Propagation-based x-ray phase contrast imaging using an iterative phase diversity technique

Through the use of a phase diversity technique, we demonstrate a near-field in-line x-ray phase contrast algorithm that provides improved object reconstruction when compared to our previous iterative methods for a homogeneous sample. Like our previous methods, the new technique uses the sample refractive index distribution during the reconstruction process. The technique complements existing monochromatic and polychromatic methods and is useful in situations where experimental phase contrast data is affected by noise.


Introduction
We have previously shown that an iterative reconstruction applied to propagation-based near-field x-ray phase contrast data can have a wider range of applicability than analytical techniques [1], such as the contrast transfer function [2,3] and transport of intensity equation [4].
The iterative technique is capable of recovering the complex transmission function of the sample-meaning quantitative determination of the sample's thickness distribution is possible -under both monochromatic [1] and polychromatic [5] illumination.
However, we have also shown [1] that loss of information in the recorded phase contrast image-for example, due to noise or source blur-has the potential to introduce artefacts-such as a cross-hatching effect-into the iterative reconstruction, thus impacting the quality of the reconstructed object image.
A potential solution to this problem can be found through the use of phase diversity techniques. These methods were originally explored to find common elements in multiple images which contained noise or suffered other aberrations in incoherent imaging systems [6]. The benefits of phase diversity can be introduced through a number of different methods including implementing additional planes of detection [7], employing different focus distances [8], varying the longitudinal and transverse position of the probe on the sample [9] and using multiple energies [10][11][12][13].
In this work, we show that a multiple wavelength phase diversity technique combined with a near-field phase contrast iterative technique is capable of recovering thickness information with fewer artefacts for homogeneous objects [14]. This new phase diverse method significantly reduces the cross-hatching artefact that was encountered in our monochromatic iterative algorithm [1].

Algorithm
Since the thickness of the object is independent of wavelength, it can be used to ensure consistency between measurements taken at each wavelength. The algorithm is similar for an individual wavelength used in [5] and shown here in equations (1)- (6). The incorporation of multiple wavelengths through the consistency of the thickness is managed as shown in equations (7) and (8). The calculated sample thickness, t, is where the subscripts here and in the following equations refer to the jth wavelength, λ, used in the measurement, k is the wavenumber, = where y 0 is the incident wavefield.
Following the operator notation often used in CDI [15][16][17], successive iterations to the wavefield, y, can be obtained using The support constraint is where I is the measured intensity and ŷ is the detector plane wavefield.
Here, we use the Fresnel propagator in its convolution form [19] to propagate between the sample and detector planes. The iterative algorithm is a solution for the lack of phase information recorded at the detector and in our implementation the initial wavefield is initiated using a random value for the phase of the wavefield.
With multiple wavelengths at each iteration we can apply equations (1) and (2) to obtain estimates for the thickness.
Then using the inverse of equations (1) and (2) with the next wavelength values, the current estimate for the next wavefield is produced, which can then, in turn, be used in equation (3). This operation can be represented by p l t j as y p y b d y = = -+ l l l l l l l l We can represent the overall phase diversity process by where Π represents the successive application of the operations and p lt M 1 sets the energy back to λ 0 . An alternative approach could be to use the average estimate for the thickness obtained from each wavelength to generate the next iterate for the wavefield.
Convergence of the algorithm can be monitored by comparing a χ 2 difference between the current and previous estimate for the diffracted intensity for any of the measured wavelengths used [1].
Any energy of interest can be used to monitor the convergence of the algorithm at the detector plane. A more robust convergence metric can also be applied by requiring that χ 2 fall below a pre-set threshold for all wavelengths used.
Convergence of the reconstructed thickness estimate, t n , can also be monitored for simulation test cases by comparing a χ 2 difference between the current estimate for the thickness with the known value [1].
The benefits of the phase diverse method compared to single and polychromatic wavelength reconstructions are demonstrated in figure 1. In this example, the test object used in the experimental demonstration was simulated under conditions that demonstrate artefacts in the reconstruction. A Gaussian source blur of 2 μm full width half maximum (FWHM) with a random signal to noise ratio of 3% was applied to the simulated phase contrast image for the test object, which has thickness of 350, 650 and 1000 nm in the regions as shown in figure 1(a). The image was oversampled on a 710×710 array. The source to sample distance, z 1 , and sample to detector distance, z 2 , were 0.76 m and 0.24 m, respectively. The pixel size at the detector was 450 nm. Figures 1(b) and (c) show the monochromatic reconstructions for λ=0.138 nm (9 keV), δ=2.67×10 5 and β=2.31×10 6 and λ=0.0954 nm (13 keV), δ=1.24×10 5 and β=1.51×10 6 , respectively, (d) shows a polychromatic recovery using a 9-13 keV spectrum in bins of 100 eV, while (e) is the phase diverse reconstruction. Although we have used wavelength notation above, we will refer to the corresponding energies for experimental convenience below. The phase diverse reconstruction provides a better-quality image, significantly reducing the cross-hatching artefact seen in the monochromatic and polychromatic reconstructions. It can also be seen in figure 1(f) that the phase diverse method converges faster than either the monochromatic or the polychromatic simulations. Table 1 shows the standard deviation of the recovered thickness divided by the mean of the recovered thickness-a percentage thickness error-for each of the reconstruction methods. Near the sharp edges of the object, the sampling assumptions used in the propagation and in the analytical formula are invalid due to the sudden change in wavefield's phase at these points [3]. Accordingly, the regions indicated in figure 1(a) are used to assess the recovered thickness. While the average values recovered over the regions are similar, the percentage thickness error for the phase diverse method is less than the percentage thickness error for either the monochromatic or polychromatic reconstructions.

Experiment
An experimental demonstration of the algorithm was undertaken using Diamond Light Source's B16 Test Beamline [18]. A conventional in-line phase imaging geometry was employed. Phase diversity was achieved by acquiring phase contrast images at two distinct energies-9 and 13 keV.
For both energies, a point source with a vertical and horizontal FWHM of 0.607 and 0.837 μm respectively was created using a Kirkpatrick-Baez mirror. z 1 and z 2 were 0.76 m and 0.24 m, respectively.
Twenty phase contrast and flat-field images were acquired at each energy, all with an exposure time of 75 s. The detector employed was a cooled 14 bit charge-coupled device camera coupled with a YAG:Ce 35 μm thick scintillator and 20×objective lens.
The sample used for the experiment is shown in figure 2(a). It was made from two overlapping patterns of Au deposited on a Si 3 N 4 window. The thickness was 350, 650 and 1000 nm in the blue, green and red regions indicated respectively.  table 1 were taken from. (b) A single energy recovery using 9 keV and the method described in [1]. (c) A single energy recovery using 13 keV and the method described in [1]. (d) A polychromatic recovery using 9-13 keV and the method described in [5]. (e) A phase diverse recovery using energies of 9 and 13 keV. (f) The convergence of the algorithm in the sample plane for 9 keV (red), 13 keV (blue), phase diverse (green) and polychromatic (purple) methods. Table 1. Comparison of recovered standard deviation divided by thickness for the simulation data using iterative monochromatic, phase diverse and polychromatic methods. The statistics were calculated using the regions highlighted in figure 1(a).

Analysis
The experimental measurements were used to create two datasets for each energy. The first dataset used all of the 20 experimental measurements of the phase contrast and flatfield images. The second dataset only used the last phase contrast measurement acquired and first flat-field measurement acquired (i.e. the phase contrast and flat-field measurements that were closest in time). The phase contrast images for both the 9 and 13 keV datasets were divided through by the relevant flat-field images to remove variations in the illumination and scintillator non-uniformity. Despite the flatfielding of the datasets, the background in the phase contrast was still not perfectly uniform due to time variations in the illuminating beam. The resultant images can be seen in figures 2(b), (c), (e) and (f).
One thousand iterations of our monochromatic iterative algorithm [1] were applied to each of the four datasets using a loose support. Propagation between the sample and detector plane was completed using the Fresnel propagator in its convolution form [19]. The detector χ 2 was used to monitor the convergence of the algorithm for each of the datasets. The thickness reconstruction for these monochromatic sets can be seen in figures 2(g), (h), (j) and (k).
The phase diverse algorithm was applied to the 9 and 13 keV phase contrast images to reconstruct the sample thickness for both the single and twenty image datasets. For both datasets, 1000 iterations were applied using the phase diverse algorithm and a loose support. The Fresnel propagator in its convolution form was again used to propagate between the sample and detector planes. A χ 2 test as described in the algorithm section was used to monitor the convergence of the algorithm for each of the datasets. The thickness reconstruction for the phase diverse method can be seen in figures 2(i) and (l).
The time variability in the illuminating beam meant that the phase contrast images were not uniform. Accordingly, the reconstructions of the object were also not uniform over large spatial regions, even where the regions are known to be of uniform thickness. This is particularly evident in the lower left part of the reconstruction shown in figure 2(j). The fact that the non-uniformity is more pronounced in this image, which was from 20 repeat exposures, then it is in figure 2(j), which is from a single exposure, is a clear indicator that the variation between exposures does not average out to a uniform result.
Fortunately, the effect that we are seeking to demonstrate here is that phase diversity can ameliorate the loss of information in the phase contrast image that results in higher spatial frequency artefacts, such as the cross-hatching shown in simulation and experimentally in figures 2(g), (i), (j) and (k). This can be verified by measuring the uniformity of the recovered object over a smaller region than used in the simulations described in the algorithm section. The regions used in each of the different thicknesses of the object are shown in figure 2(d).
The resulting measured percentage thickness error for each of these regions is shown in table 2. While we have not compared the experimental results for the regions shown in figure 2(d) with those obtained using our polychromatic approach [5], the percentage thickness error obtained in [5] was 10% or greater for each thickness but for a larger bandwidth, which improved the results. Accordingly, while a similar error is expected in the smaller regions as for the larger regions due to the uniform reconstruction obtained in that experiment, it is not expected to be any better than the 10% result previously obtained. Therefore, the experimental results are consistent with simulations described in the algorithm section. It can be seen overall that the phase diverse method gives a lower percentage thickness error in each case, consistent with the visual impression of greater uniformity of the phase diverse method given in figure 2.

Conclusion
The phase diverse method presented in this paper adds an additional tool for those attempting to undertake reconstructions in the near-field. We have shown the phase diverse technique significantly reduces the cross-hatching artefact that can be encountered with our previous monochromatic iterative algorithm. The approach could also be generalised to measurements taken with multiple polychromatic illumination conditions. Table 2. Comparison of recovered standard deviation divided by thickness for the experimental data using iterative monochromatic and phase diverse techniques. The single and twenty image data set were calculated using the regions highlighted in figure 2(d).