Abstract
The Xray darkfield signal can be measured with a gratingbased TalbotLau interferometer. It measures small angle scattering of micrometersized oriented structures. Interestingly, the signal is a function not only of the material, but also of the relative orientation of the sample, the Xray beam direction, and the direction of the interferometer sensitivity. This property is very interesting for potential tomographically reconstructing structures below the imaging resolution. However, tomographic reconstruction itself is a substantial challenge. A key step of the reconstruction algorithm is the inversion of a forward projection model. In this work, we propose a very general 3D projection model. We derive the projection model under the assumption that the observed scatter distribution has a Gaussian shape. We theoretically show the consistency of our model with existing, more constrained 2D models. Furthermore, we experimentally show the compatibility of our model with simulations and real darkfield measurements. We believe that this 3D projection model is an important step towards more flexible trajectories and, by extension, darkfield imaging protocols that are much better applicable in practice.
Introduction
The probably most studied acquisition system for Xray phasecontrast imaging is the TalbotLau grating interferometer. This system allows to measure a Xray absorption image and two additional images, namely the differential phase image and the darkfield image. The Xray darkfield measures ultrasmall angle scattering, which is caused by inhomogeneities in materials at micrometer scale^{1,2,3}.
Recently, Xray darkfield imaging has received much attention for its potential applications in medical imaging and nondestructive material testing. The investigated applications in medical imaging span a wide range. Examples are the identification of different lung diseases^{4,5,6,7}, lung cancer^{8}, the identification of microcalcifications^{9}, or the differentiation of kidney stones^{10}. Other examples are the detection of bone structures^{2} and fractures^{11} as well as brain connectivity^{12}. Also for material testing there are a wide range of application of the darkfield signal^{3,13,14,15,16}.
The origin of the observed darkfield can have various reasons, such as smallangle Xray scattering, an intrapixel differential phase contrast that cannot be resolved, or even beam hardening^{17}. While the effects are not clearly separable, we will focus on the darkfield created through smallangle scattering. Two properties of the darkfield signal are particularly interesting. First, ultrasmall angle scattering is caused by structural variations at the scale of few micrometers, which is significantly below the resolution of conventional Xray imaging systems^{18,19}. Second, a gratingbased system allows to measure the 3D orientation of elongated micrometersized structures such as fibers^{20}. Traditional absorption Xray systems have to be able to fully resolve a fiber in order to measure its orientation. In contrast to that, Xray darkfield imaging enables to deduce the fiber orientation of considerably smaller structures.
Jensen et al.^{21} and Revol et al.^{22} explored the fundamentals of the darkfield orientationdependency. In a tomographic setup, either the object or the imaging system rotates during the acquisition. During the rotation, the relative orientation between object and system changes, which leads to a variation in the signal. This signal variation allows to reconstruct the orientation of the structure. There have been several reconstruction methods proposed in previous works^{23,24,25,26,27,28}. However all of them are based on 2D projection models of the 3D structure. This means that the models rely on the reconstruction of several 2D slices and are not compatible with true 3D trajectories.
In this work, we aim to overcome this limitation by proposing a darkfield projection model over the 3D space. This allows to directly estimate the 3D structure, and to use sophisticated 3D trajectories such as a helix.
TalbotLau interferometer
The TalbotLau interferometer is a gratingbased phasecontrast setup. A sketch of the system is shown in Fig. 1. The system is an extension of conventional Xray imaging setups, where three gratings G_{0}, G_{1}, and G_{2} are placed between the source and detector. Xrays are generated by a conventional Xray tube S. This Xray tube can be operated in an Xray regime that is compatible with medical applications, such that a medical Xray detector D^{29,30} can be used. Grating G_{0} effectively separates Xrays from the large source into narrow slit sources that are individually coherent, but mutually incoherent. G_{1} imprints a periodic phase modulation onto the wave front to create an interference pattern at the detector. Both gratings G_{0} and G_{1} have periods that are in the range of few micrometers. For operation with the much lower resolution of clinical Xray detectors, the interference pattern is sampled with the G_{2} grating in front of the detector, which also has a period in the range of micrometers. The sampling at the detector can be either performed by slightly detuning the grating G_{2}, which leads to the Moiré effect^{31,32,33}, or by performing phase stepping^{29,30}. Both approaches sample points on the interference curve, which can then be fitted by a sine. In practice, two scans are performed, namely a reference scan without object in the beam path, and an object scan with the object. By comparing the sinusoidal curve of both scans, it is possible to calculate the three quantities absorption, differential phase, and darkfield. As in standard Xray imaging, absorption is defined as the change in the average intensity. The differential phase is the angular shift of the sine. The darkfield signal is given by the ratio of the amplitude of the sine over the average intensity.
For this work, it is important to note that all three signals are created by sampling the sinosoidal function in one direction. We call this direction the sensitivity direction s. The sensitivity direction is perpendicular to the grating bars.
Related work
Xray Tomography is performed by rotating either the Xray setup or the object during the acquisition. This rotation changes the orientation of the object relative to the sensitivity direction. A key difference between traditional Xray absorption and darkfield is the impact of this relative orientation: Xray absorption is independent of the relative orientation, while Xray darkfield depends on it.
This makes a major difference for the choice of reconstruction algorithm. The popular filtered backprojection (FBP) algorithm implicitly assumes that the signal strength is independent of the viewing direction — which does in general not hold for Xray darkfield imaging.
The tomographic reconstruction, in general, requires the inversion of a projection model. For the angledependent darkfield signal, several 2D projection models were proposed, which are discussed briefly in the following.
Jensen et al.^{21} first showed the angle dependency of darkfield projections. They rotated the object around the optical axis of the system, and found that the variations in visibility can be described by the first two orders of the Fourier expansion. Shortly afterwards, Revol et al.^{22} modeled the darkfield scatter by a 2D Gaussian function and showed that the logarithm of the darkfield signal can be formulated as
where ω is the rotation angle of the fiber around the optical axis, θ is the starting angle of the fiber in the xyplane (see Fig. 2(a)) and A, B are an isotropic and anisotropic contribution of the scatter, respectively. The projection models^{21,22} assume that the object is rotated around the optical axis, which limits these models to thin sample layers. Malecki et al.^{34} investigated the signal formation for the superposition of layers with different fiber orientations. They conclude that the darkfield signal can be represented as the line integral along the beam direction over the anisotropic scattering components.
In order to describe the darkfield for thicker objects, Bayer et al.^{20} proposed another projection model. They showed that the projection of a fibrous structure also depends on the azimuthal angle ϕ. This corresponds to the angle of the fiber projection in the xz plane in Fig. 2(b). They derive the darkfield signal as
The third projection model was proposed by Schaff et al.^{28} and is shown in Fig. 2(c). Here, the grating bars are aligned along the 2D trajectory, and the darkfield signal is measured along the rotation axis. Schaff et al. approximate this signal as constant with respect to the tomographic rotation, such that the the scattering strength only depends on the angle between the fiber and the rotation axis.
This approximation simplifies the reconstruction, since a normal FBP algorithm can be used. However, for the two other projection models, the resulting signal per voxel varies along the trajectory. 2D object orientations are in this case reconstructed via iterative reconstruction^{23,24,25,26,27}. Among these works, Bayer et al.^{23} proposed a method to reconstruct 2D inplane orientations of fibers. Hu et al.^{24} proposed to reconstruct the 3D orientation by combining two 2D inplane scans with different trajectories. Xray tensor tomography has been proposed by Malecki et al.^{25}, Vogel et al.^{26}, and Wieczorek et al.^{27} by combining multiple 2D planes.
Since all projection models describe the darkfield only as a function of one angle, it is only possible to reconstruct a 2D slice. The reconstruction of the full 3D distribution of oriented materials requires the combination of scans from several trajectories, which overall leads to quite complex acquisition protocols. Malecki et al.^{25} reconstructed a scattering tensor by using the model from Revol et al.^{22} and rotated the sample into a finite number of scattering directions. Hu et al.^{24} used the model by Bayer et al.^{20,23} and used two 2D reconstructions to compute the 3D fiber direction, while Schaff et al.^{28} fit a 3D ellipse to individually reconstructed 2D slices.
Previous works take different approaches to describe the 3D nature of Xray darkfield, ranging from Gaussian distributions^{21} over a kartesian basis^{26} to a spherical harmonics basis^{27}. However, to our knowledge, there exists to date no direct 3D reconstruction algorithm. One of the reasons for this may be the fact that a reconstruction method requires the inversion of a projection model, which to our knowledge has not been defined yet in 3D.
The definition of a 3D model makes it possible to use 3D darkfield trajectories. For example, the helix is a popular 3D trajectory with favorable properties in traditional absorption tomography. In this case, Tuy’s condition for absorption image can be applied, and the completeness of such a certain trajectory can be shown^{35}. In principle, a similar system can be pursued for darkfield tomography if a welldescribed 3D trajectory is available. As long as only 2D trajectories can be used, the best known acquisition schemes that fully measure the scattering orientations are still quite complex^{36}.
Contributions and organization of this work
In this work, we propose a fully threedimensional Xray darkfield projection model. Previous works are limited to descriptions of 2D projections of the darkfield signal, which limits the reconstruction to 2D scatter projections, and constrains the trajectories to 2D. In contrast, the proposed model enables the use of an arbitrary scanning geometry, and overcomes the need for combining several 2D trajectories. The proposed model allows to use established 3D scanning trajectories to acquire the 3D scatter distribution, like for example a helical geometry. Furthermore, it enables the development of novel 3D geometries that aim at optimizing the recovery of directional information for specific clinical examinations or visual inspection tasks.
Additionally, the proposed model is very general. It allows to freely choose the ray direction and the sensitivity direction. That way, it overcomes the restriction of earlier works to parallel beam geometries. Instead, it allows to model a cone beam, which is of major importance for many popular hardware designs, like for example a line scanner.
We only use the assumption that the scatter distribution of the darkfield signal is a 3D Gaussian, and we derive the general projection model from that. Furthermore, we discuss the impact of additional constraints if they are available, and demonstrate the consistency with existing 2D models. In the experiments, we show that the proposed model accurately fits predicted darkfield values from a wave simulation as well as from real experiments.
The paper is organized as follows. Section 2 provides a mathematical derivation of the proposed model, which describes the darkfield signal formation in a very general way. Afterwards, in Sec. 3, we discuss the impact of additional constraints on the model and show that our model is consistent with the 2D projection models discussed in Sec. 1.2. Experiments that link the predicted signal to simulations and actual measurements are presented in Sec. 4. We conclude the paper in Sec. 5.
Proposed Xray Darkfield Projection Model
The Xray darkfield signal measures the Xray small angle scattering of microstructures in a sample. Xray darkfield scattering has the special property that its observed magnitude depends on the relative orientation of the sample in the setup. To characterize the signal, we use the notion of isotropic and anisotropic scattering components. This notion was originally introduced for 2D projection models. A schematic sketch of this model is shown in Fig. 3. Here, the isotropic component scatters in all directions equally strongly, independent of the sample or setup orientation. Conversely, observations of scatter of the anisotropic component vary with the sample and setup orientation.
Thus, if a sample scatters purely isotropically, its signal is independent of the orientation. Such a signal can be reconstructed in a similar manner as Xray absorption. However, if a sample exhibits partially anisotropic scatter, the signal formation depends on the orientation and thus becomes considerably more difficult to reconstruct. In particular, any algorithm for 2D or 3D reconstruction has to explicitly take the directiondependent signal variation into account.
In order to model the signal formation, we introduce the notion of a fiber. A fiber is a microstructure that exhibits a mixture of isotropic and anisotropic scattering. The derivation of the model is organized as follows. First, we expose the relationship between a fiber and its associated scatter distribution in Sec. 2.1. In Sec. 2.2, we show how the fiber is projected by the Xray onto the sensitivity direction. In Sec. 2.3, we show how the projected image of the fiber is converted to a scatter distribution. In Sec. 2.4, we state the complete model, which is the actually observed darkfield signature for a sample point. Afterwards, Sec. 2.5 shows how the measured signal is expressed as line integrals.
The darkfield signal formation depends on three quantities, namely the directions of the Xray, the darkfield sensitivity direction, and the orientation of the fiber. We describe a very general model that considers all three quantities as arbitrary vectors in 3D. This generality has several advantages. It allows us to model not only a system with parallel beam and a perpendicular sensitivity direction, but instead arbitrary acquisition geometries. Examples for such more general system designs are the use of a conebeam scanning geometry, which influences the ray direction, or the use of a curved Xray detector, which results in different sensitivity directions. It also allows to model a 3D helical scanning trajectory, which requires flexibility in all these quantities.
Relationship between fiber and scatter distribution
We make the simplifying assumption that a fiber has the shape of a cylinder. More specifically, the fiber cross section is assumed to be a circle, and the height of the cylinder is assumed to be at least as long as the radius of that circle. The isotropic scattering component is mainly determined by the radius of the circle. The anisotropic scattering component is connected to the size of the cylinder, and will be more rigorously defined in Sec. 2.3.
Mathematically, we represent a fiber as a 3D vector f in \({{\mathbb{R}}}^{3}\), where the vector is parallel to the cylinder axis. The observed fiber creates darkfield scatter. Scatter is not deterministic, and therefore commonly described as a distribution.
For the following discussion, we are only interested in the relative orientations of the fiber and its associated scatter. Thus, without loss of generality, we assume that a fiber and its scatter distribution are rooted in the origin of the coordinate system. We assume that the shape of a fiber scatter distribution is a 3D Gaussian g(x), which is in line with earlier models on 2D scatter distributions^{21}. Then,
where Σ denotes the 3 × 3 covariance matrix. The shape of g(x) is completely described by Σ.
We make the mild assumption that this covariance matrix Σ can be diagonalized (which is satisfied for any nontrivial 3D Gaussian scatter observation). Then, the eigenvalues of Σ describe the scatter strength with respect to its eigenbasis spanned by the eigenvectors b_{1}, b_{2}, b_{3}. The eigenvalues correspond to the variances, i.e., the squared standard deviations along each principal axis of the distribution:
These variances have a special distribution, which comes from the particular case of a scattering fiber: the main scattering direction of the fiber is the 2D subspace that is perpendicular to f. This is illustrated in Fig. 4(a). All scatter directions within this 2D subspace are indistinguishable. As a consequence, the two largest eigenvalues \({\sigma }_{1}^{2}\) and \({\sigma }_{2}^{2}\) are identical, i.e., \({\sigma }_{1}^{2}={\sigma }_{2}^{2}\). The weakest scattering is observed in the direction of f, which is quantified by the smallest eigenvalue \({\sigma }_{3}^{2}\le {\sigma }_{1}^{2}\). This is illustrated in Fig. 4(a). The eigenvector b_{3} is associated with the smallest eigenvalue \({\sigma }_{3}^{2}\) and parallel to f. More specifically, both vectors are identical with the exception that their sign might be flipped, i.e., b_{3} = ±f. The restrictions on the eigenvalues induces that the shape of the scattering function is a oblate spheroid. A 3D sketch of the eigenvalues is shown in Fig. 4(b).
3D Fiber projection model
The darkfield signal formation depends on three geometric vectors, namely the direction of the Xray, the darkfield sensitivity direction, and the orientation of the fiber. Ultimately, we seek the projection of the fiber along the ray direction onto the sensitivity direction. This is a mapping from the 3D fiber vector onto a (1D) scalar value. A nonparallel Xray projection, e.g. from a cone beam, is modelled by a rotation of the fiber. The sensitivity direction can have an arbitrary orientation in space. To relate the fiber direction and the sensitivity direction, we introduce a virtual plane that is perpendicular to the Xray. Both the fiber and the sensitivity direction are projected onto that plane. Then, the 2D projection of the fiber onto the sensitivity direction in the plane is performed. The resulting equations show that the plane cancels, and that the projection of the fiber onto the sensitivity direction can be written as a scalar product. The mathematical details are presented below.
Let us consider a single fiber vector f. Without loss of generality, this fiber is located in the origin of our world coordinate system. The Xray darkfield projection ray r passes through that fiber, and thereby also the origin of the coordinate system.
In imaging systems, all Xrays that form one projection are typically modelled as either parallel or diverging from a central ray c. This changes the relative orientation between r and the fiber vector f. To correct for the diverging ray, we denote the angle of divergence as α, and rotate the fiber in the plane spanned by c and r in the opposite direction. The corresponding rotation matrix is denoted as R_{α}. In the case of parallel projection, R_{α} is the 3 × 3 identity matrix.
We project the fiber f onto a plane E that is perpendicular to the Xray direction r. For this projection, we use orthogonal projections instead of perspective projections of the scatter pattern. This is possible, because the projection of a fiber signature onto the detector is in the range of micrometers, but a single detector pixel is typically two orders of magnitude larger.
An orthogonal projection of a 3D vector onto a plane can be performed with an inner product between the vector and a transformation matrix consisting of the 3D coordinates of the 2D basis. We define the 2D projection plane as a plane where r is the normal vector. Since r passes through the origin, we find it convenient to choose the plane to also pass through the origin, i.e.,
with \({\bf{E}}\in {{\mathbb{R}}}^{3\times 2}\) where \({{\bf{r}}}_{1}^{{\rm{ortho}}}\) is a vector perpendicular to r, i.e., \({{\bf{r}}}^{{\rm{{\rm T}}}}{{\bf{r}}}_{1}^{{\rm{ortho}}}=0\), and \({{\bf{r}}}_{2}^{{\rm{ortho}}}={\bf{r}}\times {{\bf{r}}}_{1}^{{\rm{ortho}}}\) is the second vector spanning the plane, also perpendicular to r. This projection is visualizedin Fig. 5(a).The projection of the fiber along the ray and onto the 2D plane E is then given as product of the rotated fiber f with E, i.e.,
where \({\bf{f}}^{\prime} \in {{\mathbb{R}}}^{2}\) is now a twodimensional vector in the plane E.
The sensitivity direction s denotes the direction along which the Xray darkfield signal can be measured. It is a 3D vector with an arbitrary orientation. To relate the fiber with the sensitivity direction, we also project s onto plane E. Analogously to the fiberplane projection, we also use here an orthogonal projection. The 2D projection of s on E is
The projection of both vectors f′ and s′ on E are shown in Fig. 5(b).
To determine the alignment of the fiber f with sensitivity direction s, the inner product is computed, i.e.
Equation 9 can be simplified by noting that the inner product commutes, which leads to
since EE^{Τ} = I. Equation 11 shows that the projection of the fiber through the system onto the sensitivity direction reduces to directly computing the inner product between the fiber and the sensitivity direction.
Note that in the case of a cone beam, the rotation of the fiber by R_{α} can also be replaced by a rotation of the sensitivity direction s in the opposite direction. While we believe that the rotation of the fiber f is more intuitive, it may be preferable for an actual implementation of a reconstruction algorithm to rotate the sensitivity direction s, since s is a given quantity from the setup geometry, and f is the unknown variable.
3D Projection model for scattering
The projection of the fiber onto the sensitivity direction can be translated into the projection of the scatter. The scatter is the actually observed quantity in the imaging system. The inverse of this conversion links the observations to the unknown fiber direction.
As discussed in Sec. 2.1 the scatter distribution for a given fiber f is given as
where \({\sigma }_{1}^{2}={\sigma }_{2}^{2}\), and b_{1}, b_{2}, b_{3} are an orthogonal basis.
In Sec. 2.2 we considered the transformation from the 3D fiber to a 1D signal. We now want to describe this transformation for the scattering distribution. Since the distribution is described by an orthogonal basis, we can transform the basis vectors b_{i} individually to get the transformation.
Since we defined our projection for an arbitrary fiber f, we can use the same mapping for each basis vector b_{i}. The measured projection of the scattering component i is then given as
Under the consideration that the scattering distribution is symmetric, the variance may not depend on the sign of the basis vectors b_{i}, and its oscillation has to be of period π. In analogy to previous 2D models^{22,23}, both requirements are addressed by squaring the inner product. The projected variance is thus
Complete 3D darkfield projection model
With the individual projections of the fiber and the scattering distribution at hand, we combine both in this section to directly describe the scatter distribution for a given fiber. To this end, we use the introduced notions of isotropic and anisotropic scattering. The isotropic part results in an equal amount of scatter in all directions, while the anisotropic part depends on the relative orientation of the fiber, ray direction, and sensitivity direction. The goal is to describe the 1D darkfield scattering signal in dependency of the fiber, since the fiber is the quantity that shall eventually be reconstructed.
The observed darkfield signal is modeled as
Here, we again square the scaling factor of the anisotropic part to resemble the fact that the signal has a period of π instead of 2π. Analogously, the amount of isotropic and anisotropic scattering is also defined over the variances of the 3D scattering function. Thus, the isotropic component is given as
while the anisotropic component is
To define the anisotropic component as the subtraction from the isotropic scattering may appear counterintuitive at first glance. However, it allows to directly represent the fiber f in the model. We believe that it is useful for building a reconstruction algorithm on top of the model to have the fiber direction directly accessible, since it is the primary quantity of interest.
The derivation to use the fiber vector in the of Eq. 16 comes from the projected variance in Eq. 15. If we consider the smallest scattering component, we observe
Since the eigenvector b_{3} and the fiber vector f are collinear, we can substitute b_{3} in Eq. 19 by f. This leads to
which is used to get the darkfield model in Eq. 16.
In 2D models, the isotopic component is defined as the amount that scatters in all directions equally, while the anisotropic component is defined as an additional component in the direction perpendicular to the fiber direction. In 3D, a direct adaptation of this approach is somewhat more complicated, since the additional scatter of the fiber perpendicular to its main axis forms a 2D subspace. We argue that the concept of isotropic and anisotropic components is not really transferable to the 3D case. In 3D, one can interpret Eq. 16 as the reduction of observed scatter in the direction of the main axis of the fiber, which is mathematically correct, yet somewhat counterintuitive.
The projection of the darkfield does not only depend on the scattering strength described in Eq. 16, but also on the length of the projection rays through the fiber. Since the fiber is assumed to be smaller than one pixel, the darkfield per voxel x can be expressed as:
where C(f, α, d_{iso}, d_{aniso}) is a function describing the average length through the fiber cylinder, dependent on the fiber direction, the ray direction and linked to isotropic and anisotropic values.
Darkfield line integrals
In standard Xray projection imaging, the measured signal intensity is the line integral along the Xray beam line L. Malecki et al.^{34} showed that the superposition of darkfield signals results in a line integral along the beam direction. The darkfield signal is oftentimes modeled analogously to the BeerLambert law for Xray absorption^{37} i.e.,
Here, the line integral is only influenced by the object geometry. This expression is in many works simplified to a linear system by considering the logarithm of Eq. 22 −log(D)^{37,38,39}. We also use this model, although effects such as beamhardening can also add to the darkfield signal and thereby lead to deviations of the BeerLambert law^{38}.
Impact of Additional Constraints on the Model
The proposed projection model is very general. In this section we will show how specific assumptions allow for simplifications. In particular, we show that the model is consistent with the more constrained 2D projection models by Revol^{22}, Bayer^{20}, and Schaff^{28}. We will now show that we are consistent with these if we constrain our model to parallel beams and a circular 2D trajectory. As sketched in Fig. 2, we define for all three 2D models the sensitivity direction as s = (1, 0, 0)^{Τ}.
In a parallel beam geometry, the rotation matrix R_{α} simplifies to the identity, i.e., R_{α} = 1. Consequently, the anisotropic component only depends on the relative orientation of the fiber and sensitivity direction, thus
Revol et al. rotates the fiber around the ray direction. Thus, the fiber orientation of f in the xy plane depends on the starting angle θ and the rotation angle ω. We will denote this dependency as f(ω). The fiber is then given as
with f = (f_{x}, f_{y}, f_{z})^{Τ}. The darkfield model thus becomes d = d_{iso} + d_{aniso} \(((1,\,0,\,0)\,\overrightarrow{f}(\omega ))\) which can be transformed into the original formulation \(A+B\cdot {sin}^{2}(\omega \theta )\).
The mapping to the model by Bayer et al. can be performed in a similar manner. Here, the fiber is rotated around the yaxis. Then,
which results in the original formulation A + B · sin^{2}(ϕ).
The model by Schaff et al. constrains the sensitivity direction parallel to the rotation axis. Then, the projection of the fiber vector (as stated in Eq. 11) is given as
which is constant. It is interesting to note, however, that the model does not consider variations in intersection lengths through the fiber. In practice, the signal is only then approximately constant, if the fiber exhibits only a small elevation angle. In this case, the intersection length is nearly identical for different rotation angles ω.
In summary, the proposed model can be transformed into each of the three existing 2D models with the addition of suitable constraints. At the same time, however, the proposed model is general enough to also represent a full 3D space with an arbitrarily oriented Xray, fiber, and sensitivity.
Experiments and Results
In this section we experimentally evaluate the proposed 3D darkfield projection model. We sequentially evaluate different aspects of the model, to mitigate the combinatorial complexity of evaluating the full parameter space. The evaluated aspects of the model are

1.
Darkfield projection model (Equation 16)

2.
Darkfield signal of a single fiber (Equation 21)

3.
Darkfield measurements (Equation 22).
The corresponding experiments are described and discussed in the sections 4.1, 4.2, and 4.3, respectively. To evaluate the proposed projection model, we compare the results to simulated and real darkfield signals in Sec. 4.2 and 4.3.
Throughout the experiments we represent the darkfield magnitude in arbitrary units [a.u.]. For the prediction of actual measurements, it may be required to convert the variance of the scattering function to microradians. To this end, the scatter distribution must be scaled by the actual setup parameters^{40}.
Darkfield projection model
The formulation of the darkfield in Eq. 16 is sufficiently flexible to describe different trajectories and sensitivity directions. In this experiment, we show the dependency of the darkfield on the Xray direction and sensitivity direction. To this end, we simulate three different trajectories as shown in Fig. 6. We evaluate the darkfield for two different fiber vectors, both with larger scattering coefficient \({\sigma }_{1}^{2}={\sigma }_{2}^{2}=1\) and smaller scattering coefficient \({\sigma }_{3}^{2}=0.5\). The fiber directions are f = {1, 0, 0} and f = {1, 1, 1}, respectively.
All three trajectories have a sourceisocenter distance of 600 and sourcedetector distance of 1200. We simulate two circular 2D trajectories over 360° with an angular increment of 1.5°. Both trajectories have different sensitivity directions. The sensitivity direction s for trajectory (a) in Fig. 6 can be represented by the vector {1, 0} in the detector plane and lies therefore in the rotation plane. Trajectory (b) in Fig. 6 has the sensitivity direction along the rotation axis, i.e. the vector {0, 1} in the 2D detector plane. The third trajectory (c) is a helical 3D trajectory, also with an angular increment of 1.5° and a pitch h = 0.5. The sensitivity direction is aligned with the helical trajectory. The sensitivity direction is in all cases always chosen perpendicularly to the projection ray in order to not introduce an additional scaling factor from the inner product in Eq. 16.
The resulting darkfield signals are shown in Fig. 7(a) over the rotation angle ω. In magenta, the darkfield signals for both fibers on the circular trajectory (a) are shown. They oscillate with a regular sinosoidal. The fiber that lies within the rotation plane (magenta, dotted line) reaches the minimum and maximum theoretically possible darkfield values. The elevated fiber (magenta, solid line) creates an overall stronger signal that, due to the elevation, never reaches the minimum. This is also illustrated in the example scattering spheroid in Fig. 7(b). Here, the magenta circumference indicates the measured scatter intersection for trajectory (a) for the elevated fiber vector f = {1, 1, 1}.
For the second 2D trajectory (b), the sensitivity direction is aligned with the rotation axis. As shown in the light blue lines in Fig. 7(a), this results in a constant signal for both fiber directions. In this case, the elevated fiber f = {1, 1, 1} (solid, light blue) creates a weaker signal. The scattering strength of the elevated fiber is shown as a light blue dot on the spheroid in Fig. 7(b).
The most complex trajectory is the helical 3D trajectory (trajectory (c) in Fig. 6). The darkfield signals for both fibers are shown in black in Fig. 7(a). Due to the constant change in angle between the fiber and the sensitivity direction, the signal change is not symmetric over the 360°. While this observation holds for both fibers, it is more pronounced for fiber vector f = {1, 1, 1} (solid black line).
The experiments demonstrate how the darkfield signal depends on the Xray direction and the sensitivity direction. Furthermore, the experiment also shows that the darkfield signal behaves differently for 2D and 3D trajectories. These differences in the predicted signals demonstrate the need of a 3D projection model for performing a true 3D reconstruction.
Darkfield signal of a single fiber
To verify the proposed darkfield signal for a complete fiber (Eq. 21), we compare it to numerical simulations. We simulate the darkfield with a simulation framework for coherent Xray imaging (CXI) from Ritter et al.^{40}. The setup parameters for the simulation are chosen as follows. The G_{1} is placed at 0.01, with a period of 4.37 × 10^{−6} m, a height of 5 × 10^{−6} m and a dutycycle of 0.5. The G_{2} is placed at 0.17 m, with a period of 2.4 × 10^{−6} m, a height of 300 × 10^{−6} m and a dutycycle of 0.5. Both gratings are simulated as gold. The detector is positioned immediately behind G_{2}. The size of the focal spot is set to 10, and the pixel width is set to 50 μm. The design energy of the system is 25 μm. The simulated object is a teflon fiber (PTFE) with a radius of 1.79 μm and a length of 15 μm.
We set the fiber parameters in the model to the eigenvalues \({\sigma }_{1}^{2}=1.5\) and \({\sigma }_{3}^{2}=0.3\), which correspond to the teflon fiber parameters.
The negative logarithm of the simulated signal is shown in Fig. 8. Figure 8(a) shows the result with our model, while Fig. 8(b) shows the result from the CXI simulations. The darkfield signals are shown in the three planes spanned by the coordinate system, namely xy, xz, and yzplane. Overall, the proposed model and the wavefront simulations agree very well. The different magnitude between the two signals simulations, are due to the range chosen parameter spaces for the experiments.
The signal in the xyplane (blue, solid line) changes only slightly in both simulations. In this plane, the change between the larger and the smaller scatter eigenvalues is observed. While our model leads to a distinct sinusoidal change (Fig. 8(a)), it is more noisy for the CXI simulation (Fig. 8(b)) due to numerical instabilities.
The darkfield signal in the xz and yzplane (red and green lines, respectively) increases with the rotation angle ω, i.e., with increasing inclination of the fiber into the beam direction. Equation 16 predicts for such an inclination no increase, but instead a constant signal. However, the reason for the increasing signal lies in the increased intersection length of the ray through the fiber, as denoted in Eq. 21. It is also interesting to note that this increase is even stronger than the difference of the scatter eigenvalues. The green signal that shows the yzplane is affected by both effects, the scattering eigenvalues and intersection length through the fiber. Overall, the wavefront simulations and the predicted values of the proposed model agree very well.
Darkfield measurements
We show that the proposed complete projection model (Eq. 22) also agrees with experimentally obtained darkfield measurements. The real data consists of a carbon fiber reinforced polymer rod with a diameter of 4 mm. The fiber was measured at five different tilting (elevation) angles, namely 10, 20, 30, 40, and 50°. A projection image at each angle is shown in Fig. 9. For each tilting angle, 100 projection images are taken over a rotation of 180° at 40 kVp and 40 mA. The measurements were performed with a Siemens MEGALIX CAT Plus 125/40/90125GW medical Xray tube using a tungsten anode. The used Xray flat panel detector was a PerkinElmer Dexela 1512 with 74.8 μm pixel pitch, running in 2 × 2 binning mode for processing a faster readout resulting in a 150 μm pixel pitch. For each projection 30 phasesteps with 0.1s acquisition time were used. The gratings have a period of 24.39 μm for G_{0}, 2.18 μm for G_{1}, and 2.4 μm for G_{2}. The setup is 1.854 m long, with a G_{0 }− G_{1} distance of 1473 mm, a G_{1 }− G_{2} distance of 142 mm, and a G_{0}−object distance of 1118 mm. The darkfield is given as −log(V/V_{ref}). For the extracted darkfield signal we used the central pixels of the rod along the yellow, dashed line in Fig. 9.
To simulate the darkfield of the carbon rod, we have to estimate the corresponding parameters. We simulated the darkfield signal of the rod with a diameter of 40 pixel and used scatter parameters \({\sigma }_{1}^{2}=1.4\) and \({\sigma }_{3}^{2}=1\). However, since the real measurements bring along a lot of other unknown parameters, such as the number of fibers within the rod, the darkfield values are not quantitatively comparable.
Figure 10 shows the darkfield signal of the fiber over 180° around the yaxis for different elevation angles, where an elevation angle of 0° represents a fiber that is aligned with the rotation axis. The sensitivity direction is aligned with the rotation plane. Figure 10(a) shows the darkfield simulations, while Fig. 10(b) shows the real darkfield measurements. Please note that the rod was not aligned perfectly parallel to the detector plane on the beginning, which leads to shift in the rotation angle. We corrected our simulations accordingly to match the measurements. The simulations are in very good agreement with the real dataset. In both cases, the signal fluctuation increases with increasing tilting angle of the fiber. The influence of the path length of the projection ray through the object can be seen in the nonregular fluctuation along the rotation angles. As a sidenote, the noise in the plot showing the real data corresponds to the overall noise level of the setup. This can be observed in Fig. 9 in the image background.
Conclusions and Outlook
In this paper, we propose a Xray darkfield imaging projection model. It explicitly calculates structural quantities in 3D using the direction of the fiber, the ray direction and the sensitivity direction. To our knowledge, this is the first true 3D darkfield model.
We believe that this model is a powerful tool for further development of Xray darkfield imaging. In contrast to existing (2D) projection models, where the imaging trajectory is predefined, our model allows to image arbitrary 3D trajectories, like for example a helical trajectory. While the concrete implementation of a helix darkfield scanner is to our knowledge still subject to research, the proposed model is general enough to predict the signal formation. We also showed that the model can be simplified to any of the existing 2D models by addition of suitable constraints. We evaluated the consistency of the model with itself, with a wavefront simulation, and with experimental darkfield measurements. In future work, we will investigate an algorithm for Xray darkfield reconstruction that can make full use of 3D trajectories.
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors acknowledge funding from the German Research Foundation (DFG). Project DFG GZ: AN 257/211  MA 4898/61. L.F. and V.L. are supported by the International Max Planck Research School (IMPRS).
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Contributions
S.H. and A.M. conceived the model. S.H. developed the initial theoretical framework. L.F. and C.R. expanded the theoretical framework. L.F. performed the experiments and analyzed the data. J.B. carried out the wavefront simulations. V.L. and G.A. performed the real data measurements. L.F., S.H., and C.R. wrote the paper with input from all authors.
Corresponding authors
Correspondence to Lina Felsner or Shiyang Hu.
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