X-ray diffraction tomography with limited projection information

X-ray diffraction tomography (XDT) records the spatially-resolved X-ray diffraction profile of an extended object. Compared to conventional transmission-based tomography, XDT displays high intrinsic contrast among materials of similar electron density and improves the accuracy in material identification thanks to the molecular structural information carried by diffracted photons. However, due to the weak diffraction signal, a tomographic scan covering the entire object typically requires a synchrotron facility to make the acquisition time more manageable. Imaging applications in medical and industrial settings usually do not require the examination of the entire object. Therefore, a diffraction tomography modality covering only the region of interest (ROI) and subsequent image reconstruction techniques with truncated projections are highly desirable. Here we propose a table-top diffraction tomography system that can resolve the spatially-variant diffraction form factor from internal regions within extended samples. We demonstrate that the interior reconstruction maintains the material contrast while reducing the imaging time by 6 folds. The presented method could accelerate the acquisition of XDT and be applied in portable imaging applications with a reduced radiation dose.

ray diffraction, coherently scattered photons interfere with each other and form diffraction pattern. The number of diffraction photons, , from a small object voxel , in the direction of scattering angle covering a small solid angle Ω can expressed as: 2 2 (1 cos ) ( , ) 2 e sc r dI f q d dVdq where = 2.82 × 10 −15 , is the classical electron radius. The object is described by where is the energy of X-ray photon. The constants ℎ = 6.63 × 10 −34 and = 3 × 10 8 / are the Planck's constant and the speed of light, respectively.

Pencil beam XDT.
In this section, we relate the tomographic measurement geometry with the molecular coherent scattering profile. To reconstruct an extended object with two spatial dimensions, ( , , ), from a series of diffraction profiles, the rotation around the y-axis and translation perpendicular to the pencil beam direction are required, as depicted in Figure S1. We define the sample coordinates ( , , ) and the measurement system coordinates ( , , ). is the beam offset, the distance from the rotation center to the pencil beam; is the voxel depth along the pencil beam; and is the distance between a detector pixel and the pencil beam.
Similar to conventional CT, the coordinate transformation at rotation angle between the sample coordinates and system coordinates is (3) Figure S1. X-ray diffraction tomography geometry. Object ( , , ) is translated perpendicular to the pencil beam direction , and rotated by angle around the y-axis. is the voxel depth along the pencil beam, and is the distance between a detector pixel and the pencil beam. is the distance between the rotation center and the detector plane. The solid, black cuboid illustrates the scattering volume, and the dark blue cube represents each voxel in our forward model.
Angular dispersive setup employs a narrow band X-ray source with a central energy at .
The rotation center to detector plane distance is much larger than the size of the , with the detector distance as a parameter [1]. The XDT reconstruction problem is to recover the molecular form factor of the object ( , , ) from a series of measurements ( , ).

XDT reconstruction as parallel fan-beam CT problem.
In this section we will introduce a couple of first order approximations that are instrumental in relating the 2dimensional XDT problem with a circular cone beam 3-dimensional CT problem. Strictly speaking, the coherent scattering tomography is not a cone-beam CT, but vertical parallel fan beam projection.
In Equation (4), other than the factor 1 ( − ) 2 + 2 and the Bragg's law enforced by the Dirac delta function, the transformation resembles that of 3D parallel beam projection. Since the object dimension is on the order of 10 , which is much smaller than the detector to rotation center distance is on the order of 10 cm. The factor 1 ( − ) 2 + 2 ≈ 1 2 + 2 can be taken out of the integral. Also since ≫ , we took the first order approximation of the projection curve Er Er t q hc l t hcl l Despite being mathematically trivial, after applying both approximations, Equation (4) can be further simplified. Er . Equation (6) shows that the line integral in the ( , , ) plane is along the direction (sin cos , cos cos , sin ) , as shown in Figure S2. The projection ray intersects with the mid-plane ( = 0), at = − . The XDT reconstruction amounts to the inverse problem of a series of projections from vertical parallel fan beam.
For each point that is not in the mid-plane, i.e. ≠ 0, the beams passing through it form a sheaf. Similar to the analogy between parallel beam and fan beam CT, we can rearrange the beams to show that the parallel fan beam geometry is equivalent to circular cone beam CT geometry. It is worth noting that circular cone-beam configuration, however, does not satisfy Tuy's condition [2], and the 3D Radon transform cannot be determined completely when projection data are acquired along a circular source-detector trajectory. Approximation algorithms have achieved satisfactory reconstruction, especially when the divergence angle is small. Figure S2. Illustration of the coherent scattering tomographic projection of an object ( , , ) as a 3-dimensional parallel fan beam projection, where the projection direction deviates from the direction perpendicular to the rotation axis by an angle .
In the XDT setup, the momentum transfer measurement range is between 0.030 Å −1 and 0.250 Å −1 with step size of 0.005 Å −1 . The sampling step in the spatial domain is 0.5 .
Given the object to detector distance is 120 , the maximum angle between the midplane and intersection plane is 11.3°. In Section 2, we show the reconstruction using Feldkamp, Davis, Kress (FDK) algorithm [3]. The artefacts due to missing data in Radon space are visible in the results without regularization. For interior XDT, the object undergoes a full rotation, but the beam scans only the region of interest (ROI) that is smaller than the support of the object. From the previous section, we see that the XDT problem can be treated approximately as a cone beam reconstruction. The interior XDT problem is equivalent to a truncated cone beam problem in ( , , ) space. The 3-dimensional geometry is shown in Figure S3.

Interior XDT geometry.
From the perspective of filtered back projection reconstruction, the back projection is nonlocalized, and truncated cone beam problem therefore does not have unique reconstruction. Constraints and conditions that assure a unique solution of the ROI were developed from both practical and theoretical approaches. For truncation problem, much progress has been made based on the concept of differential back projection with prior knowledge of an interior region [4]. For XDT setup, to know the exact scattering profile within a region of the entire sample is usually not feasible. Based on compressed sensing theory, it has been proved that if an object under reconstruction is essentially piecewise constant, a local ROI can be exactly and stably reconstructed based on TV regularization [5]. Then based on the forward model matrix , the reconstruction of object is an optimization problem with TV regularization [6] [7], where ( | ′ ) is the Poisson likelihood of observing the measurement given the parameters ′ , is the weight for balancing the measurement error and the TV regularizer. The TV operator is defined as Where and represent the index along the spatial dimensions x and , respectively, and represents the index along the momentum transfer dimension . The parameter is a weight factor, balancing the difference of unit on the spatial and momentum transfer dimensions. In this section we validate our reconstruction used in XDT reconstruction by using a simulation phantom. The reconstruction performance of full sample projection and truncated projections are evaluated. Figure S4 First, we use the FDK algorithm to reconstruct the projection of the whole sample. Figure   S4  Next we perform the reconstruction using FDK algorithm to the truncated projection. Figure S5(a) shows the reconstruction results at momentum transfer 0.08 Å −1 , 0.12 Å −1 , and 0.16 Å −1 . The NMSE of the reconstruction at these momentum transfer values are 2.80%, 3.24% and 3.40%, respectively. The truncated reconstruction error is more significant than the full reconstruction due to the missing scattering information from the exterior region. The missing information has more pronounced impact near the ROI boundary. Since the simulation phantom is piecewise constant in spatial domain, from the perspective of compressed sensing, the reconstruction can be stabilized by introducing TV regularization [5]. Figure S5 Figure S5(a1) are shown in Figure S5(c). Comparing to the reconstruction from truncated projection using FDK algorithm, we can see that with TV regularization, the edge of the sample is preserved and the truncation errors close to the ROI boundary was suppressed. In conventional single energy CT image, the material contrast is based on X-ray attenuation, which reflects the electron density of the material. CT images display low contrast in liquid and soft tissues due to their similar atomic composition. XDT images reveal molecular structural information, and provide higher contrast in the momentum transfer space. In this section, we compare the material contrast between conventional CT and XDT from the perspective of material classification.

SVM classification of XDT reconstruction.
The material classification on the interior XDT reconstruction was performed using support vector machines (SVM) [8].  Figure S6(c). The material class that yields the highest posterior probability is assigned to be the material of each pixel. Figure S6(d) shows the material classification based on the experimental reconstruction of the full FOV and interior region superimposed on the total scattering intensity.

Contrast comparison between CT and XDT.
To compare the material contrast between X-ray diffraction and attenuation based tomography, we convert both CT attenuation map and XDT SVM score to a probability distribution (class | ) P f which describes the likelihood of a pixel belonging to a particular material class. Figure S7. Image histogram of attenuation based CT image shown in Fig. S6(a). Vegetable oil and ethanol has very similar attenuation coefficients, showing as one peak in the histogram.
In CT image, the scalar attenuation value is . The histogram of the CT image of the phantom was shown in Figure S7. The attenuation cross-section distribution of each material, ( |class = i), can be modeled by a Gaussian function. We pick 20 pixels from each materials to calculate the mean and standard deviation of the grayscale value . The results are shown in Table 1. XDT images have better material contrasts in the scattering profile, rather than a single value of attenuation. To compare the material classification with attenuation based CT, the SVM score was also converted to a probability (class = i| ) that describes the likelihood of each scattering profile belongs to one of the four materials. Given the SVM outputs, we can fit the score into a parametric sigmoid function,

Oil
where the parameters and are obtained by maximizing likelihood function [9]. The probability of belonging to each class (class = i| ) was evaluated using the fitted parameters at each pixels. Figure S8 compares the probability of each material in the Region 1-3 defined in Figure 2. Figure S8. Probability of material in each highlighted region marked by the numbers in (b). The probability is obtained by converting the distribution of SVMs scores to a posterior probability.

X-ray spectrum
The spectrum of our X-ray pencil beam is measured using a photon-counting detector (X-123, AMPTEK). To estimate the source irradiance, we also simulated the spectrum of a copper-anode tube operating under 35kV using XSPECT. The measured spectrum was then scaled up so that the total number of photons matches that on the simulated spectrum. Fig. S9 shows the spectrum of our source used in the Monte Carlo simulation. Figure S9. Spectrum of the source used in Monte Carlo simulation. The spectrum was measured experimentally and then scaled according to the total irradiance in the XSPECT simulation.