We describe an x-ray differential phase-contrast imaging method based on two-dimensional transmission gratings that are directly resolved by an x-ray camera. X-ray refraction and diffraction in the sample lead to variations of the positions and amplitudes of the grating fringes on the camera. These effects can be quantified through spatial harmonic analysis. The use of 2D gratings allows differential phase contrast in several directions to be obtained from a single image. When compared to previous grating-based interferometry methods, this approach obviates the need for multiple exposures and separate measurements for different directions and thereby accelerates imaging speed.
Recently, advances in x-ray phase-sensitive imaging techniques have demonstrated dramatic enhancement of contrast between materials of different indices of refraction [1, 2, 3], while x-ray diffraction or dark-field techniques reveal information about microscopic structures unseen in traditional absorption images [4, 5]. Phase-grating-based Talbot interferometry methods [6, 7] have demonstrated quantitative differential phase contrast (DPC) [8, 9, 10] and diffraction  images at high spatial resolution and sensitivity using conventional x-ray tubes and detectors. However, its speed is limited by the need to scan an anal yzer grating. Techniques based on precisely aligned coded apertures have also shown phase-contrast effects in the transmission image , but more than one exposure is needed to separate phase-contrast information from absorption.
For a variety of applications that require speed, we developed a variant grating-based method to obtain absorption, differential phase-contrast, and diffraction from a single raw image, called the spatial harmonic method [13, 14]. Here, a single transmission grating of lower density is placed in the beam of an x-ray tube  (Fig. 1). The imaging device consisted of a 50 W tungsten target x-ray tube of focal spot size operating at exposure, an x-ray camera consisting of a 16-bit water-cooled CCD array of matrix size of and pixel size of , coupled to a phosphor screen via a fiber optic taper, and a radiography Bucky grid that acted as the transmission grating. The source-camera distance was , and the grid was placed exactly midway between the two. Imaged objects were placed at from the source and magnified eight times by the cone beam geometry. The grid was an off-the-shelf square Bucky grid of density ( period). Structurally, it was a stack of two orthogonal linear grids made of alternating lead and aluminum stripes. In this setting, the fractional Talbot distance was approximately for a parallel beam, and it extends beyond infinity in the cone beam configuration. Thus, the x rays simply project the grid pattern achromatically onto the camera. The camera resolution was high enough to directly resolve the grid lines. The speed of this approach has permitted in vivo imaging of live animals (manuscript in preparation).
In general, a linear grating can only detect refractive bending of x rays in the direction perpendicular to the grating fringes. Kottler et al. showed that measurements in two orthogonal directions are often necessary to retrieve good quality and artifact-free images . Clauser’s patent  proposed the use of 2D Talbot gratings for this purpose but required stringent period and phase matching of the detector elements to the interference patterns to make such measurements. Our spatial harmonic method offers the possibility to do so in a single image without special requirements on the camera.
The key question in a single-shot method is how to quantify small changes in the locations and amplitudes of the grating fringes in a pixelated image, when a grating period covers just a few and likely a noninteger number of pixels. Fortunately, this problem has been solved in the framework of the discrete Fourier transformation , which converts an image into its spatial frequency spectrum (SFS) (Fig. 2). When a 2D square grating overlays the sample, the SFS of the raw image contains a number of sharp peaks at (, ), where M and N are integers and P is the period of the grating projection image. We call these the harmonic peaks and label them (M, N). The spatial frequency content of the sample projection image is duplicated at each peak, which provides a harmonic spectrum.
Now we make the basic assumption that the harmonic spectra do not overlap (detailed below). Then, the inverse Fourier transformation of a subregion centered at a peak yields a corresponding harmonic image (Fig. 2):9], similar to Shack–Hartmann wavefront sensing of optical as well as x-ray light [18, 19]. Specifically, the shifts of the vertical and horizontal fringes lead to proportional phase shifts of the and harmonic images, respectively. Therefore, DPC is provided by their phase maps . It is important to note from Eq. (1) that the resolution of the diffraction and DPC images is the projected period of the grid.
In practice, the SFS is modified by the point spread function (PSF) of the phosphor screen in the x-ray camera. The PSF is measured with a narrow transmission slit, and its effect can be corrected by the step (SFS of raw transform of PSF) for spatial frequencies where the Fourier transformation of the PSF does not diminish. Additionally, imperfections in the grating give rise to extraneous values in both the amplitude and phase of the harmonic images. These are removed by the calibration step (calibrated harmonic -only harmonic image).
The basic assumption that the harmonic spectra are well separated is met when the cone beam magnification factor G of the sample is sufficiently large such that . This is because the bandwidth of each harmonic spectrum is given by spot size)], and the separation of the harmonic peaks is .
Figure 3 shows DPC and diffraction images of glass beads. The two DPC images from the , harmonics highlight different segments of the boundaries of the beads. This has also been observed in previous studies using linear gratings in different orientations .
Figure 4 shows the complete set of absorption, DPC, and diffraction images of the paw of a dehydrated rat hind limb specimen, which are obtained from a single raw image. The and DPC images emphasize horizontal and vertical interfaces, respectively, while interfaces at from horizontal have approximately equal signal in both images.
In these experiments, the surface radiation dose at the sample was , comparable to a human chest lateral radiograph . Clinical x-ray tubes have approximately 1000 times the power of our compact tube and, therefore, can acquire the image in less than . The main limitation of this method is that (camera pixel size) , such that the camera can resolve the grating pattern. Therefore, the camera resolution limits the density of the grating, and, in turn, the achievable level of phase contrast . One solution to this limitation is to convert high-density grating fringes into a lower density Moiré pattern by the use of a second transmission grating placed on the camera surface . The ultimate solution may be high-resolution x-ray cameras. CCD arrays coupled with high-resolution x-ray scintillators have demonstrated submicrometer resolution . This technology may eventually provide large-format detectors of several-micrometer resolution that can be used for many applications.
In summary, the spatial harmonic method is able to acquire DPC and diffraction images in a single exposure without the need for scanning and allows measurements in multiple directions simultaneously with the use of 2D transmission gratings. Taken together, these characteristics are well suited for fast imaging applications.
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