Modulation transfer function (MTF) evaluation for x-ray phase imaging system employing attenuation masks

Objective. Attenuation masks can be used in x-ray imaging systems to increase their inherent spatial resolution and/or make them sensitive to phase effects, a typical example being Edge Illumination x-ray phase contrast imaging (EI-XPCI). This work investigates the performance of a mask-based system such as EI-XPCI in terms of Modulation Transfer Function (MTF), in the absence of phase effects. Approach. Pre-sampled MTF measurements, using an edge, were performed on the same system implemented without masks, with non-skipped masks and finally with skipped masks (i.e. masks in which apertures illuminate every other pixel row/column). Results are compared to simulations and finally images of a resolution bar pattern acquired with all the above setups are presented. Main results. Compared to the detector’s inherent MTF, the non-skipped mask setup provides improved MTF results. In comparison to an ideal case where signal spill-out into neighbouring pixels is negligible, this improvement takes place only at specific frequencies of the MTF, dictated by the spatial repetition of the spill-out signal. This is limited with skipped masks, which indeed provide further MTF improvements over a larger frequency range. Experimental MTF measurements are supported through simulation and resolution bar pattern images. Significance. This work has quantified the improvement in MTF due to the use of attenuation masks and lays the foundation for how acceptance and routine quality control tests will have to be modified when systems using masks are introduced in clinical practice and how MTF results will compare to those of conventional imaging systems.


Introduction
Over recent decades, two (2D) and three dimensional (3D), i.e. tomographic, x-ray phase contrast imaging (XPCI) methods have indicated significant advantages over conventional absorption x-ray imaging in the imaging of objects with small absorption differences against the background they are immersed in, especially in applications requiring high spatial resolution (Bravin et al 2012, Hagen et al 2017, Mayo and Endrizzi 2019, Momose 2020. XPCI methods take advantage of the phase-shift effects caused by a sample (supplementary to the absorption effect), leading to an increase in contrast when compared to conventional x-ray imaging (Olivo and Castelli 2014).
Edge illumination (EI) is a non-interferometric XPCI technique that has been successfully implemented with both synchrotron and conventional laboratory sources (Olivo and Speller 2007a, Munro et al 2012, Diemoz et al 2013a. The method has been extended to quantitative computed tomography (CT) and has more recently successfully demonstrated its capabilities in biological tissue imaging using a compact version of the system (Hagen et al 2014a, Havariyoun et al 2019, Massimi et al 2021a, 2021b. It has proven its ability to provide reliable quantitative results with spatially and temporally incoherent x-ray sources, reduced exposure times, higher x-ray energies, relatively flexible setup requirements and large fields of view (FOV) , Astolfo et al 2017, Olivo 2021. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Figure 1 illustrates a schematic of the laboratory-based implementation of EI-XPCI. Relative to a conventional x-ray imaging system the setup differs only in the introduction of two masks, made of a series of alternating apertures and absorbing septa, between the source and the detector.
The sample mask, placed before the sample, creates a series of secondary beamlets each of which propagates towards a single pixel detector column. The detector mask, aligned with the detector pixels, blocks a predetermined fraction of each beamlet-effectively creating insensitive regions between adjacent pixels. Placement of a sample between the two masks has two effects: attenuation of the beamlet intensity and beam refraction i.e. a local change in the beam's direction of propagation.
Lateral misalignment of the two masks (along x in figure 1) allows for: a change in the detected intensity of the beamlets and a variation of the refraction sensitivity of the system (Olivo and Speller 2007b). A process called 'dithering', where multiple frames are acquired whilst the sample is shifted by sub-pixel steps, is often used to further improve resolution and accuracy of the retrieved quantitative information (Hagen et al 2014b, Diemoz et al 2015. The optimal number of dithering steps depends on the source size, sample mask slit width and sample structures (Diemoz et al 2014); for a correct recombination, the size of the dithering step needs to be a submultiple of the sample mask period. The ultimate attainable resolution is equal to the aperture size; if such a maximum resolution is targeted, compliance with the Nyquist-Shannon theorem requires a dithering step of half the aperture size. Higher resolution images obtained by recombining frames obtained through dithering can suffer from artefacts if a detector with a line spread function (LSF) significantly greater than a pixel, described as 'signal spill out' or 'signal diffusion', is used. 'Skipped' masks, where every other detector column is illuminated, can be implemented in these cases to mitigate the problem; in such a setup, data from skipped pixel columns are 'dumped' (Ignatyev et al 2011). For a more detailed explanation of the method's principles and how phase effects are exploited, the reader is referred to a recent review (Olivo 2021).
To fully characterize the performance of an imaging system several key characteristics, some specific to XPCI, need to be investigated. Previous work on the EI technique has investigated angular resolution, sensitivity, spatial resolution, contrast and noise properties (Diemoz et al 2013b, 2013c, 2016, Hagen et al 2020a. However, the modulation transfer function (MTF), commonly used in conventional x-ray imaging systems to characterize their spatial frequency response (Samei et al 1998), has not yet been studied. While the MTF and PSF of some of the detectors used in EI setups have been investigated in previous studies, there has been no previous work on, and comparison of, the effect of the different mask types on the MTF (Konstantinidis et al 2012, Endrizzi et al 2013. Previous work has either been limited to simulations or has only reported on experimentally measured Edge Spread Functions (ESFs) (Diemoz et al 2014, 2015, Massimi et al 2021a. MTF evaluation is a key characterisation procedure that forms part of the acceptance tests of digital x-ray imaging devices as per the International Electrotechnical Commission's (IEC) standards (IEC 62220-1-1:2015) as well as routine medical physics tests performed for quality control and beyond (IPEM 2010). The MTF is significantly altered by the introduction of the masks, hence we found it needed to be discussed in detail before mask-based XPCI could be translated to clinical practice. To the best of our knowledge, MTF evaluation of other XPCI systems have focused on CT systems (Li et al 2013) apart from one work which involved simulated results for 2D in-line phase contrast imaging (Sun et al 2017). In this work, we evaluate via simulations and experiments the pre-sampled MTF of EI, and how this affects the resolution of the system. The response of a digital imaging system with discrete sampling does not only depend on the properties of the detector but also on the input signal and its location relative to the sampling grid of the detector. To avoid aliasing effects caused by the discrete sampling of the input signal the edge is angled to increase the effective sampling rate and obtain a non aliased response, pre-sampled (i.e. effectively sampled at a frequency higher than the pixel pitch) MTF (M D et al 2009(M D et al , IEC 2015. The relevance of this work goes beyond EI-XPCI and is more generally applicable to methods using pre-sample masks, either to gain access to some degree of phase-based image contrast or specifically to increase the resolution (

Materials and methods
The source used was a Rigaku Micro-Max 007 rotating anode (Rigaku Corporation, Japan) with a molybdenum anode and an effective focal spot size of approximately 70 μm. The detector was a CMOS-based flat panel C9732DK-11 (Hamamatsu, Japan) with directly deposited CsI and a 50 by 50 μm pixel size (pixel pitch (p) of 50 μm) with a PSF with an approximately 100 μm Full Width Half Maximum (FWHM) (Endrizzi et al 2013). The source to detector distance is 85 cm, while the source to sample distance is 70 cm.
Images were acquired without masks, with non-skipped masks and finally with skipped masks. For the nonskipped masks, the aperture size/period were 12 μm/38 μm and 20 μm/48 μm for the sample and detector mask, respectively. For the skipped masks, the same quantities were 10 μm/79 μm and 17 μm/98 μm, respectively. The masks have been manufactured by Microworks GmbH (Karlshrue, Germany) and are made by gold electroplating on a 400 μm thick graphite substrate. The thickness of the gold layer is >120 μm from specifications. Dithering was performed in the skipped masked acquisitions to match the pixel size of the nondithered, non-skipped mask acquisition.
The source was operated at 40 kVp for all acquisitions. A 45 μm Molybdenum filter, commonly used in breast imaging x-ray systems, was added in front of the source for the no-masks acquisitions to prevent detector saturation. Acquisitions with the non-skipped mask setup were performed both with and without the additional filtration to allow for a direct comparison (in terms of beam hardness) with the no-masks and skipped masks setups, respectively. Phase retrieval was not performed on the images. The scope of this work is to establish how the introduction of (skipped and non-skipped) attenuation masks affects the detector's response in terms of input and output signal in the spatial frequency domain; e.g. it should be noted that a phase contrast image simply has a different distribution of spatial frequencies compared to an attenuation one, and these individual frequencies would be transferred through the system according to the same MTF. As an overall system parameter, it is inherently related to the point spread function (PSF) of the detector, x-ray source size and, for the Edge Illumination x-ray phase contrast imaging (EI-XPCI) system, attenuation masks. Furthermore, masks can be added also to increase the spatial resolution of an imaging system, which makes the presented results also relevant to some non-XPCI areas (Hagen et al 2020b). All images acquired were dark noise and flat field corrected.
Images were acquired with a 1 mm thick tungsten edge at a slight angulation (1.5°-2°). Due to the presence of the detector masks, the edge cannot be placed in contact with the detector in cases where the masks were employed. Hence, the edge was placed at the usual sample position of the system for all acquisitions. It is known that MTF measurements are affected by geometry parameters and source blurring. However, as all measurements were taken with the edge at the same position and magnification was considered when calculating pre-sampled MTFs, this allowed a fair comparison among the three setups. Note that this resulted in all presented graphs being referred to the (de-magnified) sample coordinates. Transmission through the edge was measured to be below <2% for all setups. To obtain the pre-sampled MTF, an oversampled edge spread function (ESF) approach was implemented (Samei et al 1998). The ESF was smoothed with a median filter (window size of 5); according to reference (M D et al 2009), window sizes up to 17 can be used with median filtering to reduce noise in the tails while preserving as much as possible the integrity of the data (M D et al 2009). Furthermore, a median filter of length 5 pixels is routinely used by the UK's National Health Screening (NHS) Breast Screening Programme for MTF evaluation (Marshall 2023). The ESF was then differentiated to obtain the line spread function (LSF), which in turn was Fourier-transformed to obtain the pre-sampled MTF. Finally, a Hüttner Type 18UL resolution lead (0.03 mm thickness) bar pattern with line pairs ranging from 1 lp/mm to 20 lp/mm was imaged under the same conditions as the edge for a qualitative evaluation.
Simulations were performed using a wave optics simulation of the EI setup (Vittoria et al 2013). The edge shape was modelled as a step function considering the attenuation and refractive properties of tungsten. As the simulation involves a one-dimensional representation of the setup (i.e. single detector row) oversampled ESFs were obtained by dithering the simulated edge. As dithering was also employed experimentally, it was possible to obtain measured pre-sampled MTFs using this same method for comparison. It should be noted that this method provides a more accurate ESF, and therefore pre-sampled MTF results, because it eliminates the need to determine the edge angle and the perpendicular distances from the edge to the centre of each pixel.

Results and discussion
3.1. Effect of non-skipped masks Figure 2 shows the experimentally acquired tungsten edge images from which the LSFs and MTFs (in the horizontal direction, i.e. perpendicular to mask apertures) displayed in figure 3 were obtained. Note that the signal spill out effect of the detector is made evident by the gradual grey level change of the edge on the mask-less image ( figure 2(a)). The red arrows on the non-skipped mask image ( figure 2(b)) indicate the appearance of step like effects. These can be explained by the introduction of the masks producing insensitive regions on the detector, hence causing a reduction in the width of the detector's PSF. Figure 3 shows the experimental LSFs (figure 3(a)) and MTFs ( figure 3(b)) obtained for the mask-less and non-skipped mask setups. Although the envelope of the LSF does not change, we notice the presence of secondary 'lobes', caused by the step like effects highlighted in figure 2(b), at higher spatial distances in the latter case. The number of lobes introduced is dependent on the intrinsic signal 'spill-out' (i.e. the width of the LSF) of the used detector. For the Hamamatsu detector, this was previously reported to be 49% and 6% to the first and second neighbouring pixel if only one pixel was illuminated (Olivo et al 2009). The secondary lobes repeat in space; their FWHM matches the sample mask's aperture and the spacing between the lobes is equal to the pixel pitch (p). This effectively corresponds to a convolution of the detector LSF without the mask with a comb function with period equal to p. When this is Fourier-transformed to obtain the MTF curve ( figure 3(b)), this results in a sampling effect with period 1/p on the broader MTF corresponding to the narrow, aperture-driven LSF. This leads to a suppression of this MTF curve at frequencies that do not match the 1/p sampling period. This is further demonstrated by the simulation results displayed in figure 4. The simulation confirms the introduction of secondary lobes on the LSF (figure 4(a)) when non-skipped masks are used, the intensity and number of which depend on the width of the original LSF. The simulated MTF results ( figure 4(b)) also support the experimental results, with additional lobes appearing with frequency 1/p. Results also indicate that if all the secondary lobes were to be removed from the LSF (i.e. removing all the signal spill-out), leaving only the central one, an MTF corresponding to the Fourier transform of the narrower, aperture-driven LSF would be obtained, with a smooth curve at high frequencies and no lobes. Indeed, removing all secondary lobes corresponds to an  'infinite' repetition frequency of the narrow LSF, and therefore to a perfectly sampled MTF. This indicates that the use of skipped masks, where only every other column is illuminated, should increase the sampling frequency of the MTF by a factor of two, leading to a better overall frequency response.
It should be noted that the discrepancy in terms of number and intensity of the lobes (e.g. the second lobe having a peak value of 0.2 (figure 3(a)) instead of the expected 0.06 based on the detector's specifications) between simulations and experimental results (figures 3 and 4) is due to the difference in how the oversampled ESF was obtained in the two cases, i.e. slanted edge versus single row dithered edge respectively. Figure 5 shows the results obtained using both methods. Retrieving the ESF from a single pixel row of a dithered acquisition (10 dithering steps) provides results very close to the simulation ( figure 5(a)) and to the results expected from the detector's specifications (the second lobe's peak value is now close to the expected one of 6%). Conversely, retrieving the ESF using the slanted edge method and multiple pixel rows leads to a broadening of the lobes and to a non-zero intensity between them. This can be attributed to a number of factors including but not limited to the additional inaccuracies in estimating the angle and position of edge transitions (in turn caused by the step effects on the edge images), to small variations in the sensitivities of individual pixel columns and to small mask misalignments. Figure 6 depicts LSF (left) and pre-sampled MTF results (right) for the non-skipped and skipped masks. Note that the skipped-masks image was acquired with 2 dithering steps to achieve the same pixel size as a nondithered non-skipped mask acquisition. The secondary lobes on the LSF (figure 6(a)) due to signal spill-out into the first neighbour are no longer present when using the skipped masks, as every other pixel column is masked completely and the corresponding data is dumped. The third lobe, however, is still present, as the second neighbour is not covered and this still receives some degree of signal sharing occurring from the nearest  'unmasked' pixel columns. As anticipated, this leads to significantly heightened MTF values (figure 6(b)) at values midway through the lobes previously observed for the non-skipped case, confirming a doubling of the frequencies at which the MTF is correctly reproduced (i.e. with lobes appearing every 1/(2p) as opposed to 1/p). Since tertiary lobes (signal spillage into second neighbouring pixel columns) are still present, there is still a degree of spatial repetition of the 'narrow' LSF, meaning that the 'ideal' red curve shown in figure 4(b) is still not obtained; in order to obtain such an MTF curve, all signal spill-out should be eliminated. However, as it can be seen the presence of only the (less intense) tertiary lobes in the spatial domain has led to a smaller suppression of the pMTF in the spatial frequency domain. Figure 7 shows images of the bar phantom imaged with the EI setup with no masks (figure 7(a)), non-skipped masks ( figure 7(b)) and skipped-masks (figure 7(c)). Also in this case, two dithering steps were used in the skipped-mask case to obtain the same pixel size as in the other two setups. The three bar groups visible in the images are, from left to right, 8.9, 12.5 and 16.6 lp mm −1 , respectively. The improvement in resolution from left to right is evident. This is expected based on the LSF and MTF results shown previously. Attention should be given to the 12.5 lp mm −1 (central) group: the non-skipped image (figure 7(b)) shows some improvement at this frequency when compared to the no-masks case (figure 7(a)), but the change is much more significant in the skipped-mask image (figure 7(c)).

Effect on resolution
This should be expected as 12.5 lp mm −1 is one of the 'supressed' frequencies in the non-skipped mask case which is recovered in the skipped case, as visible in figure 7(c). Furthermore, although there is an improvement compared to the no-masks and non-skipped masks cases, the suppression of frequencies both above and below 12.5 lp mm −1 (i.e. at 8.9 lp mm −1 and 16.6 lp mm −1 ) is clearly visible in the skipped masked acquisition. Again, this could be expected from the MTF curve for this setup ( figure 7(b)). The effects of aliasing, light and dark bands with varying spacing and orientation to the bar patterns, can be seen in all images (Albert et al 2002). The change in orientation of the bands occurs between the firstand second-line pair groups in the image acquired without masks and between the second and third line pair groups in the images acquired with both masks. The spacing of the aliasing patterns increase from figures 7(a)-(c). Both these effects are a consequence of the increase in sampling frequency discussed above and of the effective change in resolution of the detector due to the introduction of the sample masks. We also note that the smaller aperture and larger period of the sample and detector skipped masks (i.e. less photons reaching the sample and detector) means there is trade off between increased resolution and exposure time. The resulting increase in exposure time can only be overcome by utilising a higher flux x-ray source. This, however, does not translate into a dose increase, since the mask is placed upstream of the imaged sample, an aspect discussed in detail in previous work (Olivo et al 2013, Hagen et al 2014c. The choice between non-skipped and skipped masks depends primarily on whether the used detector has a significant or negligible signal spill-out: skipped masks are needed in the former case, while nonskipped ones can be used in the latter. If the same resolution level (as determined by the size of the pre-sample aperture (Diemoz et al 2014)) is desired, then the use of skipped masks will require double the exposure time.  (128)) obtained with the EI-XPCI system with: (a) no masks, (b) non-skipped masks, (c) skipped-masks. From left to right the line groups have a spatial frequency of 8.9, 12.5 and 16.6 lp mm −1 , respectively.

Conclusions
Simulation and experimental results of the LSF and MTF of the EI-XPCI setup have been presented for the nonskipped and skipped mask setups and further supported with images of a resolution bar pattern. It has been shown that the introduction of masks leads to an improvement in MTF and hence spatial resolution and contrast transfer function of the system. The introduction of the masks, specifically the sample mask, introduces a selective improvement in MTF performance driven by the sample mask aperture size and pixel pitch p. The use of skipped masks doubles the improvement in MTF in terms of frequencies being correctly reproduced. This work has led to further understanding of the impact of sample mask aperture size, pitch and detector PSF in terms of spatial frequency response of the EI-XPCI system. In particular, it has shown how signal spill-out and the consequent creation of spurious peaks in the LSF leads to the suppression of certain spatial frequencies in the MTF, and how these can be partially restored by reducing the impact of signal spill-out through skipped mask designs.
This can be used in future experiments to identify the most appropriate system properties for the imaging task in hand; for example, by making sure that the imaging system is sensitive in a given range of spatial frequencies, which in turn can be important for the correct reproduction of certain object features. Furthermore, in the light of a future clinical adoption of EI-XPCI, this work lays the foundation for how acceptance and routine quality controls tests will have to be modified, and how results will compare to conventional systems. As made clear by e.g. Figure 4(b), the MTF of a system employing masks looks very different from that of a system that does not; the same figure provides both a direct comparison with a system using the same detector with no masks, and the 'ideal' MTF the system with masks could aspire to if all spill-out effects were eliminated: in practical terms, the MTF of a system with masks will always lie between these two extremes.