Analyzer-free hard x-ray interferometry

Objective. To enable practical interferometry-based phase contrast CT using standard incoherent x-ray sources, we propose an imaging system where the analyzer grating is replaced by a high-resolution detector. Since there is no need to perform multiple exposures (with the analyzer grating at different positions) at each scan angle, this scheme is compatible with continuous-rotation CT apparatus, and has the potential to reduce patient radiation dose and patient motion artifacts. Approach. Grating-based x-ray interferometry is a well-studied technique for imaging soft tissues and highly scattering objects embedded in such tissues. In addition to the traditional x-ray absorption-based image, this technique allows reconstruction of the object phase and small-angle scattering information. When using conventional incoherent, polychromatic, hard x-ray tubes as sources, three gratings are usually employed. To sufficiently resolve the pattern generated in these interferometers with contemporary x-ray detectors, an analyzer grating is used, and consequently multiple images need to be acquired for each view angle. This adds complexity to the imaging system, slows image acquisition and thus increases sensitivity to patient motion, and is not dose efficient. By simulating image formation based on wave propagation, and proposing a novel phase retrieval algorithm based on a virtual grating, we assess the potential of a analyzer-grating-free system to overcome these limitations. Main results. We demonstrate that the removal of the analyzer-grating can produce equal image contrast-to-noise ratio at reduced dose (by a factor of 5), without prolonging scan duration. Significance. By demonstrating that an analyzer-free CT system, in conjuction with an efficient phase retrieval algorithm, can overcome the prohibitive dose and workflow penalties associated grating-stepping, an alternative path towards realizing clinical inteferometric CT appears possible.


Introduction
The past 30 years have witnessed extensive efforts to improve the contrast attainable in x-ray images by including phase shift information in the image formation process (Momose et al 2003).Contrast due to the phase shift between rays traversing two different tissues is often orders of magnitude larger than the contrast imparted by differences in x-ray attenuation, the primary contrast mechanism in conventional x-ray imaging.This makes phase contrast (PC) an interesting imaging technique for objects with low transmission contrast, such as tumors in soft tissue organs like the liver.
Phase shift is caused by the refraction of x-rays, which depends on the real part of the refractive index, δ, whereas the attenuation depends on the imaginary part, β.Traditionally, PC imaging methods have relied on the coherence of synchrotron light or microfocus sources for illumination.The earliest methods use gratings configured in a traditional interferometry setup (Davis et al 1995, Chapman et al 1997).Later, a grating-free technique using only propagation was suggested (Snigirev et al 1995, Wilkins et al 1996).Many variants of such grating-and propagation-based methods are demonstrated to successfully yield high-contrast images.
For clinical imaging, however, a synchrotron or microfocus source is highly inconvenient, and polychromatic sources with low coherence are preferred.The only PC methods that are demonstrably viable using this type of source is grating-based phase contrast (GBPC) (Pfeiffer et al 2006) and edge illumination (Olivo and Speller 2007).In addition to the phase contrast, information on small-angle scattering may also be obtained (Pfeiffer et al 2008).Grating-based interferometry was recently applied to human conventional CT (Viermetz et al 2022).
The basic GBPC setup is shown in figure 1(a).A first grating, the source grating or G 0 , is placed in the beam after its emergence from the source.Rays then propagate through a phase grating (G 1 ), which is deployed at a distance s 1 distal to G 0 .G 1 is designed so that its absorption is minimal and its purpose is to impart a phase-shift pattern to the wavefront.The object can be placed either proximal or distal to the phase grating.If there is no phase shift from the object, radiation passing through G 1 reproduces its intrinsic grating or fringe pattern at the Talbot distance.However, in the presence of an object with any non-zero phase gradients, the pattern is perturbed.By recording this distorted fringe pattern, the phase shift due to the object is calculated.
To record the fringe pattern, a micrometer-scale resolution is needed.However, common energyintegrating CT detectors have a pixel size of 0.25-2 mm (Danielsson et al 2021).These detectors typically employ a ceramic optical scintillator that is segmented into pixels and coupled to photodiode arrays.Detectors with smaller pixels require a greater proportion of their total area to be occupied with septa (filled with reflective material) that are needed to optically-isolate each pixel.This leads to a trade-off between geometric detection efficiency and spatial resolution.Consequently, resolution in the tens of microns is impractical using this technology.Photon counting CT detectors are available with pixel sizes of 0.1-0.5 mm; these directly convert photon energy to charge inside a semiconductor.The potential of conventional detectors to achieve <100 μm resolution is limited due to charge sharing and crosstalk between pixels and K-escape fluorescence from atoms such as Cd and Te that constitute the detector material.
Since contemporary x-ray detectors are not able to record the fine details of the fringe pattern, a third grating, the analyzer grating (G 2 ), is placed proximal to the detector.By shifting the fringe pattern relative to the analyzer grating (either by moving one of the gratings, the source, or the object) in a process called phasestepping, the fringe pattern is sampled at a sufficiently small interval, and the phase information required for image formation is reconstructed from this collection of measurements (Pfeiffer et al 2008).
Using an analyzer grating has several drawbacks.First, it is an absorbing grating placed after the object.This means that approximately half the photons that would otherwise contribute to the image are absorbed by the grating.Consequently, the patient must receive approximately twice the radiation dose to maintain the image contrast-to-noise ratio (CNR).Second, although at constant sample dose, the noise in the differential phase image is independent of the number of phase steps (Mechlem et al 2020), the object may move between exposures and create motion artifacts.At least three images, acquired at three different offsets, are required (Weikamp et al 2005).In addition to augmented dose and motion artifacts, continuous-scan CT imaging becomes impossible.Many publications have suggested ways to overcome this limitation.One way (Bevins et al 2012) is to introduce a small misalignment in the setup, creating a Moiré pattern, from which the phase information can be extracted.Phase information and dark-field signals are easily recovered by Fourier analysis, but at the expense of lower resolution in one direction.The method was recently improved (Marschner et al 2016) through the use of helical tomography.Another technique utilizes a sliding window where the grating is moved slightly for each projection angle (Zanette et al 2012).This works well if there are many projection angles.One method exploits complementary angles in full-scan CT, and uses a different grating position for each (Zhu et al 2010), while another employs a modulated phase grating (Xu et al 2020).A promising way of enabling CT imaging is described by von Teuffenbach et al (2017), where a statistical iterative reconstruction algorithm reconstructs projection images recorded with a sliding window, or a moving source (Miao et al 2013).All of the above-mentioned work, with the exception of that by Xu, utilizes an analyzer grating, since the fringes cannot otherwise be resolved.However, recent work in the soft x-ray regime (Cartier et al 2016) uses a 25 μm pixel detector combined with charge-sharing analysis to enable 1 μm resolution, and thus obviates the need for an analyzer grating and a phase stepping procedure.
In this work, we extend the work of Cartier et al to hard x-rays utilized in medical imaging, and present an analytical as well as a numerical study of phase retrieval and CT reconstruction of PC images.This is possible due to recent developments in detector technology (Sundberg et al 2020) where the potential to reach 2.5 μm resolution is demonstrated by calculating the charge distribution in a pixel of a silicon strip detector irradiated in an edge-on geometry.First, we show analytically how the new configuration enables an improvement of the CNR for equal dose by moving the G 2 grating from hardware to software.We then compare simulated analyzerbased CT images to analyzer-free CT images, and show, for a clinically practical CT geometry, that the removal of the analyzer grating reduces the dose for phase contrast by a factor of 5, which can enable clinical application of phase contrast CT at doses comparable to contemporary absorption CT.The removal of the analyzer grating not only simplifies the setup and reduces cost and dose, but also relaxes demands on mechanical stability and alignment, which have been reported as issues that limit GBPC (Horn et al 2018).This is a potentially exciting development for GBPC that could enable rapid imaging with lower dose and reduces impact from patient motion, making it suitable for eventual transition to clinical use.

Principle of analyzer-free x-ray interferometery
The setup for an analyzer-free grating interferometer for hard x-rays is shown in figure 1(b).This configuration is identical to analyzer-based imaging, excepting that the analyzer and detector are replaced by a high-resolution detector.The necessary spatial resolution of the detector depends on the imaging geometry, but a suitable candidate was recently presented (Sundberg et al 2020).This detector concept realizes a point-spread function (PSF) with 2.5 μm full-width at half-maximum (FWHM) is presented.The model includes detector noise and a realistic model of the detector response.Due to the edge-on geometry utilized by the detector, the efficiency is similar to that of detectors fabricated from high atomic number materials such as CdTe (90% at 60 keV).The primary Compton interactions of the x-ray photons are registered, and a portion of the secondary scattered Compton photons are absorbed in thin sheets of tungsten, which are interposed between the silicon wafers (Danielsson et al 2021).
If the fringes are sampled at a high enough resolution, the phase information can be extracted by using a Hilbert transform (Cartier et al 2016).Here, we propose a different approach, using a virtual grating.This is applied numerically after the image is acquired.We now show analytically that, since this virtual grating can be made ideal, and with any duty cycle, analyzer-free x-ray grating interferometry can improve dose efficiency.

Theory
The standard deviation of electron density in grating-based x-ray phase imaging is given (Mechlem et al 2020) by: S is the sensitivity of the setup, v is the visibility of the phase-stepping curves and b, the number of detected counts.We derive the relationship between the number of counts in analyzer-based imaging, b AB and analyzerfree imaging, b AF as G 2 is the only factor that differs between the two configurations.Furthermore, since the phase signal is linearly dependent on the electron density, , PS e s s µ r where PS s is the standard deviation of the phase signal, we can now express the ratio of the CNR for the two methods as with v AF and v AB denoting the visibilities of analyzer-free and analyzer-based imaging, respectively.The visibility is defined as: C max and C min are the maximum and minimum values of the phase-stepping curve.Phase stepping curves can be derived analytically but differ between the two imaging configurations.The analyzer-based case was derived previously (Thuering and Stampanoni 2014) but only while ignoring higher order Fourier coefficients.It is trivial to expand this to the general case where all terms are included: where b b , 0 2 and a a , 0 2 denote the transmission through the grating bars, and the grating duty cycle, respectively, for the G 0 and the G 2 gratings.
For the analyzer-free case, the PSF of the detector, x PSF , D ( ) also affects the phase-stepping curves.For simplicity, we assume this PSF to be Gaussian with a standard deviation of w, yielding: We express the detected image intensity as: where I x ( ) is the interference pattern intensity and x PSF S ( ) models discrete pixel sampling, for which we use a top-hat function spanning the full-width of the sampling interval s.
To arrive at the stepping-curves C x , ( ) we need to apply the virtual analyzer grating I x .G 2 ( ) In the analyzerfree case, this grating has zero transmission through the bars, so b 0, 2 = yielding: where L 2 is the grating period.Since I x where I G 0 is the intensity transmission function of the G 0 grating and I G 1 is the fully coherent interference pattern from a point at the source, we arrive at a final expression for C x ( ) of: This expression can be efficiently evaluated in the spatial frequency domain via multiplication of the Fourier coefficients.An expression for the visibility in the analyzer-free case is achieved by applying equation (2), to yield: Where we use L to represent the period of the virtual G 2 grating.Combining (10) and (5), yields the ratio between the visibility of the two methods.To a reasonable approximation, it is sufficient to include only the first and third term of the sum.A commonly used G 0 grating in GBPC imaging is fabricated with approximately 300 μm thickness and a duty cycle of 0.5, so b b 0.073

=
This means that we can acquire images at 2.18 times higher CNR for the exposure, or, since image CNR 2 is proportional to dose (for Poisson-distributed quantum noise, see e.g.Barrett and Swindell (1996)), we can reduce dose by almost 5 times and preserve CNR.Next, we will use a numerical model to simulate the impact of noise on image quality for a relevant sample.

Method
In order to numerically compare the performance of the two methods depicted in figure 1, we developed an x-ray interferometry CT image formation model implemented in Python.We employ this model to compare the CNR and image quality of the two methods.
The CT image-formation model can be broken down into four parts: (1) a model describing how projection images are formed in the x-ray interferometric setup.(2) Generation of the raw image data captured in an x-ray CT interferometry acquisition using (1).In the traditional analyzer-based case, this is a collection of phasestepping images (including no-object reference) acquired at each angular projection view.In the analyzer-free case, these are replaced by a single high-resolution image acquired at each view angle.(3) Reconstruction of the absorption, phase, and small-angle scattering signals from the simulated raw data projections using (2).( 4) Tomographic reconstruction of the object's local linear absorption, phase, and small-angle scattering coefficients.A flow chart of the model is shown in figure 2.

X-ray interferometric projection-image model
Our projection image simulation starts with a monochromatic wavefield originating from a point at the x-ray source origin .0 Y The field incident on the object is obtained by multiplying this wavefield by the complex transmission of the G 1 -grating T x G 1 ( ) and then Fresnel free-space propagating it to the object.The field at the detector is calculated in a similar manner: the field at the object is multiplied by the complex transmission of the object and then propagated to the detector.The interference pattern at the detector I x x is the ideal diffraction-limited pattern from the spatially-and temporally-coherent point source.
A source with a realistic 'micro-focus-like' spatial distribution usually limits the achievable imaging resolution.Furthermore, we have shown that the choice of G 0 -grating affects the spatial coherence in the setup and hence impacts the visibility of the interference pattern.To include these effects, the optical image I i is modeled by a convolution of the point-source image I x c ( ) with the G 0 -masked source intensity distribution, where, M s s s s  the optical image is multiplied by the analyzer transmission function T x , G 2 ( ) which is shifted according to the phase-stepping process.These approximations are reasonable for low scattering objects (Paganin 2006) and are commonly applied in modeling x-ray wave propagation (Goodman 1968).
The detector model is different for the two methods.For the analyzer-based case, the detector is considered as ideal, with 2 mm wide pixels; 400 pixels yield an 800 mm detector.The detected image, I x , ,AB ( ) is obtained by binning to 2 mm pixels and applying Poisson-distributed noise based on image quanta.To ensure that the comparison with the analyzer-free case is relevant, we do not add dark noise or electronic noise.The detector model for the analyzer-free case is more complex, and a Gaussian function is used to represent the detector PSF (Sundberg et al 2020).The detector PSF is convolved with the optical image to yield the detected image, and then binned to a pixel size of 3.2 μm; a detector size of 800 mm corresponds to 250 000 pixels.Since we do not currently know the exact efficiency of the detector, we set it to ideal as in the analyzer-based case.
All free-space propagations are performed by numerical Fresnel propagation, and the divergent geometry is accounted for via the Fresnel scaling theorem (Langer et al 2020, Paganin 2006).For human-size phantoms and micrometer-sized detector sampling, the matrices involved in the calculations become impractically large, requiring extensive computational resources.We therefore limit the image formation calculations to that of a single phantom slice with a given thickness, t.
The input defining the phantom cross-section is in the form of a rasterized image where pixel values represent different materials and densities.The amplitude and phase of the object transmission function are calculated by projection through the cross section and then interpolated to the sampling needed in the free-space propagation.The x-ray and material properties are obtained from an online library, Xray DB, which is based on available data, using data and methods published in Chantler (1995).For the phantom tissues in this study, the elemental compositions were set according to Woodard and White (1986) and added to the library.The library calculates the β values from the mass absorption coefficient.

Phase retrieval and tomographic reconstruction
As described earlier, phase retrieval for analyzer-based imaging is affected by phase-stepping, projection images being calculated for the number of phase steps.When all images are modeled, the simulation calculates the Fourier coefficients in the stepping direction and assigns these to the absorption, phase and small-angle scattering.The absorption data is retrieved from the average signal of the phase-stepping curves, and the phase, , j of the intensity curves is integrated to give the phase data, , F since x j ~ ¶F ¶ (Weitkamp et al 2005).The amplitude of the curves is related to the dark-field signal, but it is not currently implemented in our model.The processing steps have been described previously and are well-known in the field (Pfeiffer et al 2007, Bech et al 2010).
For the analyzer-free case, it is possible to reconstruct the phase from a single projection image.We use a two-step process in which the high-resolution image is split into a set of N lower resolution images, which are calculated by applying a virtual analyzer grating at N different offsets and binning pixels into a larger macropixel.Mathematically, we can express this as: is the virtual grating with values of either 0 or 1.I x d d ( )is the acquired high-resolution image.The N lower resolution images, in the macropixel co-ordinate system u, I u n ( ), are used to reconstruct the phase, absorption and dark-field images in the same way as with the analyzer-based method.Figure 3 provides a schematic illustration of these functions.
After acquiring views at an adequate number of projection angles, and the two image signals are retrieved, we use filtered back projection (Barrett and Swindell 1996) with a cosine filter to reconstruct the local absorption and phase coefficients of the phantom cross section.
Regardless of the use of analyzer grating, choosing the design parameters for a Talbot-Lau x-ray interferometer; such as grating periods, Talbot order, grating and detector locations; must satisfy several constraints to ensure setup viability (Weitkamp et al 2005, Pfeiffer et al 2006, Pfeiffer et al 2007).These constraints ensure that different incoherent contributions to the interference pattern overlap.Within these constraints, the setup can be optimized for specific imaging tasks, such as optimized phase gradient or dark-field signal sensitivity for a specific object to be imaged.In the case of the sample positioned after the G 1 grating, the sensitivity increases with increasing object-detector distance and smaller grating periods.For CT, the total setup length, from source to detector, has a practical limit.For our simulations we have chosen an inverse geometry with a total length of 2 m. 'Inverse' means that the G 1 grating precedes the object (Donath et al 2009) and is preferred for integration in CT systems (Viermetz et al 2022).Additional simulation parameters are listed in table 1.The geometry chosen balances imaging performance with ability to fit within a practical clinical CT imaging suite.
In this work we have chosen to sample the wavefront such that it is sampled at 1/4 of the detector sample element size (2 times the Nyquist frequency), resulting in low phase difference between sampling elements.
We use the numerical model to simulate images of three types of objects.The first is composed of a water cylinder (β = 2.398e-12, δ = 6.398e-8) of 50 mm diameter placed inside of an adipose cylinder (β = 1.318e-12, δ = 6.086e-8) of 300 mm diameter, surrounded by air (β = 2.996e-15, δ = 7.048e-11).This simple object enables evaluation of CNR for different absorbed radiation dose values.Contrast is evaluated in tomographic reconstructions of phase-retrieved projection images of the object.For a cylinder, a 2D image is sufficient.The second object contains six cylinders that model water, with diameters 60 mm, 50 mm, 40 mm, 30 mm, 20 mm and 10 mm.These cylinders are placed in an identical 300 mm diameter adipose cylinder (as in the CNR phantom above) and surrounded by air.An illustration of these two phantoms appears in figures 6(c)-(d).The third object, shown as an inset in figure 6(b), is composed of adipose material surrounded by air, but has a diamond shape to allow us to evaluate how well the phase information is retrieved for different number of photons per pixel.All tomographic reconstructions are based on 180 regularly-spaced projections over a 360 degree rotation of the object.The effects of a polychromatic source can be modeled by numerical integration over the source spectrum but are small in comparison to other effects, as long as dE/E is < 10% (Pfeiffer et al 2007).This can be achieved by filtration (using, for example a metal foil) or by exploiting the energy resolution of the photon counting detector.(Having the ability to partition the detected spectrum into two or more bins where m denotes the mean signal and s the noise value in the regions with water and adipose.The noise power is calculated as the variance of the signal values, and the CNR is averaged over ten reconstructions.To verify its expected linear dependence on dose, we compute CNR 2 for different dose values applied to the same object.Dose is calculated from the total number of absorbed photons in the object, multiplied by the energy and divided by the object mass in one pixel.This is a fair approximation as we are comparing the same objects for the two methods.

Results
A projection plot of the first object described above (single water cylinder in adipose) is shown in figure 4. As expected, high intensity values are achieved at the interface between air and adipose.The projections are then calculated into phase and absorption values and subsequently tomographically reconstructed into the images shown in figures 5 and 6, where two dose values are given as examples, for both methods and both objects.
Figure 5 shows the absorption and figure 6 the phase CT reconstructions.The images exhibit no phase-wrapping artefacts or numerical errors.More importantly, the noise follows the same behavior which has been described in other phase CT simulations (Raupach and Flohr 2011).As the quantum noise increases in the phase data, it is  especially the low frequency noise which becomes dominant.It is also clear from the images that the analyzerfree method provides better CNR.
A comparison of CNR is shown in figure 6(a), where CNR 2 values calculated for images corresponding to different absorbed doses are plotted, for both analyzer-based and analyzer-free configurations.On average, a 5 times lower dose is required to attain equal CNR in the analyzer-free versus analyzer-based reconstructions.Conversely, for equal dose, analyzer-free CNR 2 is 5.3 times higher than the analyzer-based cases.This improvement is slightly higher than expected, given the analytical values presented above.This is likely due to the fact that the analytical expression uses a Gaussian function for the PSF, whereas the numerical model uses a combination of a Gaussian and a Lorentzian.This leads to a slightly higher visibility, and therefore also contrast.
Another way to demonstrate the improved performance of HRD phase contrast is to calculate how many photons are necessary to retrieve the phase in a projection image, since this is related to the robustness of the fidelity of the reconstructed images: when a high level of noise is present in the phase-stepping curves, a curve fit is not possible.The phase error is thus small for higher photons counts but increases as the photon count drops and the curve fit becomes increasingly difficult.A comparison between phase error variation in 100 projection images per 55 μm pixel in analyzer-free and analyzer-based phase contrast imaging appears in figure 6(b).The requisite number of photons in the analyzer-based case for equal phase error is about 2.5 times, which is in good agreement with the CNR difference between methods.This suggests that analyzer-free imaging is more robust against phase-error-induced artifacts.

Discussion
Improved dose efficiency is crucial for phase contrast x-ray imaging.It has been shown theoretically (Raupach and Flohr 2011) that, when considering many biological samples imaged with standard x-ray sources and detectors, phase contrast does not lead to a reduction of the absorbed dose versus absorption contrast; phase contrast retrieval depends on the coherence length divided by the pixel size squared and Raupach et al have argued that this limits the clinical use of the technique, since pixel sizes in CT systems are large relative to coherence length.Several improvements in reconstruction algorithms have since reduced this gap, but there remain limited applications where phase contrast leads to higher contrast-to-noise ratio for equal dose compared to absorption contrast.Coherence length needs to be increased through, for example, implementation using long imaging beamlines or small-focal-spot sources, yielding impractical imaging geometries and exposure times.By removing the analyzer grating as shown here, the gap between absorption and phase contrast is significantly reduced, potentially opening up new possibilities for phase contrast and dark field imaging.The application of a virtual analyzer grating is straightforward to implement in an image signal processing chain.Its low computational overhead is highly advantageous considering the very large amount of data generated (count events for an incident flux of >10 8 photons mm −2 s −1 on the 2.5 gigapixels of a 1 × 0.2 m 2 detector) in a short amount of time (a high-resolution clinical CT system might acquire 4000 views during each 0.23 s rotation).The virtual grating allows groups of pixels to be ignored, and the outputs of others to be compressed; operations that can be executed rapidly by front-end electronics.Power consumption per detector unit area is expected to be higher than in a conventional CT detector.However, the reduced capacitance due to smaller pixel size will serve to offset the additional local power needed by the logic required to apply the virtual grating.Compared to the G 2 grating, which must be fabricated with the aid of a synchrotron source, detector manufacture can leverage microfabrication infrastructure available for building integrated circuits.Similar detector development has been demonstrated, for example, in the implementation of monolithic CMOS in fully depleted silicon (Peric et al 2021) and for detectors developed for high-energy physics (Neubüser et al 2023).
The work presented here can also be extended to dark-field x-ray imaging, for which we expect the same order of magnitude improvement in dose efficiency as observed here.Dark-field imaging already shows great promise for lung (Velroyen et al 2015) and gout imaging (Braig et al 2020), clinical imaging milieus where the small-angle scattering is high and further dose reduction could further increase the attractiveness of this technique.
Future work will include more detailed modeling of the detector, including energy dependence and electronic noise, and will proceed as such detectors are further developed and characterized.It has been shown (Thuering and Stampanoni 2014) that the visibility in analyzer-based stepping curves decreases with increasing source polychromaticity.Since a high-resolution detector as described in our analyzer-free method will be energy-resolving, we expect this to be less of an issue, but the thresholds of the energy bins need to be optimized and characterized.It is likely that extension to PC imaging of task-optimal energy bin weighting schemes such as those presented by Yang et al (2023) will help to optimize visibility and CNR by adaptively controlling dE/E (the core of the monochromicity assumption).
Finally, analyzer-free imaging has already been proven experimentally (Cartier et al 2016).Future work will compare the Hilbert transform method for phase retrieval to our virtual grating method in terms of dose efficiency and computational power.It will also be interesting to apply statistical iterative methods (Teuffenbach et al 2017) and the use of an absorptive phase grating (Huang et al 2009).

Conclusion
We have presented an analytic and numerical analysis of analyzer-free phase contrast x-ray imaging based on a proposed clinical-CT-capable high-resolution detector.Our simulations demonstrate that dose can be reduced by more than a factor of 5, or equivalently, that CNR can be increased by 2.5 for equal dose.Artifacts are also suppressed at lower photon counts.In addition, the method greatly simplifies the imaging setup and likely also the CT system cost, since the largest grating (typically fabricated from Au) is removed.We also proposed and demonstrated a means of reducing the volume of requisite output data from the detector by introducing a sparse readout scheme that can be implemented in front-end hardware.We believe that this can be a significant step toward clinical realization of phase contrast x-ray imaging.

Figure 1 .
Figure 1.(a) Schematic illustration of a conventional analyzer-based phase imaging setup for CT imaging.(b) Imaging geometry of an analyzer-free interferometer for CT imaging.
the geometric magnifications of the source and the G 0 grating.I s is the source intensity distribution and T x G 0 ( ) is the G 0 grating transmission function.If an analyzer grating is used,

Figure 2 .
Figure 2. Flow chart illustrating the steps in the numerical model.Two separate and independent paths are shown for the two methods.For CT applications, an additional step is required in which the output phase images from different projection angles are tomographically reconstructed.

Figure 3 .
Figure 3. Illustration of the fringe pattern on the detector, I x d d ( ) (top), the virtual grating, G x , nL N 2 d 2 ( ) and the integrated function I u n ( ).

Figure 4 .
Figure 4. Plot of a calculated projection together with the true phase of the object.High intensity values are seen when the material changes from air to adipose.

Figure 5 .
Figure 5. Simulated images of absorption contrast CT slices for two dose levels.The object consists of a water cylinder inside an adipose cylinder, surrounded by air.

Figure 6 .
Figure 6.(a) Plot of contrast-to-noise ratio squared (CNR 2 ) versus absorbed dose for analyzer-free and analyzer-based x-ray interferometry.(b) Standard deviation of reconstructed phase versus photons per pixel in the absence of an object, for analyzer-free and analyzer-based imaging.(c) Example of simulated CT phase images of the object used in the CNR calculations for the two methods and for two different absorbed doses.The ground truth in the upper right also shows where the signal and noise is evaluated.(d) Ground truth (lower left) and four simulated phase contrast CT images of a resolution target.The smallest cylinder is 10 mm in diameter.

Table 1 .
Values for the parameters used in the simulation of the two phantoms.
Zambelli et al (2010)E/E both prospectively and retrospectively.)Wehavetherefore conducted this study at a single energy.The CNR is calculated as described inZambelli et al (2010)