimization, this process is called statistical phase retrieval (SPR).It is discussed in more detail in [26].Conventional sliding-window-based demodulation uses the SPR algorithm and is explained in the following.Furthermore, we introduce three extensions of the algorithm that change its implicit filter characteristics and improve the demodulation of sliding window measurements.

A. Conventional Sliding Window
First, one applies the same change of variables as defined by equation ( 3) and further introduces: The forward model is now linear in the new set of sample parameters M , C, and S: Generally, all parameters depend on the time t and the individual pixel ⃗ x.However, demodulation in the conventional case is performed for each pixel separately.The pixel dependency is thus omitted in the following.Furthermore, it should be noted that we assume that the interferometer state R M , R C , R S is known from previous processing steps and only the sample parameters need to be determined.Demodulation is achieved by defining windows of consecutive shots centered around a respective central shot t c and performing a least-squares optimization in each window.The goal is to find the sample parameters that minimize the cost function C window .This cost function is defined as: with weights w that can be used to introduce a distance weighting in the window to weight shots less towards the borders of the window.
In each window the sample parameters M (t), C(t), and S(t) are approximated to be constant and are denoted by M , C, and S. Consequently, the model function y window assumed in each window is given by: This model is linear in the sample parameters.Therefore, the least-squares optimization is a linear regression which possesses an analytical solution that yields the optimal values of M , C and S: The full demodulation of the data sinogram is achieved by iterating over all shots and pixels and at each step processing the window centered around t c .The sample parameters at the current shot are then given by the least-squares optimization results.This type of processing implicitly employs a low-pass filter that is defined by the window-size and window-weights w.Constant weights, for example, perform an implicit moving average filter during the sample data extraction.By changing the weights, the filter response can be adapted.Generally, a wider weight kernel performs worse at object edges but is more robust against noise and bad phase sampling.For choosing a suitable window size, one can analyze the width of the magnitude of the Fourier transform of the weight kernel w(t).Ideally, it should fall below the noise level between DC and the frequency of the first phase stepping carrier peak.By changing the window size, this cutoff frequency can be shifted to adapt for the different bandwidths of the sample signals.In our experience, it is favorable to shift the cutoff closer to the carrier peak than towards DC due to the stronger attenuation signal and thus its wider bandwidth above the noise level.

B. Local Polynomial Sliding Window
An alternative way of changing the implicit filtering of the sliding window processing is to combine it with a so called Savitzky-Golay filter [27].This filter locally approximates the data by fitting polynomials up to a given order within a specified window.Since the Savitzky-Golay filter is also based on a sliding window approach, it can be integrated easily in the sliding window phase retrieval algorithm.For this, we extend equation ( 9) and represent each sample parameter locally as a linear combination of known basis functions and their a priori unknown coefficients: For simplicity, the basis functions α m , β n , and γ k are chosen to be polynomials of t.By selecting different polynomial orders the filter roll off and pass band response can be adjusted [28] and made more suitable for the task of signal demodulation.Intuitively, the local polynomial fitting of the sample channels can also be thought of as approximating the movement of the sample's projection within a window.Since this approach improves the wrong assumption of a stationary projection within one processing window, movement-induced-crosstalkartifacts are reduced.The model function for the polynomial SPR y poly window assumed in each window is given by: Again, this model is linear in the basis function coefficients.Therefore, a linear regression yields the optimal coefficients for the respective window: The sample parameters at the window's center position t c are then given by evaluating equations (11) at t c using the least-squares optimization results a opt m , b opt n , g opt k .

C. Dual-Pass Sliding Window
Experience shows that crosstalk-artifacts are mainly caused by the varying attenuation within the window.Including an estimate of this attenuation variation into the phase retrieval algorithm is therefore expected to reduce the crosstalk substantially.Therefore, a second approach to avoid the emergence of crosstalk-artifacts is to perform a dual pass procedure and execute the sliding window algorithm twice.In the first sliding window iteration, a guess for the transmission channel T is found.Next, an attenuation corrected measurement y ′ is calculated: Finally, the sliding window algorithm is applied again to y ′ and D ′ and Φ ′ are extracted.In this iteration, the extracted mean values M ′ are discarded and set to 1 for the inverse variable transformation (5), resulting in As this approach explicitly performs a separation of the DC modulation information and the phase stepping carrier terms, crosstalk-artifacts between the attenuation and the modulation channels are mitigated.Furthermore, it enables one to use different algorithmic parameters during each sliding window step, for example, it is possible to use a smaller window size for the first pass, to extract a sharp attenuation signal, and then use a larger window to extract the dark-field channel when the resolution is of lower priority and a stronger filtering is beneficial to combat noise.Similarly to the local polynomial approach, this algorithm should be insensitive to slow attenuation changes because in the second pass the attenuation is no longer incorrectly assumed to be stationary within one window.Therefore, it should perform well in windows where the attenuation changes slowly enough to be correctly identified during the first pass.However, in windows where the attenuation changes too rapidly, e.g.due to a sharp edge, the first-pass attenuation does not correctly describe the attenuation and crosstalk-artifacts can still emerge at the position of the edge.

D. Patch-Wise Sliding Window
Furthermore, as means to mitigate crosstalk-artifacts the sliding window concept from [22] can be extended from pure temporal to spatio-temporal domain.This is done by defining patches of pixels on the detector and consecutive shots in the measured sinogram.In each patch one optimizes the cost function C patch : while assuming the general model:  Here, the basis functions can again be chosen to be polynomials in ⃗ x and t to combine the patch-wise and polynomial approach or can be simple constant functions.Apart from a higher statistical sampling, this approach has the main advantage that the phase information of neighboring pixels can also be exploited during the optimization step.Therefore, a spatial modulation of the measurement can be used to enhance the effective phase sampling leading to further separation of the sample signals in the frequency domain.This is because the phase stepping carrier peak is then not only shifted along the time direction in the Fourier space of the data sinogram but also orthogonally in spatial domain like in the carrier fringe approach.Therefore, the signals are separated along an additional axis.Consequently, this method benefits from a careful adjustment of the Moiré fringe pattern on the detector that results in a gradient in the interferometer phase ϕ(⃗ x).Similar to linear per-shot stepping, this "spatial stepping" benefits from dense fringes and thus a steep phase gradient over the detector.

IV. SIMULATION STUDY
To evaluate and demonstrate the performance of the introduced extensions of the conventional sliding window approach for CT imaging, a simulation study was performed.The results are depicted in Fig. 2. Here, the FORBILD head phantom is used as a sample in all three channels [29].To isolate the effect of channel crosstalk more clearly, in each channel the phantom was rotated with respect to the other two channels.
The attenuation phantom was scaled such that the smallest transmission T value was approximately 0.23%, the smallest dark-field value D was 46% and the maximum differential phase shift Φ was 0.2 rad.These values are motivated by phantom measurement results.The simulation was performed assuming a cylindrical detector and a cone beam geometry, the same as used for reconstructions at the setup described in [18].
In all examples, 2400 shots were acquired in an axial scan assuming a continuous rotation with a total acquisition time of 1.5 s.Additionally, An interferometer visibility of 20% was assumed.The simulated fringes were horizontal with a period of 5.3 pixels in z-direction.The simulated system geometry matches the Philips Brilliance iCT that was used as a basis for our experimental setup.
To demonstrate the capabilities of the various demodulation approaches under non-ideal conditions, a non-optimal phase sampling was chosen for the simulation.Based on the phase movement observed by [18], [25], the phase was modeled as an oscillation with an amplitude z = 1.5π and a frequency of f ϕ = 175 Hz at a gantry rotation time of 1.5 s.For the demodulation of the "conventional SW" and the "polynomial SW" cases a window size of 15 was used.To reduce the movement-induced crosstalkartifacts, in all examples von Hann window weighting [30] was performed in equation (8).Additionally, in the polynomial case the basis functions α m (t) were chosen to be monomials in t up to third order and β n (t) and γ k (t) up to second order.For the "dual-pass" example, first a demodulation with a window size of 9 was performed to estimate the transmission.In the second pass, a window size of 15 was used on the attenuation corrected data according to equation (14).For the first pass' narrower weight kernel this means that its frequency response falls below 10% right at the first carrier peak to maximize the retrieval of high-frequent transmission information.Then, in the second pass for the wider window size of 15 the frequency response drops below 10% approximately halfway between the phase stepping carrier peak and DC to allow for a better retrieval of C and S. Furthermore, for the "patch-wise" example the horizontal fringe pattern was exploited to achieve additional spatial phase sampling and patches of 11 shots and five pixels in axial direction were used for demodulation.Finally, for the "combined" example, the "polynomial" case was treated as a first pass and used for the attenuation correction.In the second pass, a patchwise demodulation was performed with the same parameters as in the "patch-wise" example.The parameters of the sliding window algorithm for each test case are summarized in Table I.
For quantitative analysis the normalized root-mean-squared errors (NRMSE) were calculated for each channel and example.To remove the influence of the FBP reconstruction error, the ground truth images were first forward projected and afterwards reconstructed using the FBP implementation.For the calculation of the NRMSE values given in Table II these ground truth reconstructions were used.
To asses the influence of noise, the study was repeated and Poisson noise was applied to the simulated data with a per pixel incident count rate of 1 • 10 6 .The results of this noisy test case are summarized in Fig. 3 and Table III.

V. EXPERIMENTAL RESULTS
To further demonstrate the capabilities of the shown algorithmic extensions for real measurements, they were applied to data from the dark-field CT prototype presented in [18].At this setup, the dark-field channel is of primary interest, and it is therefore the focus of the following analysis.The measurement consists of an axial scan of a thorax phantom that was performed with a tube voltage of 80 kVp and a current setting of 550 mAs.Over a rotation period of 1.5 s a total of 4800 projections were acquired.The effective detector size that is covered by gratings is 32 by 590 pixels.To be able to exploit the spatial modulation of the phase, a horizontal fringe pattern was tuned on the detector.A section of this pattern is depicted in Fig. 5.The phase stepping carrier was obtained by the blankscan processing described by [26], it is driven by multiple oscillations.The primary high-frequency phase oscillation (see equation ( 18)) has a frequency of approximately f ϕ = 175 Hz.Three demodulation techniques were tested for this data set.First, a conventional sliding window phase retrieval was performed with a window size of 21 and von Hann window weights.Next, a patch-wise demodulation with 5 pixels in axial direction and a window size of 21 was performed.Additionally, polynomials of the first order in t were used to model the movement of the sample's projection in the channels C and S, respectively.For the DC channel M , the sample's change in axial (z-)direction was modeled with the following set of basis functions: {1, t, z, z 2 , tz, tz 2 }.No window weighting was used in this case due to the high noise level.In the third case, the dual-pass approach was applied with a first pass window size of 17 with von Hann window weights.The second pass was performed with a patch-wise demodulation with 5 pixels in axial direction and a window size of 21 without any window weights.Apart from a beam hardening correction that is described in [25] no major post-processing was applied after reconstruction in all test cases.The resulting dark-field channels are depicted in Fig. 4 along with the reconstructed attenuation channel derived from the dual pass demodulation.

VI. DISCUSSION
Multiple observations can be made by analyzing the simulation results in Fig. 2 and Table II.First, there are strong crosstalkartifacts for the results of the conventional sliding window demodulation.In the dark-field channel of the conventionally demodulated case the occurring crosstalk-artifacts are especially severe at the high contrast edges of the transmission image which produces strong streaking artifacts.In the phase channel, the artifacts are less localized, but the overall reconstruction quality is poor as well.
By contrast, all the presented extensions of the sliding window approach reduce the occurring artifacts substantially.Each of the approaches posses individual strengths and weaknesses and in our opinion have use-cases depending on the setup parameters: While the dual-pass approach shows slightly stronger crosstalk in the simulation example than the polynomial demodulation, it is the easiest to implement and is more robust against noise.The polynomial approach yields visually better results for the noise free simulation and the best NRMSE value in the attenuation channel of all simulation examples.However, for the dark-field and phase channels it is more susceptible to noise and therefore less suited for noisy data.This becomes evident, e.g., in the results of the noise study and the high NRMSE value of the dark-field channel in Table III which can be attributed to the high noise level in the reconstruction.This poor behavior for noisy data is a well known problem of polynomial regression methods as they tend to overfit noise.The patch-wise approach achieves better results for the dark-field channel with less streaking for both the noise-free and the noisy simulations.However, it also features two drawbacks.First, the higher statistical and phase sampling power comes at the price of reducing the resolution in axial direction, due to the wider demodulation kernel.Second, sharp edges of the sample in axial direction introduce additional crosstalk-artifacts similar to sharp edges in angular direction.This, however, can be mitigated by combining the patch-wise with the polynomial and/or dual-pass approach, which can be seen in the combined example.Such a combined method was therefore applied to the experimental data.
Generally, it should be emphasized that all the presented techniques can be combined for best results.For example, it is possible to perform the first pass step with the polynomial approach and then in the second pass use the patch wise approach to extract the dark-field and phase, benefitting from additional spatial phase sampling.As presented in the "combined" case, this yields the best visual results for the simulation.
For the experimental data, it is immediately visible in Fig. 4B  and C that both advanced demodulation techniques suppress the occurring crosstalk-artifacts successfully and reduce streaking considerably.This is especially noticeable at sharp edges far away from the iso-center, like the patient couch, that produce a high-frequency signal in the sinogram [31].Furthermore, the overall image quality benefits from the stronger implicit filtering of the patch-wise demodulation and the reconstructions appear less noisy.However, this stronger filtering also leads to a loss of resolution.This can be observed, for example, at the vertebrae of the phantom.In the attenuation, it is clearly visible that in the shown slice the vertebrae is hollow.However, due to its change in axial direction it appears more solid in the second test case depicted in Fig. 4B because this approach averages adjacent slices.The dual-pass approach can mitigate this problem by not employing a patch-wise demodulation in the first pass.The guessed attenuation is therefore better suited to describe changes in axial direction during the secondpass.Consequently, for this example the patch-wise dual-pass demodulation yields better results.
The demodulation quality of the proposed algorithm in the

VII. CONCLUSION
In summary, this paper addresses the challenge of improving sliding window phase retrieval to enable fast data acquisitions for clinical phase contrast and dark-field CT.A continuous tomographic data acquisition shows the sample under a different angle for each projection.This leads to movement-induced crosstalk-artifacts when trying to demodulate the signal with a conventional sliding window approach.We introduced a new interpretation of the phase retrieval problem for a sliding window acquisition as a demodulation problem.In this interpretation, the occurring crosstalk is attributed to partially overlapping modulation side bands.To mitigate this effect, we presented three advanced extensions of the sliding window SPR algorithm.Since our proposed model extensions still lead to a linear optimization problem, demodulation can be performed in a fast and highly parallelized manner.For each test case for the experimental data, our respective implementation of the proposed algorithm took less than 30 s to demodulate the full data sinogram.In both, the simulation study and on experimental data we could show an improvement in image quality with the proposed demodulation techniques.While the focus of this paper lies on tomographic applications, it should be noticed that the concepts presented Fig. 5. A: Measured intensity y on a detector section that features Moir é fringes due to a de-tuning of the interferometer gratings G2 and G1.B: The corresponding phase ϕ shows a phase gradient in axial direction that can be exploited for additional phase sampling during demodulation.
here also apply to other scanning-based techniques that employ grating-based imaging and a continuously moving detector or sample.

Fig. 1 .
Fig. 1.A: Example Fourier spectrum of a test signal acquired with linear phase stepping.The modulation channels C and S are decoded in the side bands around the modulation peak at f ϕ = 600 Hz.The DC modulation M causes a response in the low frequency bands.B: Fourier spectrum of a signal acquired with vibration induced phase stepping with vibration frequencies of 200 Hz and 1 Hz.The emerging spectrum features multiple carrier peaks whose side bands contain the information on the modulation channels S and C (see equation (3)).

Fig. 2 .
Fig. 2. Comparison of different sliding window demodulation methods.The rows from top to bottom feature reconstructions of a phantom's attenuation and the reconstructed dark-field and phase channels.Each column features a different demodulation method as outlined in TableI.In the attenuation channel, difference images to the ground truth are depicted.

Fig. 3 .
Fig. 3. Comparison of different sliding window demodulation methods for noisy data.Except for the application of Poisson noise to the simulation, all parameters have been kept the same as in Fig. 2.

Fig. 4 .
Fig. 4. Reconstructed dark-field channels that were acquired with A: the conventional sliding window algorithm.B: a combination of the proposed polynomial and patch-wise demodulation approach.C: a combination of the proposed dual-pass and patch-wise demodulation approach.D: Reconstructed attenuation channel with levels (-1000, 160)[HU] E: Sinogram section of the dark-field channel depicted in A. F: Sinogram section of the dark-field channel depicted in C.