FIELD

The present application relates to X-rays, and, more particularly, to obtaining X-ray images using scattering.

BACKGROUND

X-rays can penetrate high density materials, making it the radiation of choice for non-invasive evaluation of bone. With the increasing demand for the diagnosis and treatment of orthopedic diseases, X-ray techniques have advanced continuously from planar radiography to dual-energy radiography, CT densitometry and other approaches. These techniques use attenuation contrast where X-rays passing through the body are variably attenuated according to local material density and elemental composition. Because X-ray is a form of high-energy light, many contrast mechanisms that are standard in visible-light microscopy have also inspired parallel development in X-ray, such as dark-field, phase-contrast, fluorescence and confocal.

The dark-field, or scattering-based contrast, arises from microscopic density fluctuations of X-rays that are deflected from a straight line (i.e., that scatter). While traditional X-rays produce a simple absorption contrast image, the dark-field X-ray image technology captures the scattering of the radiation within a material to give a clear, defined image that can show subtle inner changes in bone, soft tissue, or alloys. Small angle X-ray scattering is fundamentally different from absorption and refraction in that it is caused by spatial variation of electron density on the atomic to sub-micrometer scale, which leads to dispersion of the transmitted X-ray beam.

Imaging the distribution of X-ray scattering remains a difficult task due to requirements for specialized X-ray optical components and/or brilliant sources. The majority of available techniques use raster or line scans of narrowly collimated beams, which are produced with single-crystal filters or pin-hole and slit apertures.

Recently, non-raster simultaneous scattering imaging using polychromatic X-ray tubes has been proposed which takes advantage of scattering-induced blurring of sharp features as a result of the disruption of X-ray spatial coherence. One such technique employs a high-density X-ray phase grating to phase-modulate a monochromatic component of the transmitted beam, which results in intensity oscillation at proper distances from the phase grating due to interference effects. This oscillation was resolved by placing a second high-density intensity grating over the detector surface, and taking a series of X-ray exposures for different positions of the second grating. Unfortunately, the X-ray gratings require unique fabrication and are expensive. Additionally, precise distances need to be established between the two gratings for proper imaging. Finally, multiple exposures need to occur, making the overall process time consuming, expensive.

Additionally, a recent publication (US2007/0183563 Al) mentioned that by using a detector with elements less than 1/3 of the grid pattern pitch, it is possible to determine the phase shift in one measurement. US2007/0183580 Al further elaborates on this technique and specifies that the detector elements are an integer fraction of the grid pattern pitch so that sub-groups of the detectors can report x-ray intensities of different portions of a grid period, from which the phase shift of the grid pattern is measured. Such detectors are highly challenging to realize, and are not able to cope with varying grid pitches or patterns.

It is additionally known in the art to remove the effects of scattering with the use of grids, gratings, or other masks of periodically arranged opaque areas.

Specifically, a mask or multiple masks of periodically arranged opaque areas are placed in the x-ray path, such that periodic dark shadows are created on a recorder surface either by direct geometric shadowing or by wave-interference effects. The shadow areas only receive x-ray which is scattered in the object. The signals of these shadow areas are subtracted from the raw image to yield an image free of the effects of scattering.

Nonetheless, the above variations require exacting procedures or are expensive making the prior art ill-suited for today's routine x-ray imaging applications, including non-destructive testing (e.g., component inspection without damage), security screening, and medical diagnostic exams. SUMMARY

The present invention overcomes the drawbacks of the prior art by allowing scattering images to be obtained using a single exposure and using commercially available gratings.

In one embodiment, a scattering imaging method uses an intensity grating to modulate the intensity of a beam of an X-ray radiation source. A detector captures a raw image from the modulated intensity pattern. A scattering image can be automatically generated from the detected modulated intensity pattern.

In another embodiment, both a scattering image and a phase-contrast image are obtained in a single exposure. Additionally, exact positioning of the grating is unnecessary, as the method works for any non-zero distance between the grating and the detector so long as the grating is far enough from the x-ray source to cause sufficient modulation of the x-ray intensity. Thus, the speed and ease of implementation makes it suitable for non-destructive testing, security screening, and medical diagnostic exams.

The foregoing and other objects, features, and advantages of the invention will become more apparent from the following detailed description, which proceeds with reference to the accompanying figures. BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a perspective view of an apparatus, according to one embodiment, for obtaining scattering images.

FIG. 2 is an embodiment of a method for generating a scattering image.

FIG. 3 shows further details of a method that can be used in conjunction with the method of FIG. 2 for generating a scattering image.

FIG. 4 is an embodiment of a method for generating a phase-contrast image.

FIG. 5 is an example showing absorption and scattering images of a tree branch.

FIG. 6 shows a view of a Fourier spectrum using both vertical and horizontal gratings. FIG. 7 shows images obtained using the method according to one embodiment.

FIG. 8 shows an attenuation image and scattering images obtained using vertical and horizontal gratings.

FIG. 9 is an embodiment of an intensity grating using X-ray opaque and

X-ray transparent lines.

FIG. 10 is an embodiment of an intensity grating having hexagonal X-ray transparent elements.

FIG. 11 is an illustration of reduction of artifacts through relative movement of a subject during imaging.

FIG. 12 is a flowchart of a method for reducing artifacts through movement.

FIG. 13 is another embodiment of an apparatus for removing scattering due to water.

FIG. 14 is a flowchart of a method for reduction of water scattering.

FIG. 15 is an illustration showing the reduction of water scattering.

DETAILED DESCRIPTION

A method is disclosed to form an X-ray scattering image from a single fundamental exposure. An X-ray phase-contrast image can also be derived from the fundamental image.

General Aspects

Figure 1 shows an apparatus 10 having an X-ray source 12 and an X-ray detector 14. A grating 16 having a periodic alternating pattern of X-ray transparent and X-ray opaque regions is interposed between the X-ray source 12 and the X-ray detector 14, either before or after an imaged object 18, such that the image at the detector surface contains a periodically modulated intensity pattern. The grating is a low-density intensity grating used to create a pattern on the detector surface. A distance is maintained between the detector surface and both the grating and the object, such that X-ray scattering in the object blurs the grating pattern and X-ray refraction in the object deforms the pattern. The distance need not be precise and is not fixed. Preferably, the X-ray image is recorded at a resolution smaller than one third of the period of the grating pattern. Using a computer or processor 20, the detected image is Fourier transformed and separate sub-images of complex values are formed by inverse Fourier transformation of separate regions surrounding the spectral peaks in the Fourier spectrum. These sub-images can be normalized to reference sub-images (which are acquired without any samples) to yield harmonic images.

The intensity ratio between a K^-order harmonic image and an N^-order harmonic image provides an image of X-ray scattering distribution. The phase of a K^-order harmonic image represents gradients of the index of refraction in the direction perpendicular to the grating lines.

The grating can have different patterns of opaque and transparent regions, such as, linear, rectangular or hexagonal patterns. Other patterns can also be used whose Fourier transformation contains multiple peaks that are arranged at different angles around the origin.

In another embodiment, a small relative movement of the sample and the imaging system in the direction that is perpendicular to the grating lines during the X-ray exposure can be used to remove artifacts at sharp interfaces in the sample. This can be further refined by modulating the X-ray intensity level during the movement. One technique for causing such a relative movement is to use a stage 22 that moves the object during the exposure. Such a stage can be used in any of the embodiments described herein.

FIG. 2 is a flowchart of a method for obtaining a scattering image. In process block 200, an intensity grating is positioned between an X-ray source and X-ray detector. The X-ray source emits a beam of X-ray radiation towards a subject. In process block 202, an X-ray detector captures (detects) a single image of the subject by detecting an X-ray beam that has passed through the intensity grating positioned before or after the subject. The intensity grating modulates an amplitude of the X-ray beam with substantially no phase modulation. The result is that a modulated intensity pattern is generated and detected by the X-ray detector. In reality some phase modulation occurs due to passage of the X-ray radiation through opaque regions on the intensity grating. However, the amount of phase modulation is kept to a minimum. In process block 204, a scattering image is automatically generated using processing techniques carried out by the computer 20. The scattering image can be derived from the detected image using only a single exposure and using an intensity grating, rather than a phase grating. By using an intensity grating, commercially available gratings are readily available.

Additionally, the distances between the grating and the detector need not be precise in order to obtain the scattering image.

FIG. 3 shows an embodiment of a method for generating the scattering image. In process block 300, a raw image in the spatial domain is captured using an X-ray detector, such as a camera. The raw image is converted into the spatial frequency domain image using well-known techniques, such as Fourier transforms, Laplace transforms, etc. In the spatial frequency domain image, various peaks are visible, which corresponds to integer multiples of the basic spatial frequencies of the shadows of the grid projected on the detector surface. In process block 302, an N*- order peak is selected from the image in the spatial frequency domain. In one example, the N*-order peak is a zero-order peak, but any desired harmonic can be selected. A spatial frequency domain filter, such as a Hanning or Fermi filter, can be applied to an area around the nth-order peak. An inverse transform is performed to obtain an N^-order harmonic image in the space domain using the area around the Nth-order peak. In process block 304, calibration is performed on the N^-order harmonic image by dividing that image with an N^-order harmonic image of a reference image that was taken without a subject. Such a reference image only needs to be taken once and stored on the computer 20 for later use. The calibration can include taking a ratio of corresponding portions of the N^-order harmonic image and the reference image. In process block 306, a K^-order harmonic is selected from the image in the spatial frequency domain and an area around it is multiplied with a filter and is inverse transformed in order to obtain a K^-order harmonic image in the space domain. In process block 308, calibration is performed on the K^-order harmonic image in the space domain using the same calibration technique described above. In process block 310, a scattering image is automatically generated by calculating a ratio between the N^-order harmonic image and the K^-order harmonic image.

FIG. 4 is a flowchart of a method for obtaining a phase-contrast image. The reference image is provided in process block 400. The reference image is taken without a subject, as already described. In process block 402, a K^-order harmonic of the reference image is obtained. Such a K^-order harmonic is obtained through a transformation of the reference image into the spatial frequency domain, selecting the K^-order peak, and filtering and inverse transforming an area around the Reorder peak to obtain a reference image of the K^-order harmonic in the space domain. In process block 404, the phase of the K^-order harmonic of the reference image is subtracted from the phase of the K^-order harmonic of the image being processed in order to obtain the phase-contrast image.

FIG. 5 shows the advantage of obtaining the scattering image. The top photo shows an absorption image of a cedar branch, while the bottom photo shows the scattering image. The scattering image shows structures that are not visible in the absorption image. For example, the central core of the cedar branch running the length of the stem is clearly visible in the scattering image, but absent from the absorption image. This bright band running down the center of the scattering image corresponds to the pith of the stem. Additionally, the combination of absorption and scattering data can help distinguish different materials that appear similar in absorption images.

FIG. 6 illustrates the effects of using a vertical grating versus a horizontal grating. The result is that the alignment of the peaks in the spatial frequency domain is rotated ninety degrees. X-ray scattering is the broadest in a plane perpendicular to fibers in the subject and results in different scattering image intensities between vertical and horizontal grating placement.

Particular sample apparatus and method

In a particular example apparatus, the X-ray radiation source was a fixed- anode tungsten-target tube operating at 50 kVp/0.6mA (SB-80-lk, Source-Ray Inc., Bohemia, NY, USA). The tube has a beryllium window, and the grating mask effectively adds 1.7 mm of aluminum filtration. No other filters were used. The half-value layer (HVL) of the X-ray cone -beam is 1.3 mm aluminum (ionization chamber dosimeter, PTW, Freiburg, Germany). An example exposure lasted 10 seconds and delivered 0.19 mGy incident radiation at the sample, equivalent to a thoracic X-ray exam.

The X-ray camera used was a 16 bit CCD camera of matrix size of

2048x2048, pixel size 30 μm, and a Gd ₂ O ₂ SiTb phosphor screen for X-ray-to-light conversion (PI-SCX-4096, Princeton Instruments, Trenton, NJ, USA). The screen conversion efficiency is 16%, and the camera CCD array had a quantum efficiency of 33% for the green emission of the screen. Its resolution of 30 μm can resolve the projected grating period of 254 μm on the camera surface. In order to reduce the dark-current noise accumulated over the 10 second exposure time, the camera was cooled with 5 ⁰ C chilled water. Although a large-format X-ray camera is used, digital X-ray cameras and flat panel detectors are becoming widely used commercially and clinically, and can readily perform this type of imaging.

A grating was used that had 200 lines-per-inch (lpi) parallel radiography anti-scatter grating of 10:1 grating ratio (MXE Inc., Los Angeles, CA, USA) and it was interposed between the source and the camera, with the sample placed immediately down-beam from it. Both the scattering length scale and the exposure time were factors when determining the geometry of the device. The scattering length scale, d, is the maximal length scale of the electron density fluctuation in the sample that still gives rise to appreciable scattering signal. For example, particles of radius d' scatters X-rays into a cone. For a given device geometry, the scattered cone from a single X-ray appears as a dispersed pattern on the X-ray camera. When this pattern is superimposed on the grating shadows, those X-ray photons that fall at radial distances greater than the period of the grating shadows will blur the shadows, which lead to the scattering signal. For a given device layout, the size of the dispersion pattern on the camera is inversely related to the size of the particle and the X-ray wavelength. The size of the structure whose scattering can be detected by this means is limited to an upper threshold which is dependent on the X-ray wavelength λ, the grating period Po, the distances between the source and the grating D ₁ , between the grating and the sample D ₂ , between the sample and the camera D ₃ (FIG. 1), and the order of the harmonic peak n, as

O K d'≤λn Dj D ₃ Z[P(K D ₁ + D ₂ + D ₃ )],

where the upper threshold is defined as the scattering length scale of the nth-order harmonic image of this specific device setup: d _n - ^ ^D ^ . (1)

P ₀ D ₁ + D ₂ + D ₃

Generally, the larger this length scale, the higher the scattering signal. Since the sample-to-grating distance is small, D ₂ ~ 0. Then, for a given total length of the layout, the scattering length scale is maximized when the grating and sample are placed midway between the X-ray tube and the camera. Additionally, the scattering length scale increases with the camera-to-source distance, but at the cost of exposure time. With a camera-to-source spacing of 1 meter, it is desirable to have 10 seconds of exposure. The distance between the grating and the camera was then 0.5 m.

A further consideration regarding the scattering length scale is that the spectrum of a tungsten-target tube operating at 50 kVp peaks at approximately 27 keV and has a full-width-half-height of approximately 20 keV. Therefore, the scattering length scale is not a single value, but a distribution of values corresponding to the distribution of X-ray energies. Based on the peak of the energy spectrum, the first-order scattering length scale observed was approximately 100 nanometers.

If x-ray tubes in medical diagnostic radiography systems are used, then to reach the same signal level as the present study, the exposure time should be two orders of magnitude shorter than the current 10 seconds. This is in line with current chest X-ray exams.

The resolution of the camera or flat panel detector should be sufficient to sample the first-order modulation of the grating. By the Nyquist sampling theorem, the detector pixel should not be larger than one-third the period of the grating shadow. Using current clinical flat panel detectors (e.g., Siemens Pixium 4600 of 0.143 mm pixels and 43 x 43 cm field of view), the period of the grating shadow should be greater than 0.143mm x 3 = 0.429 mm. Then the final scattering image resolution for a subject placed midway between the x-ray source and the detector is

0.429/2 = 0.215 mm. A tradeoff with lower density gratings and higher tube voltages is that the scattering length scales will be reduced relative the present experiment, i.e. the range of sizes of the microscopic structures whose scattering effect can be seen is reduced. As a result, the scattering signal per unit thickness of sample will be lower, but this is offset by the fact that human bones are larger than the rat and pig bone samples in the present study, and therefore provide more scattering overall.

FIG. 7 shows actual images taken in conjunction with the method for obtaining the scattering and phase-contrast images, from a single exposure. The raw image from the camera is shown in FIG. 7a. The raw image is Fourier transformed to reveal several distinct peaks (FIG. 7b), including the zero^-order peak at the center containing radiation un-modulated by the grating mask, and the first and higher-order peaks from the grating shadows. Similar to the use of physical apertures in dark-field visible-light or electron microscopy, these peaks are individually selected with mask filters in the Fourier space. The areas around the selected peaks are then inverse Fourier transformed to yield the zero*, first and higher-order images separately (FIGs. 7c and 7d). As a result of the filtering the resolution of these harmonic images, and ultimately of the attenuation and scattering images, was reduced from that of the camera to the period of the grating. With the 200 lpi grating the image resolution was approximately 127 μm. The scattering image (FIG. 7f) was obtained by dividing the magnitude of the first harmonic image by the magnitude of the attenuation image. The phase contrast image (FIG. 7e) is obtained by using the phase of the first-order harmonic image and subtracting of phase of the corresponding no-sample reference image.

In order to remove native imperfections of the grating, the harmonic images can be normalized (calibrated) with reference images acquired without samples (not shown here).

FIG. 8 illustrates X-ray attenuation and scattering images of the hind limb of a rat. Differences in scattering intensity are visible between perpendicular and parallel alignment of the grating and the bone axes. In the dense superficial cortical bone of the tibia, the scattering signal is brighter when the bone axis is parallel to the grating. FIG.8a shows regular attenuation. FIG. 8b shows a scattering image of the same limb with vertical placement of the grating. FIG. 8c shows a scattering image with horizontal placement of the grating. FIG. 8d shows a graph of scattering versus attenuation. Pixels are divided into four bins according to their attenuation values. Mean values of scattering of the bins, with corresponding standard deviations, are plotted. For all bins, scattering is greater when the grid is parallel to the tibia than when it is perpendicular.

Gratings

Different types of intensity gratings can be used. Preferably, the gratings do not affect phase, although some phase modulation is acceptable.

FIG. 9 shows a horizontal grating 900. The grating has a periodic alternating pattern of X-ray opaque lines 902 and X-ray transparent lines 904. Example materials include lead for the X-ray opaque lines and aluminum or plastics for the X-ray transparent lines. Other materials can be used that are well-known in the art. The grating is structured so that the period P of grating shadows on the surface of the X-ray detector is equal or less than P/3 of the detector pixel spacing.

FIG. 10 is an illustration of a hexagonal intensity grating 1000 that generates a Fourier spectrum shown at 1002. The hexagonal grating can be made of similar material that was described above. Other shapes can be used, such as square, rectangular, circular, etc.

It is desirable that the gratings modify amplitude and only minimally impact phase of an X-ray beam passing through the grating. Gratings of square or hexagonal cells enable imaging of multiple scattering directions in a single exposure, while gratings of narrow passing slits more evenly distribute the transmitted X-ray energy among the Fourier harmonic peaks and allow scattering imaging at two or more length scales simultaneously.

Removing artifacts near edges with motion blurring

A further consideration is that very sharp edges in the sample can interfere with the fine grating shadows, or equivalently in the Fourier spectrum, adjacent peaks may overlap. This can be prevented by slightly moving the sample in the direction perpendicular to the grating lines over a distance equal to the grating period during the X-ray exposure. The result is that sharp edges in the sample itself can be blurred to widths that are similar to the grating period, while the grating shadows are not affected. The motion-blurring effectively limits the extent of each peak in the Fourier spectrum to less than half the distance between the peaks, and thus avoiding any chance of interference between the peaks.

Using FIG. 1 as an example, the stage 22 can be moved at a constant speed, during the exposure time T, over a distance Δ in the y direction (for a vertical grating) or the x direction (for a horizontal grating), or in a circular motion (for a hexagonal grating). Additionally, the X-ray tube current during the exposure time can vary and is denoted as I _s (t). The resulting sample image F(x _d ,y _d ) on the detector surface is related to the image without motion F ₀ (XdJd) by the relationship

F( ^χ _d , y _d ) = -\_ _Al2 F ₀ ( ^χ _d > y _d +— ^~ B^ ₂ — VΛjOdu, (2) where

A rT/2

C = - f I _s (t)dt.

J J-T /2 ^s

By two-dimensional Fourier transformation of eq.(2) we have

f(k _x , k _y ) = f _o (K ,k _y )M(k _y ), (3) where/ and /o are the Fourier transforms of the original and blurred sample images, and M(k _y ) is the band-limiting function:

M(k _y ) = ^{Dl + D} ' r ² I, (±-T)e*»du _d , (4) y (D ₁ + D ₂ + Z) ₃ )C -U' ² ' A _d ^d

where

D _{1 +} D _{2 +} D _{3 A}

D ₁ + D ₂

is the sample motion projected onto the detector. If the tube current I _s remains constant over the exposure time, then M(k _y ) is a sine function:

M (^ ) = sin c(^— -). (5) When this function is substituted into eq.(3), can suppress the part of the sample image spectrum outside the extent of ±π/Δ _d . If Δ _d is set to the period of the grating shadow P, this step reduces the overlap of adjacent harmonic peaks in the Fourier spectrum when the grating is in place. It should be noted that only the sample is moved during this process, and the grating shadows are not affected.

It is also clear from the above discussion that the band-limiting function M(k _y ) can be improved by modulating the tube current I _s over the time of exposure. A Gaussian modulation of the tube current is more desired than a constant tube current.

FIG. 11 shows two images. A first static image has artifacts around the edges, while the sample that was moved has reduced artifacts.

FIG. 12 shows an embodiment of a method for reducing artifacts. In process block 1200, an X-ray beam is emitted for a period of time (the time can vary depending on the application), such as between 1 and 15 seconds. In process block 1202, the subject of the X-ray is moved relative to the apparatus. For example, the stage (FIG. 1) can be used to move the subject during the period of time. The movement can be linear, circular, or other motions. Alternatively, the apparatus can be moved while the subject remains static. Another optional feature is to have the X-ray beam vary in intensity (process block 1204) over the period while the relative movement is occurring. In process block 1206 the artifacts are removed using a ratio of harmonic images, such as outlined in FIG. 3.

Reducing Scattering of Water

FIG. 13 shows how the apparatus can be modified in order to reduce scattering caused by water. A first grating 1302 is placed before or after the subject 1304. A second grating 1306 is placed near the X-ray detector (e.g., camera). The first g rating has a different orientation that the second grating so as to obtain adequate separation in the Fourier spectrum. The resultant Fourier spectrum shows a peak 1308 associated with grating 1302 and a peak 1310 associated with grating 1306. These different peaks can be used to cancel the scattering effects of water. As human tissue is predominately water, reducing water scattering provides a clearer image of the more interesting aspects of the tissue being analyzed.

FIG. 14 shows a flowchart of a method for reducing scattering caused by water. In process block 1400, a first intensity grating is placed before or after the subject. In process block 1402, a second intensity grating is placed adjacent to the X-ray detector. A raw image is then taken and a Fourier transform is performed to place the raw image in the Fourier spectrum. In process block 1404, a N^-order peak and K^-order peak are selected such that one peak is associated with one grating and the other peak with the other grating. The areas around the peak are determined and the same filter can be applied to both. In process block 1406, the inverse transforms are performed and a ratio is taken (process block 1408) in a manner similar to as previously described in order to obtain a scatter image with the scattering of water reduced or removed.

FIG. 15 shows the results of the water-suppressed scatter image.

Theory

The zero^-order image provides the distribution of the attenuation of transmitted X-rays through the sample:

D ₀ (x, y) = - In[Z ₀ (x, y) I I _Og (x, y)] , (6) where / ₀ is the magnitude of the zero^-order image, / _Og is the magnitude of a zero*- order image without any samples, (x, y) are the coordinates in the image, and the natural log yields values that are linearly related to the sample thickness. This is a regular radiography attenuation image. The higher-order images are frequency- modulated versions of the zero^-order image and are therefore more severely attenuated by the blurring effect of scattering in the sample. The magnitude ratio of a high-order image to the zero^-order image measures the degree of blurring of the grating shadows. The expression for an image that is solely dependent on X-ray scattering in the sample is as follows:

/ _n (x, j) / / _ng (x, j)

D _n (x, y) = -\n[— —*- -], (7)

I _o (x, y) / I _Og (x, y)

where I _n is the magnitude of the nth-order harmonic image, / _ng is the magnitude of a nth-order image without any samples. The normalization (calibration) relative to the no-sample reference images remove features of the grating itself, and the natural log makes the values linearly related to the thickness of the sample. Scattering images of different orders reflect structures of different length scales. In practice, the first- order peak usually has the highest intensity and provides the best signal-to-noise ratio.

Besides magnitude, the phase of the high-order images is influenced by the slight bending of X-rays from refraction. X-ray refraction in the direction perpendicular to the grating shifts the grating shadows, resulting in phase shifts in the high-order images. Phase-contrast images can be obtained concurrent with the scattering images as

P _n (x, y) = φ _n (x, y) - φ _n0 (x, y), (8) where φ _n is the phase map of the nth order image, and φ _n0 is a reference phase map without any samples.

The above processing steps were automated with a program written in IDL image processing language (ITT, White Plains, New York), but other programming languages can be used.

In the cortical bone of the rat hind limb, the change of the scattering image intensity with the grating orientation can be explained by its structure. The current model of compact bone is a lamellar formation of bundles of mineralized collagen fibrils with crystalline material between them. The bundles are aligned in each layer, and the size of a fibril is about 80 nm, on the same order as the length scale observed by the scattering image. When X-rays pass through the ordered fibrils of an individual layer, the angular distribution of scattering is the broadest in the equatorial plane. The accumulated effect through multiple layers is broader dispersion in the direction perpendicular to the layers, or the periosteal surface. Therefore, when the stripes of the grating mask are parallel to the bone surface (bone axis), their shadows are maximally blurred by X-ray dispersion, resulting in the highest scattering image intensity.

The notion that the anisotropy of the scattering image reflects the ordered structure of compact bone is in agreement with previous small-angle X-ray scattering measurements of bone sections. These measurements show that the azimuthal scattering distribution of a single fibril bundle has an ellipsoidal shape with the short axis in the direction of the bundle, and the scattering distribution of a plate-shaped mineral particle is an ellipsoid with the long axis perpendicular to the plate.

To explain the granular pattern of scattering and PC images in trabecular bone, we note that its structure is a porous matrix of mineralize material and soft tissue. Thin section X-ray scattering measurements show that the mineralized scaffold is similar to cortical bone in its scattering distribution, but the collagen bundles are locally aligned with the wall of the trabeculum and, thus distribute randomly over distances greater than the pore size, which is approximately 100 μm(27). For this reason the scattering intensity for a given grating orientation should appear heterogeneous at the scale of 100 μm and beyond. The image pixel size of 127 μm in the present study can resolve this heterogeneity, which leads to the granular appearance. The values of the phase-contrast images are determined by spatial gradients of material density at the scale of the image resolution. In the porous matrix, the calcified compartment is denser than the soft-tissue compartment, leading to local density gradients in random directions at the scale of the pore size of 100 μm.

For better resolution with clinical X-ray units, a solution is to decrease the distance between the grating and the detector. For example, a grating period and final resolution of 0.2 mm is feasible if the grating is placed at 25 cm from the detector and 75 cm from the X-ray tube. The tradeoff is that the patient may not fit into the 25 cm space and need to be positioned in-front of the grating. Without the screening of the grating the radiation exposure of the patient may increase by approximately a factor of two. Given all of the above considerations, the optimal device parameters and layout for human imaging need to be determined experimentally.

Lastly, the thickness of the sample affects Fourier scattering imaging and phase-contrast imaging in different ways. With increasing sample thickness the transmitted X-rays become more dispersed, and the scattering signal increases proportionally with the sample thickness in the same fashion as regular X-ray attenuation. In contrast, the phase signal originates from the refractive bending of the X-rays at tissue interfaces, and depends on the number of interfaces the X-rays pass through and the direction of the density gradients at these interfaces. Generally the phase signals from different segments along an X-ray path do not add constructively, and it becomes weaker when the X-ray becomes more dispersed. For these reasons, scattering imaging is more suited for thick samples.

In summary, the technique of Fourier X-ray scattering radiography is able to acquire attenuation and scattering images in bone in a single exposure, which indicates material density and fine structure respectively.

Further theoretical description of Fourier scattering imaging

Referring to FIG. 1 for the layout of the imaging device, the coordinates (x, y) in the sample are within planes perpendicular to the central axis of the X-ray beam, and the coordinate z is along the beam axis. Positions in the sample are labeled r ₀ , and positions on the detector plane r^ The period of the grating shadows on the detector surface is P, and the X-ray wave length is λ.

In the absence of a sample, the projection image of the grating onto the detector surface is a periodic pattern of dark and bright bands. Its two-dimensional Fourier transformation, or spectrum, contains a number of discrete peaks corresponding to the harmonics of the basic grating frequency. In the following description the projection image of the grating is denoted G, its spectrum as g. The projection image of the sample alone without the grating is denoted F, its spectrum as /. The image of the sample with the grating present is denoted as F _g , its spectrum f _g . Clearly F _g is the raw data obtained in the experiment.

When the grating stripes are aligned in the X direction, the grating spectrum consists of discrete harmonic peaks at multiple intervals of the grating period: g(k) =∑g _n δ(k _y -—)S(k _x ) (9) n P

where (Ic _x , k _y ) are the coordinates in the Fourier space, and g _n is the amplitude of the nth-order harmonic peak. Practically, only the low order peaks have appreciable amplitudes.

Considering the image of the sample with the grating in place but without any scattering, it is simply the product of the sample and grating images:

F _g (x, y) = F(x, y)G(x, y). The above relationship means that in the Fourier domain, the spectrum of the grating-modulated sample image /^ is the convolution of the spectrum of the sample / with the spectrum of the grating g. Because the grating spectrum consists of discrete harmonic peaks, f _g is effectively the superposition of a number of copies of 5 the sample spectrum positioned at the harmonic peaks. Mathematically the 2D Fourier transformation of eq.(6) yields f _s (k) = f(k) * g(k) =∑gj(k _x ,k _y -^).

^F (11)

With slight motion blurring the sample spectrum becomes band-limited such that it does not extend to spatial frequencies beyond the density of the grating:

0 f(k) = 0 if \k _y \ > π/P. (12)

Then the harmonic peaks of f _g do not overlap:

. , _{1 S} ,„ , 2m. , . _τ 2m π 2m π _Λ

f _g (k) = gJ(K,k _y -—) for k _y e [_ -- _ + -]. ^

The nth-order harmonic image is the two dimensional inverse Fourier transformation

, . , _r 2τm π 2m π _Λ

on the interval \ , 1— 1

P P P P =

g _n F(x, y)exp(-i2my/P).

(14)

Therefore,

H _n (x, y) = g _n F(x, y). (15)0 This means that in the absence of scattering, all harmonic images are identical to the band-limited sample image and weighted by the amplitude of the harmonic peaks in the spectrum of the grating. To avoid edge ringing artifacts, we also applied a Hanning filter of the form filter = [COs[P Jk _x ² + (k _y - ^) ² ] + l} / 2 (16) to the inverse Fourier transformation in eq.(14) when calculating the harmonic images.

The above relationships are modified by the scattering of X-rays in the sample. Since large-angle inelastic Compton scattering reduces the flux that reaches the detector, it is observed as attenuation. In the following discussion we consider the effect of small-angle forward scattering by the elastic (coherent) photon-electron collisions.

To further describe the effect of coherent scattering, the sample is viewed as constructed by adding thin layers that are perpendicular to the beam axis Z, incrementing from the grating side to the detector side. The layer thickness dz is sufficiently small so that X-ray photons either go through without any collisions or be scattered once. Prior to the addition of a layer, the image intensity at position r _d on the detector surface is

F _g (r _d ) = \ l(r, r _d )d ² r (17) where r denotes the (x, y) coordinates on the layer surface, and I(r, r _d ) is the X-ray that exists at r and strikes r _d . The addition of the thin layer dz causes a minute change of the image intensity. Here we assume that coherent scattering is confined to small forward angles, then the addition of layer dz results in both attenuation and scattering: dz ^J HD ₃ - Z) D ₃ - Z

(18) where r and r' are points on the layer surface, r _d an r/ are points on the detector surface, qo is the wave vector of the X-ray, σ(q, r) is the differential scattering cross- section per unit volume for scattering wave vector q and at point r, μ _a is the attenuation coefficient due to absorption and inelastic Compton scattering, and μ _s (r) is the total coherent scattering cross-section at point r, which is the integral of the differential cross-section:

μ _s (r) = \ σ(Ω, r)dΩ. (19) The first integral in eq.(18) is the X-rays that strike r _d on the detector without experiencing any events in the layer, and the second integral is the X-rays that were scattered from various locations within the layer before striking r _d . Again under the assumption that scattering is limited to small angles, the X-rays that strike r _d on the detector surface all exited the layer dz in a small area about a point r ₀ , within which the absorption and scattering coefficients are uniform. Then eq.(18) is simplified to

(20)

In eq.(20) X-ray attenuation is represented by the first term, and the effect of scattering is expressed by the second term. We can now look at the effect of the layer dz on the nth-order harmonic image in eq.(14). Since the sample is band- limited, the harmonic modulation of _e -' ^y2πn/p varies much more rapidly than the demodulated image H _n (x, y). Then, from eq.(20), d ^H " ^{(rd }} - -μ _β (r _o )H _g (r _d ) - J— -^{1 - S^ ^2* " ]σ(

dz (A - ZY D 3

(21)

The relationship between the harmonic images and the microscopic structures of the sample lies in the dependence of the coherent-scattering cross- section σ on the electron charge density distribution under Born approximation:

where r _e is the classical electron radius, and R(u) is the auto-correlation of the electron density distribution: \

Equation (22) means that the angular distribution of the scattered X-rays at any location in the layer is proportional to the three-dimensional Fourier transformation of an auto-correlation function of the electron charge density distribution at that location. Substituting eq.(22) into equation (21) yields

where R _∑ (x, y; r ₀ ) is defined in terms of the auto-correlation in eq.(19) in the vicinity of r _o :

R ₂ (x, y) = -R(x,y,w)]dw.

Equation (24) relates the scattering effect to electron charge density variation and thereby the microscopic structure of the sample. For the zeroth-order image of n = 0, dz (26) Therefore, the zeroth-order image is simply a conventional attenuation image. Summing all the layers that constitute the sample by integrating eq.(24) over the thickness of the sample yields

(27) where GJj _d ) is the nth-order harmonic image without any samples, and the X-ray path is a straight line connecting the source with the point r^ on the detector. Equation (27) means that the sample harmonic images should be normalized with the no-sample reference harmonic images in order to remove any features associated with the device itself. We can separate the effect of scattering from absorption by the ratio between the nth and zeroth-order harmonic images: f ^l O ^'J ' "θ V j x-ray path *

(28)

Equations (27) and (28) are the basis for obtaining attenuation and scattering images from the harmonic images. The normalization reference images G _n are acquired for a given device setup beforehand and are used for all samples if the setup is not altered. Scattering length scale

The above discussion, summarized by eq.(28), shows that the scattering signal from the nth-order harmonic is linearly dependent on an electron density autocorrelation function of the form

R _n = jjdxdy JdZ ₁ jdz ₂ p(x, y, y + d _π ,z ₂ )l

unit area unit length unit length

(29) where d _n , the scattering length scale, is expressed as

DΛn

d = -

P

Because P is the period of the grating shadows on the detector surface, it is related to the period of the grating itself by a geometric magnification:

P = P ₀ (D ₁ + D ₂ + D ₃ ) I D ₁ , ₍₃₀₎ which leads to the final expression of the scattering length scale.

Physically eq.(29) means that the scattering signal depends on the difference between the local electron charge density distribution and the same distribution shifted by the distance d _n in the direction perpendicular to the grating lines. If the sample is made of structures that are much larger than this scale, then the charge density changes little over this distance and the difference between the two copies of the density distribution is small, leading to low scattering signal; if on the other hand the structures are smaller than the length scale, than the shift results in appreciable changes of the density distribution and significant scattering signal.

As an example, consider a sample made of spherical particles of radius d' and charge density p which are randomly distributed such that the overall density of the material is p/10. The scatter decreases quickly when the particle radius exceeds the scattering length scale.

Interdependencies of hardware specifications and system layout

When designing the Fourier imaging device, we can consider grating blurring by the focal spot size of the X-ray tube. If the focal spot size is s, then an infinitely sharp edge at the grating plane will be blurred to , D ₂ + D ₃

s =— ^L s

A

at the detector surface. In order for the grating shadows to be clearly visible on the detector surface, s ' should be smaller than the period of the grating shadows given in eq.(30). The relationship that needs to be satisfied by all components is then

^ A ÷ V A p ₍₃₁₎

Since Po is also the final image resolution, one may be tempted to minimize it by reducing the distance D ₃ . However, the tradeoff is that the scattering length scale expressed in eq.(l) will be reduced proportionally, resulting in lower scattering signal.

Another consideration is that the detector pixel size should be less than /73 in order to sufficiently sample the first-order harmonic peaks:

D, + D ₇ + D,

camera pixel < P ₀ — . (32)

3D ₁

These relationships and the scattering length scale can be considered together when scaling the device for human imaging.

It should be recognized that any of the methods described herein can be executed using a computer-readable medium having instructions thereon to perform the methods.

In view of the many possible embodiments to which the principles of the disclosed invention may be applied, it should be recognized that the illustrated embodiments are only preferred examples of the invention and should not be taken as limiting the scope of the invention. Rather, the scope of the invention is defined by the following claims. We therefore claim as our invention all that comes within the scope of these claims.