We describe a direct quantitative phase reconstruction approach using an X-ray laboratory-based source. Using a single phase-contrast image from each tomographic projection we show that it is possible to modify the filter term in a filtered back projection reconstruction to take account of the broad spectrum from a laboratory source. The accessibility of conventional X-ray laboratory sources makes this method very useful for quantitative phase imaging of homogeneous and weakly absorbing objects.
©2012 Optical Society of America
Phase contrast is a class of imaging that has been developed since the 1930s by Zernike  to improve visualization of transparent objects in optical microscopy. Development of X-ray phase contrast began in the 1980s  for imaging objects with low electron density, such as low Z and low density materials. Phase retrieval in propagation based x-ray imaging is the inversion of a phase contrast image to quantitatively obtain the phase of a complex wave field at the exit surface plane of a sample, from which the decrement of the real part of the refractive index from unity of a sample is often referred, typically using the projection approximation . This component of the sample's refractive index is referred to here as the phase of the sample.
The phase of sample recovered by phase retrieval is the linear projection of the sample phase through a given projection of that sample. Accordingly, standard filtered back projection methods can be used to reconstruct the 3D distribution of the phase of the sample . Under certain conditions it has been demonstrated that the phase retrieval step can be incorporated as part of the filter in filtered back projection to obtain the 3D distribution of the phase of the sample in a single step . These conditions have been extended from a requirement that the sample be non-absorbing , to cases where the sample is weakly absorbing using a semi-empirical approach  or knowledge of the sample composition . Algorithms based on the contrast transfer function approach  or which use the phase-attenuation duality  have also been demonstrated.
These approaches have typically been implemented using monochromatic light from a synchrotron source. However, it has long been a feature of propagation based phase contrast methods that the broad band spectrum delivered by a laboratory source can be used where the source size is sufficiently small [10, 11]. Furthermore, it has been demonstrated that quantitative phase retrieval can be performed using a laboratory source . This approach uses spectrally weighted values for the material properties corresponding to the sample phase and attenuation . In this paper, we demonstrate that a similar approach can be used to incorporate the phase retrieval step within the tomographic reconstruction process. This provides a single-step process for the quantitative phase retrieved reconstruction of a three-dimensional complex refractive index distribution.
2. Reconstruction algorithm
In the Transport of Intensity approximation  for an homogeneous object (one that consists of a single material) that is weakly absorbing it is possible to express the monochromatic intensity, I, at coordinate (x,y) for a given projection, θ, measured in a plane at distance, z, from the sample, assuming that the phase of the unpropagated field is sufficiently slowly varying in x and y, as [3, 7]:
To extend Eq. (1) to take account of a polychromatic source we note that the measured intensity is the integral over the source spectrum of the monochromatic intensity as modified by the detector response and any wavelength dependent effects in the optical path, such as air scatter or absorption, which can be represented by the combined function, Dλ. The Dλ function for the experimental set-up used in this paper is shown in Fig. 1 . as dotted lines. Accordingly,Eq. (2) in terms of the effective polychromatic materials coefficients,Eqs. (4) and (5) for the intensity) and δpoly is defined similarly as the polychromatic counter part of δλ. This expression confines the wavelength-dependent terms to expressions that can be calculated knowing the composition of the sample.
Following earlier work  we have previously shown  that an expression in the form of Eq. (6) can be combined with the approach used in the filtered back projection inversion of the Radon transform to perform a phase retrieved 3D reconstruction of the density fraction distribution throughout the object. This results in the following expression:Eq. (5), andEq. (8) becomes identical to the one derived by Bronnikov [5, 14]. As with the earlier monochromatic approach  Eq. (8) has the advantage of stability due to the denominator having no zero values. The benefit of this approach is that for a weakly absorbing homogeneous object in the TIE regime the density distribution throughout a sample can be obtained using a single reconstruction step by obtaining a single measurement of intensity at each projection.
3. Experimental results
To demonstrate our method polystyrene (C9H12) micro-spheres of 74µm diameter with a nominal density of 1.05g/cm3 were imaged using a laboratory-based micro X-ray Computed Tomography system (Xradia Inc.). The X-ray source was a closed X-ray tube containing Tungsten as a target material and operated with a tube voltage of 40kV. In this energy, the x-ray source has the size of about 6µm. The spectrum of the source was measured using an energy sensitive detector (XR-100T-CdTe, AMPTEK Inc.). Figure 1 shows the μ and δ values for polystyrene in the same energy range of the measured source spectrum. The absorption and phase coefficients, μpoly and δpoly, were calculated using the spectral weighting distribution used in Eq. (4) that is also shown in Fig. 1. In this case, μpoly is 140.6 m−1, which is equal to the linear attenuation at 10.9keV (μ10.9keV) and δpoly is 1.69x10−6 (δ11.9keV). The calculated values of μpoly and δpoly correspond to different energies. This is to be expected as μ and δ have a different energy dependency of E−3 and E−2 respectively . This highlights why a very broad-band spectrum cannot be understood as a single effective energy whereas in the narrow band approach  we can use one effective energy.
The sample was placed at a distance of 70 mm from the source and 83.5 mm from the imaging detector. Based on the Fresnel scaling theorem , this is equivalent to an effective propagation distance of 38mm. The imaging detector used was different from the energy sensitive detector mentioned above. The imaging detector comprised of a 250µm CsI(Tl) scintillator coupled to a CCD camera by a 4 × objective lens. The CCD camera (Andor Technology) has 2048 × 2048 pixels with a physical pixel size of 13.5 µm. 2 × binning was used in the data acquisition to reduce the exposure time. The total image magnification from this set-up is 8.8, and an effective pixel size is 3.03 μm.
An image of the polystyrene sample for one projection is shown in Fig. 2(a) . The image contains 512 × 120 pixels. Each image was corrected for the dark current and for the non-uniform illumination in the imaging system, determined by taking a background image of the beam without the sample present. The edge enhancement typical of phase contrast imaging is clearly visible as fringes on the edge of the spheres. Each projection image was recorded in 10s. In order to collect a three-dimensional data set, the sample was scanned by acquiring 451 projections at equal angles through an angular range of 180°. The distribution of the term from Eq. (8) is shown in Fig. 2(b). We used μpoly 140.6 m−1, δpoly 1.69x10−6 and z 38mm to calculated Eq. (8).
Using the acquired projection data sets, we then performed the filtered back projection tomographic reconstruction based on Eq. (7) using the filter shown in Fig. 2(b). The 3D reconstruction result is shown in Fig. 3(a) . Figure 3(b) shows a slice of 512 × 512 pixels from the reconstruction with the corresponding plot in Fig. 3(c) along the dashed horizontal line in Fig. 3(b). The plot shows the quantitative value of the function ρf(x,y,z). The retrieved values of 1 ± 0.05 in the area around the center of the spheres are slightly below from the expected value of ρf = 1. The dimension of the spheres can be measured determined by the width of the features in Fig. 3(c) using the known magnification. It is in agreement with the manufacture's specified diameter of 74 ± 11.8µm.
In Fig. 4(a) , we compare the results obtained by choosing a single effective energy and using the corresponding values for μ and δ. An effective energy of 16.2 keV was calculated by making a weighted sum of the energy spectrum. The density fraction calculated using the effective energy is shown with a dotted line. Another comparison is also made by using the energy of the maximum spectral intensity (8.4 keV). The density fraction calculated using this energy is presented as a dashed line. From the plot it is obvious that both the peak and the effective energy approach do not yield satisfactory results. Figures 4(b) and 4(c) show the capability of the method to handle a non-homogeneous sample using the same set-up. Polyimide film with a density of 1.45g/cm3 is inserted around the polystyrene spheres. Polyimide (Kapton, C22H10N2O4) has the value of μpoly 285.1 m−1 and δpoly 2.19x10−6 in this energy range. The same filter using the polystyrene parameter shown above in Fig. 2(b) was been used for the reconstruction process. The plot in Fig. 4(c) shows that the polystyrene spheres were reconstructed with the correct density value while this is not the case for the polyimide film. The second peak from the left in the plot originated from the kapton film and its density fraction deviates from the expected value of 1. However, the reconstruction slice in Fig. 4(b) still shows a good qualitative result. The breach of the weakly absorbing and slowly varying phase requirement can be seen in Fig. 4(b) as resulting in an artefact in the elongated direction of the kapton film. This is due to the fact that the film is no longer weakly interacting with x-rays in some angles of projection.
Without a priori knowledge of the sample, our method still behaves well in the sense that there are no artefacts produced when a weakly absorbing sample is used. However the value will not be quantitative.
We have demonstrated a direct reconstruction algorithm for quantitative phase tomographic imaging using an extremely broad source spectrum for homogeneous and weakly absorbing samples. We need only a single distance projection data set collected from a laboratory polychromatic X-ray source. This method may find widespread application for quantitative phase imaging since it makes efficient use of X-ray laboratory sources that are more widely accessible than monochromatic synchrotron X-ray sources.
The authors acknowledge the support of the Australian Research Council through the Centre of Excellence for Coherent X-ray Science.
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