Complementary coded apertures for 4-dimensional x-ray coherent scatter imaging

Shuo Pang; Mehadi Hassan; Joel Greenberg; Andrew Holmgren; Kalyani Krishnamurthy; David Brady

doi:10.1364/OE.22.022925

1. Introduction

X-ray imaging systems have various applications in medical diagnosis [1], security [2], and non-destructive testing [3]. Photo-absorption is the most commonly used mechanism to provide contrast. Though the absorption-based systems are capable of fast tomographic imaging, they cannot provide accurate material identification. Other contrast mechanisms have been explored, including fluorescence [4], phase contrast [5], Compton scattering [6], and coherent scatter imaging [7]. Among these approaches, x-ray coherent scattering is one of the mechanisms that can combine two important aspects: tomographic imaging and accurate material identification. Several x-ray coherent scatter imaging systems have been developed based on energy dispersive x-ray diffraction (EDXRD) geometry, such as coherent scatter computed tomography (CSCT) [8] and selected volume tomography (SVT) [9]. In these systems, each object voxel is measured at a single scattering angle in each snapshot, which limits the collection efficiency of the scattered x-ray signal; therefore, the image acquisition process is slow.

New approaches using coded aperture x-ray scatter imaging (CAXSI) have been proposed [10], [11], using 2D energy-integrating detectors. These systems abandon the detector side collimators, which are critical parts to locate the scatter location in SVT and CSCT systems. A 1D detector array or single pixel detector (0D) with energy resolutions can also be applied in CAXSI [12], [13]. These systems place a detector mask between the object and the detectors. Each scatter point projects the mask pattern to the detectors. By carefully designing the pattern of the mask, the scatter point location can be identified, and the coherent scatter form factor is recovered. Instead of placing a detector-side mask, we have previously demonstrated using structured illumination introduced by a source-side mask to obtain spatial information [14].

All the coded aperture based systems mentioned above represent imaging objects in two spatial dimensions. In this paper, we propose an x-ray coherent scatter imaging system that takes advantage of the cone beam from an x-ray source to image objects with three spatial dimensions using complementary coded apertures. A coded aperture is placed on the detector side to introduce multiplexed measurement on an energy-sensitive detector array; a complementary source-side coded aperture selectively illuminates the object to decouple the ambiguity due to the parallelization by the opening of the cone beam. The system is capable of providing three dimension spatial information and reveals the coherent scattering form factor in the momentum transfer space.

This paper is organized as follows. Section 2 describes the mathematical model for the imaging system and specifically explains the role of the coded aperture induced structured illumination. We describe the setup of the imaging system and experimental methods in Sections 3. In Section 4, we show the imaging results of the system and discuss the resolution of the system. Finally, we summarizes our conclusions and discusses potential improvements in Section 5.

2. Theory

In our coherent scatter imaging system, coded apertures introduce measurement parallelism through implementing structured multiplexing on energy-sensitive detector array [13]. The elimination of the source and detector side collimations can greatly improve the scattered x-ray photon detection efficiency and opens the possibility for compressed measurement, which could further increase the throughput [15].

2.1 Mathematical model for the system

The system setup and the coordinate system are shown in Fig. 1. The scattering object voxel coordinates are denoted as r, (x, y, z) and the detector pixel, with energy bin centered at E’, located at r’ (x’, y’, z’). The measured irradiance spectrum, b, is given by

b (E, r^{'}) = \int^{​} \int^{​} \frac{1}{r_{s c}^{2}} \cos θ_{s c} S (E, E') d Φ_{s c} d E',

where dΦ_sc is the radiant flux from a voxel within the whole scatter volume.

r_{s c} = (x^{'} - x, y^{'} - y, z^{'} - z),

is the vector from the scattering voxel to a detector pixel. θ_sc is the angle between the scattering vector to the normal vector of the pixel. In the experimental setup, the detector normal vector is perpendicular to the xy plane (0, 0, 1), and thus

\cos θ_{s c} = \frac{| z^{'} - z |}{‖ r_{s c} ‖}

. S(E, E’) is the spectral response of the pixel. For the energy channel of the detector, the spectral response can be modeled as a Gaussian function centered at

S (E, E^{'}) = \frac{1}{Δ E \sqrt{2 π}} exp (- \frac{{(E - E^{'})}^{2}}{2 Δ E^{2}}) . Δ E

is the standard deviation of the energy channel, i.e. the energy uncertainty of the measurement.

Fig. 1 Geometric setup for the coded aperture x-ray coherent scatter imaging system.

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The radiant flux from a voxel of the scatter, dΦsc is given by

d Φ_{s c} (E) = d Φ_{i n} (E) T (r, r') \frac{d σ_{c o h}}{d Ω} d z d q,

where

d Φ_{i n} (E) = \frac{I_{i n} (r, E)}{{‖ r ‖}^{2}} \cos θ_{i n} d x d y

, is the spectral power on the scatter voxel illuminated by the x-ray source. I_in(r, E) is the spectral irradiance from the x-ray source. The source is located at the origin of the coordinate, r₀ = (0, 0, 0). θ_in is the angle between the incident vector to the normal vector of the xy plane of the scatter voxel, and

\cos θ_{i n} = \frac{| z |}{‖ r_{} ‖}

. Here we assume the attenuation induced by the object is weak. For optically thick objects, a self-attenuation term can be included.

T(r, r’) is the coded aperture transmission function, defined by the detector side mask t(x,y), which is located at plane z = l. We define the ratio of the object to mask distance and the object to the detector distance, which has a value between 0 and 1, as the scale factor

β (z) \equiv (z^{'} - l) / (z^{'} - z),

where z’ is the z coordinate of the detector. For a given detector mask, t(x, y; z = l), the detector mask transmission function is:

T (r, r') = t (x' β + x (1 - β), y' β + y (1 - β)) .

The mask transmission function at z can be treated as the “scaled” detector mask function, t(x, y), with the scale factor, β. Inversion is straightforward if t(x, y) is orthogonal in scale [16]. Harmonic functions are orthogonal in scale. We choose to align the line-detectors along the Y direction, and use one-dimensional harmonic mask along the Y direction. To simplify the discussion, here the mask function

T (r, r') = (1 - \cos (2 π u y)) / 2

, where u is the frequency of the harmonic mask. The mask transmission function

T (r, r') = (1 - \cos [2 π u (y' β + y (1 - β))]) / 2 .

The differential cross section, $\frac{d σ_{c o h}}{d Ω}$ , describes the amount of x-ray coherently scattered by the material within the voxel at energy E into a unit solid angle,

\frac{d σ_{c o h}}{d Ω} = \frac{r_{e}^{2}}{2} (1 + \cos^{2} θ) f (r, q) = \frac{r_{e}^{2}}{2} (1 + \cos^{2} θ) n (r) f_{0} (r, q),

where θ is the scatter angle,

\cos θ = \frac{r \cdot r_{s c}}{‖ r ‖ ‖ r_{s c} ‖}

. r_e is the is the classical electron radius, and f(r, q) is the spatially dependent coherent scattering form factor. n(r) is material density, and f₀(r,q) is the coherent scatter form factor of the material.

Coherent scattering obeys Bragg’s Law:

q = \frac{E \sin θ / 2}{h c} .

Combining the above equations, and considering that the detector is a one dimensional array along the y axis, Eq. (1) becomes,

\begin{array}{l} b (E, y^{'}) = \frac{r_{e}^{2}}{4 \sqrt{2 π} Δ E} \iint I_{i n} (r, E) \exp (- {(E - \frac{q h c}{\sin \frac{θ}{2}})}^{2} / 2 Δ E^{2}) \frac{| z^{'} - z |}{{‖ r_{s c} ‖}^{3}} \frac{| z |}{{‖ r ‖}^{3}} \\ (1 + \cos^{2} θ) f (x, y, z, q) (1 - \cos [2 π u (y^{'} β + y (1 - β))]) d r d q . \end{array}

Let

U (E, y^{'}, r, q) \equiv \frac{r_{e}^{2}}{4 \sqrt{2 π} Δ E} I_{i n} (r, E) \exp (- {(E - \frac{q h c}{\sin \frac{θ}{2}})}^{2} / 2 Δ E^{2}) \frac{| z^{'} - z |}{r_{s c}^{3}} \frac{| z |}{r^{3}} (1 + \cos^{2} θ)

As the object is translated along the x direction with speed v, the detector readout, g, is a function of time.

\begin{array}{l} g (E, y^{'}, t) = \iint^{​} U (E, y^{'}, r, q) f (x + v t, y, z, q) \\ (1 - \cos [2 π u (y^{'} β + y (1 - β))]) d r d q, \end{array}

where at time point t, the detector took a snapshot measurement, and the whole frame sequence is described by Eq. (9),

U (E, y^{'}, r, q)

can be treated as a snapshot transformation kernel that maps the object form factor to the measurement space. The object movement encodes the translational direction (x) information in the temporal dimension.

2.2 Complementary structured illumination combined with the detector-side coded aperture

We have previously shown that multiplexed scatter imaging systems of lower dimensionality [12], [13] can employ a harmonic detector mask to encode the object’s range (z) information. In these systems, the “scale transformation” induced by the harmonic mask has the transformation kernel $T (y^{'}, β) = 1 - \cos [2 π u (y^{'} β)]$ , where u is the mask frequency.

When the object is extended and fully illuminated in the vertical axis (y), the kernel becomes the transmission function described by Eq. (5). An object that spatially extended in Y direction will introduce a continuous phase shift, $2 π u y (1 - β)$ , to the harmonic function. This phase shift couples the scale factor β and the vertical coordinate y. When integrating the whole object along y, as in Eq. (9), the kernel is no longer orthogonal under a “scale transformation”.

We can eliminate this phase shift term, and ensure the orthogonality by forcing the phase term to be multiples of 2π. We achieve this by simply introducing a source-side mask and selectively illuminating the object. In the central object plane z = z_c and for a periodic mask along y, the source only illuminates the voxels at horizontal coordinates, y, given by

y = \frac{m}{u (1 - β_{c})} + y_{0}, m = 0, \pm 1, \pm 2, \dots

where

β_{c} = (z' - l) / (z^{'} - z_{c})

, is the scale factor at object central plane. y₀ is a constant displacement. Equation (10) shows that the illumination mask is also periodic. If the source side mask is located at z = d. The frequency of the source side mask, u_s, is given by

u_{s} = u (1 - β_{c}) \frac{d}{z_{c}} .

3. Materials and methods

3.1 Experiment setup

The system setup is shown in Fig. 2(a) and 2(b). We defined the origin of the coordinates of the imaging system as the x-ray tube focus, as described in Section 2. The x-ray tube (XRS-225, Comet AG) operated at 140 kVp, with a current of 5.7 mA. Two layers of lead plates with a thickness of 3 mm served as collimators to limit the illumination volume. The source-side coded aperture mask was made of bismuth-tin alloy (58% bismuth and 32% tin). The mask was placed in between the two lead plates, at z_s ≈20 cm, as shown in Fig. 2(b1). The source-side mask had a period of 4 mm and ~40% duty cycle (1.6 mm openings of each period). The mask pattern was fabricated by a 5-axis CNC machine.

Fig. 2 Scatter imaging system setup. (a) System schematics. (b) Photos of the system setup (b1), which includes the detector side mask (b2) and source side mask (b3).

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The detector side mask was made of tungsten powder, which was placed at z ≈90 cm. The pattern was a square wave function with a period of 4 mm in y and 1.5 mm thick in z. We used a 3D printer (Objet Eden 333) to create the mold of the mask. The opaque area was filled with tungsten powder and sealed with epoxy. The tungsten mask’s feature size along y is 2.0 mm. The mold had a 0.3 mm plastic backing in the tungsten filled area. The total area of the mask was 60 mm (x) by 300 mm (y). The mold material attenuation of the x-rays was negligible in the experiments. Figure 2(b2) shows the photo of the detector side mask.

The detector array consisted of three energy sensitive detector modules (ME100, Multix Detection) located at z_d ≈1.1 m. Each module had 128 pixels with 64 energy channels ranging from 20 to 170 keV (~2.2 keV energy step). The three detectors were aligned along y axis to form one row at x = 36 mm. The full width at half maximum (FWHM) of each energy channel was measured to be ~7 keV. A belt-driven linear stage (A-BLQ0295-E01, Zaber Technology) translated the object along the x axis. The center of object space was located at z = 55 cm plane. The illumination area on the object plane was 30 mm (x) by 100 mm (y). We set the detector frame rate to be 10 fps. One snapshot measurement sums up 50 frames. We translate the object at a speed of 1 mm/second, corresponding to 5 mm of object movement along x axis during one snapshot.

To demonstrate the imaging methods and evaluate the performance of the system, we used the system to image objects made of aluminum, graphite and Teflon. The aluminum and graphite objects in the experiments were in powder form. The powders were contained in molds of different shapes fabricated by the 3D printer. The container thickness was 0.5 mm, so that the scatter from the printing material can be neglected. The Teflon objects were fabricated by machining a Teflon plate.

3.2 System simulation and reconstruction

Given the measurement model described in Section 2, each energy-sensitive measurement is represented in g. The measurements also contained a noisy background, μ_b, due to unaccounted Compton and multiple scattering. We assume the actual measurements, y_noise, can be approximated as a Poisson process. Under these assumptions the measurement is given by

y_{n o i s e} = Pois (g + μ_{b}),

where Pois(•) is an instance of Poisson observations. With the experimental measurement y, the noisy measurement of the background y_b = Pois(µ_b), and the calibrated model, we can estimate the object, f, by a maximum likelihood estimator (MLE) given by

\hat{f} = \arg \min_{f^{'}} (- \log P (y_{n o i s e} | f^{'}, y_{b})),

where P(v|u) is the Poisson likelihood of observation v given parameters u. The maximum likelihood estimate of f is obtained using an iterative method [17]. The forward model is based on a ray-tracing method [11], and both the model and the MLE reconstruction are implemented in Matlab code.

4. Results and discussions

4.1 Imaging of a “point” graphite sample and resolution analysis

We apply the imaging system to image a graphite cube with an edge length of 8 mm. The graphite cube is sufficiently small, so that only one fan beam of x-ray illuminated the object during the imaging process. With the system model described in Section 2.1, we simulated one snapshot (Fig. 3(a)) measurement, which is in good agreement with the experimental measurement (Fig. 3(b)). The measurement shows the modulation induced by the detector side mask. Graphite has only one major momentum transfer peak at q = 0.15 A⁻¹, which makes it effectively a point object in momentum transfer coordinate as well. According to Bragg’s Law, shown in Eq. (7), at one momentum transfer value, q, the lower energy x-ray is scattered to larger angle, resulting in the bell-shaped curve in the pixel-number to energy plot. The graphite cube was translated in X direction during the imaging process. The system took 5 frames for reconstruction. The step-size of the translation between each frame is 5 mm.

Fig. 3 Imaging of an object graphite point object. (a) Simulation of one snapshot coherent scattering measurement. (b) The experimental results. (c) The reconstruction of an point graphite cube of 1 cm³ .

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Figure 3(c) shows the reconstruction of the graphite cube. The pixel values in Fig. 3(c1)-3(c3) are the sum of the form factor profile along the momentum transfer coordinate of each voxel. The reconstruction sampling was 5 mm in x and y, 10 mm in z, and 0.01 A⁻¹ in q.

The x-ray diffraction (XRD) reference form factors were acquired using Panalytical XPert PRO diffractometer with an 8 keV Cu source and a scan time of 20 minutes. The samples were prepared separately and not for imaging purposes (i.e., their locations were known a priori). The high resolution XRD data only serves as form factor references.

By differentiating Eq. (7), the estimated uncertainty of the coherent scattering form factor can be expressed as:

\frac{Δ q}{q} = \sqrt{{(\frac{Δ E}{E})}^{2} + {(\frac{Δ θ}{θ})}^{2}}

The first term of Eq. (14) the normalized uncertainty, ΔE/E, is the uncertainty due to the energy resolution of the x-ray detector. The second term of Eq. (14) is the normalized angular uncertainty, which is determined by the location uncertainty of a voxel of the object.

The resolution, Δx, is determined by the detector size and the translation step size. $Δ x = \sqrt{Δ x_{m}^{2} + d_{x}^{2}}$ , where Δx_m is the translation step size, and d_x is the pixel width. Δy is determined by the height of the detector pixel, d_y. $Δ y = d_{y}$ .

The modulation introduced by the sinusoidal detector-side mask provides the resolution in the z direction. Assume for a point-scatterer that the maximum number of modulations that can be observed during the imaging process is N. In the absence of noise, two points along the z axis are discernable when the number of observable modulations have a difference of ΔN > 1.

Given the minimum detectable energy E_min, dictated by the detector limitation, the maximum number of modulations can be observed is $N = 2 u (l - z) \sin θ_{m a x} = 4 h c u (l - z) q / E_{m i n}$ , where θ_max is the maximum scatter angle from a single momentum transfer peak, q. According to the Bragg’s Law, the maximum detectable scatter angle is determined by the minimum detectable energy of the detector E_min. Here we assume the length of the detector does not limit the number of observable modulations. Setting ΔN = 1, we can derive the range resolution as:

Δ z = \frac{E_{m i n}}{4 h c u q}

Equation (15) shows that the finer range resolution can be achieved for masks with smaller period and larger momentum transfer peaks (larger diffraction angle).

As the object translates through the illumination volume, the normalized angular uncertainty, δθ/θ, is minimum when the object is scattered at the maximum scatter angle, and its value is given by:

\frac{δ θ}{θ} = {[{(\frac{E_{m i n}}{2 h c q})}^{2} ({(\frac{Δ x}{z - z'})}^{2} + {(\frac{Δ y}{z - z'})}^{2}) + {(\frac{Δ z}{z - z'})}^{2}]}^{1 / 2}

In our system setup, the motion step size, Δx_m, was 5mm, and the pixel’s height and width were both 0.8 mm. The resolution along the x axis, Δx, was calculated to be 5.6 mm. It is worth noting that this is not the limit of the x resolution. We could achieve higher resolution by reducing the motion step size. The minimum detectable energy, E_min, was 21 keV. The mask frequency, u, was 0.25 mm⁻¹. Using the value of E_min and u for Eq. (15), we calculated the object range resolution, Δz = 1.1 cm, at momentum transfer q = 0.15 A⁻¹. The distance from the object center to the detector, z-z’, was about 0.5 meter. According to Eq. (16), the angular uncertainty was 0.07.

The energy sensitive detector has a full-width at half-maximum (FWHM) of 6 keV in energy, and ΔE/E ~0.1. Compared with the angular uncertainty of the system, the energy measurement uncertainty was a little larger, and is the limiting factor for the resolution of the coherent scattering form factor. Plugging all the parameters calculated above into Eq. (14), the uncertainty of the momentum transfer, Δq, is 0.018 A⁻¹.

In Fig. 3(c4), the reconstructed object momentum transfer peak has a FWHM of about 0.03 A⁻¹, corresponding to a normalized momentum transfer uncertainty ~0.2. The calculated momentum transfer uncertainty is the lower bond of the reconstruction resolution. Another factor that was not included in the analysis above is the opening width of the illumination mask. We include a more detailed analysis in Section 4.2. It is worth mentioning that the reconstruction results showed a small ambiguity in the yz plane, which is the result of the object’s extension along the y axis.

4.2 Imaging extended objects with structured illumination

In Section 2.2, we showed that the structured illumination can decouple an object’s dependence of scale transformation on y. To demonstrate the effect of the source-side mask, we next use the system to image an object with extent along the y direction, with and without structured illumination. The object is a container of dimensions 2 mm (x) by 50 mm (y) by 5 mm (z) filled with aluminum powder. Because the object’s extent in x was smaller than the reconstruction pixel width, the object can be considered as a 3D object (two dimensions in y and z, and one dimension in q). We placed the object at x = 54 mm, a snapshot measurement of the object is sufficient for the 3D imaging. The detector integration time was 2 seconds.

Figure 4(a) and 4(b) show the experimental measurement for the scenario without and with structured illumination, respectively. Due to the extension of the object in the y dimension, the detector-mask modulation was “blurred out” for the system without using the structured illumination. When the structured illumination described in Eq. (11) is employed, the range modulation is maintained. Figure 4(c) and 4(d) show one y-slice of the reconstruction in the zq plane. For the structured illumination case, the object location along the z axis was well-resolved, and the scattering form factor is reconstructed along the corresponding row. Without structured illumination, the range information was coupled in the scale transform, resulting in an ambiguity in the zq space. In essence, structured illumination reduces the object’s components from the null space of the system model, which improves the measurement signal to noise ratio significantly.

Fig. 4 Comparison between system with and without the illumination masks. (a) Experimental measurement of an aluminum object extended in y direction without structured illumination. (b) Experimental measurement of the same object with structured illumination. (c) Reconstruction of the object without structured illumination in q, z space. (d) Reconstruction of the object with structured illumination.

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Here, we would like to further discuss the effect of the duty cycle of the periodic source-side mask. The duty cycle is related to the feature size of structured illumination, which was analyzed in Ref [14]. We define the duty cycle as the ratio of the opening width to the period of the mask. In Section 2.2, we assumed the opening is infinitely small. As the opening gets larger, more scatter signal can be detected. However, the larger opening can also reduce the visibility of the range-encoded modulation. We simulated the momentum transfer resolution, Δq, as a function of the code’s duty cycle in the noise-free case. Figure 5 shows the simulation results. The momentum transfer resolution monotonically increases as the duty cycle increases. Figure 5(b) shows two instances of the cross-correlation coefficients map of the extended object with duty cycles of 50% (Fig. 5 (b1)), and 12.5% (Fig. 5(b2)). We can observe a narrower peak in the correlation coefficient map for small duty cycle. In the extreme case where the duty cycle is 100%, i.e. no structured illumination, the momentum transfer information is lost completely.

Fig. 5 (a) Form factor resolution, Δq, as a function of illumination duty cycle. (b) Cross-correlation coefficients map of the extended object with duty cycles of 50% (b1), and 12.5% (b2).

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4.3 Imaging of 4D objects

To demonstrate the 4D imaging capability, we used the system to image the letter “D” and letter “U” made from graphite powder and Teflon, respectively. The letter “D” was placed at z = 536 mm and the letter “U” was placed at z = 616 mm. Figure 6(a) shows the reconstructed scatter intensity at each voxel (i.e. the sum along the momentum transfer axis, q). The graphite is rendered in red, and the Teflon is rendered in white.

Fig. 6 Imaging of 4-dimensional object (a) 3D rendered pseudo-colored image of two object, a Teflon letter “D” and a graphite letter “U”. (b) The photograph of the two objects. (c-d) The interpolated correlation map of the object of the letter “U”(c), and the letter “D” (d). (e-f) the reconstructed form factor of the letter “U” made of graphite (e) and the form factor of the letter “D” made of Teflon (f).

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Applying the periodic structured illumination maintains the range resolution of the system at the expense of losing the measurements from the volume blocked by the mask. Figure 6(b) shows the photos of the object with the Gafchromic^TM film in the background. The dark area on the film roughly indicated the illuminated area by the source-side mask. The duty cycle of the mask was 40%, to ensure the normalized momentum transfer resolution, Δq/q was better than 0.2. From the measurements, we reconstructed the 4D form factor f(x,y,z,q) from the illuminated voxels. The illuminated voxels’ y coordinates satisfy $\frac{m - c / 2}{u (1 - β (z))} + y_{0} < y < \frac{m + c / 2}{u (1 - β (z))} + y_{0}$ , where m is an integer, and c is the duty cycle, which is 0.4 for our source-side mask. y₀ is a phase shift constant, which is same as that in Eq. (10). The form factors from the voxels that were blocked by the source-side mask were linearly interpolated from the adjacent illuminated voxels along the y axis. We calculated the correlation map by comparing the estimated form factor at each location to that obtained by XRD [14]. Figure 6(c) and 6(d) shows the correlation map at z = 62 cm and z = 54 cm, respectively. We set a threshold of 0.5 in the correlation map for material identification, which determines the color shown in Fig. 6(a).

The form factors shown in Fig. 6(e) and 6(f) are the summation of the form factors profiles from all pixels in these two planes, which are in good agreement with the XRD references.

5 Conclusions

In summary, we have demonstrated an x-ray scatter tomographic imaging system using complementary coded-apertures. The system is capable of imaging a spatially 3-dimensional object and its coherent scatter form factor. Complementary coded apertures on both the source side and the detector side introduce structured multiplexing on an energy-sensitive detector array. We have demonstrated the imaging capability and studied the effect of the illumination mask. The coherent scatter form factor resolution of our system is affected mainly by the detector energy resolution and the duty-cycle of the structured illumination. We have demonstrated a normalized momentum transfer resolution, Δq/q, of 0.2. A finer mask and detectors with better energy resolution could improve the system resolution.

The reported system uses simple periodic codes on both sides of the object. This setup selectively illuminates certain parts of the object in order to maintain the range resolution. One simple solution to achieve full-volume imaging is to introduce multiple arrays of detectors and multiple source-side masks. The source-side periodic masks should have a phase shift that covers the entire volume of the object. A pair of slanted periodic masks with same rotation angle can also be applied for full volume imaging. Instead of using one array of energy-sensitive detectors, it is possible to implement a 4D snapshot tomographic imaging system based on complementary masks by using a two-dimensional array of energy-sensitive imagers and multiple x-ray sources. Furthermore, instead of designing the coded aperture for 4D imaging, the pattern can be designed to enhance the sensitivity/resolution for specific features.

Finally, we note that this scheme could be combined with tomographic imaging based on attenuation. The compensation due to the object attenuation can improve the reconstruction quality for highly absorptive or scattering objects. We believe this method has a range of potential applications, including biological and industrial imaging [18], [19].

Acknowledgments

This work is support by the U.S. Department of Homeland Security, Science and Technology Directorate Explosives Division under contract HSHQDC-11- C-00083. The authors would like to thank Dr. Ken MacCabe for helpful discussions.

References and links

1. D. J. Brenner and E. J. Hall, “Computed tomography-an increasing source of radiation exposure,” N. Engl. J. Med. 357(22), 2277–2284 (2007), http://www.nejm.org/doi/full/10.1056/nejmra072149. [CrossRef] [PubMed]

2. G. Harding and B. Schreiber, “Coherent X-ray scatter imaging and its applications in biomedical science and industry,” Radiat. Phys. Chem., 56 (1)–(2), 229–245, (1999), URL:http://linkinghub.elsevier.com/retrieve/pii/S0969806X99002832. [CrossRef]

3. C. Teichert, J. F. MacKay, D. E. Savage, M. G. Lagally, M. Brohl, and P. Wagner, “Comparison of surface roughness of polished silicon wafers measured by light scattering topography, soft-x-ray scattering, and atomic-force microscopy,” Appl. Phys. Lett. 66(18), 2346 (1995), http://scitation.aip.org/content/aip/journal/apl/66/18/10.1063/1.113978. [CrossRef]

4. J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, “Fluorescent computer tomography: a model for correction of X-ray absorption,” IEEE Trans. Nucl. Sci. 38(6), 1721–1727 (1991). [CrossRef]

5. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

6. P. Zhu, G. Peix, D. Babot, and J. Muller, “In-line density measurement system using X-ray Compton scattering,” NDT Int. 28(1), 3–7 (1995), http://www.sciencedirect.com/science/article/pii/0963869594000088. [CrossRef]

7. G. Harding, “X-ray diffraction imaging--a multi-generational perspective,” Appl. Radiat. Isot. 67(2), 287–295 (2009), http://www.ncbi.nlm.nih.gov/pubmed/18805014. [CrossRef] [PubMed]

8. G. Harding, “X-ray scatter tomography for explosives detection,” Radiat. Phys. Chem., 71 (3)–(4), 869–881, (2004), URL: http://linkinghub.elsevier.com/retrieve/pii/S0969806X04003081.

9. P. G. Lale, “The Examination of Internal Tissues, using Gamma-ray Scatter with a Possible Extension to Megavoltage Radiography,” Phys. Med. Biol. 4(2), 159–167 (1959), http://iopscience.iop.org/0031-9155/4/2/305. [CrossRef] [PubMed]

10. K. MacCabe, K. Krishnamurthy, A. Chawla, D. Marks, E. Samei, and D. Brady, “Pencil beam coded aperture x-ray scatter imaging,” Opt. Express 20(15), 16310 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-20-15-16310. [CrossRef]

11. K. P. MacCabe, A. D. Holmgren, M. P. Tornai, and D. J. Brady, “Snapshot 2D tomography via coded aperture x-ray scatter imaging,” Appl. Opt. 52(19), 4582–4589 (2013), http://www.ncbi.nlm.nih.gov/pubmed/23842254. [CrossRef] [PubMed]

12. J. A. Greenberg, K. Krishnamurthy, and D. Brady, “Snapshot molecular imaging using coded energy-sensitive detection,” Opt. Express 21(21), 25480–25491 (2013), http://www.opticsexpress.org/abstract.cfm?URI=oe-21-21-25480. [CrossRef] [PubMed]

13. J. Greenberg, K. Krishnamurthy, and D. Brady, “Compressive single-pixel snapshot x-ray diffraction imaging,” Opt. Lett. 39(1), 111–114 (2014), http://www.ncbi.nlm.nih.gov/pubmed/24365835. [CrossRef] [PubMed]

14. J. A. Greenberg, M. Hassan, K. Krishnamurthy, and D. Brady, “Structured illumination for tomographic X-ray diffraction imaging,” Analyst (Lond.) 139(4), 709–713 (2014), http://www.ncbi.nlm.nih.gov/pubmed/24340351. [CrossRef] [PubMed]

15. A. Mrozack, D. L. Marks, and D. J. Brady, “Coded aperture spectroscopy with denoising through sparsity,” Opt. Express 20(3), 2297–2309 (2012), http://www.opticsexpress.org/abstract.cfm?URI=oe-20-3-2297. [CrossRef] [PubMed]

16. D. J. Brady, D. L. Marks, K. P. MacCabe, and J. A. O’Sullivan, “Coded apertures for x-ray scatter imaging,” Appl. Opt. 52(32), 7745–7754 (2013), http://www.opticsinfobase.org/viewmedia.cfm?uri=ao-52-32-7745&seq=0&html=true. [CrossRef] [PubMed]

17. W. H. Richardson, “Bayesian-Based Iterative Method of Image Restoration,” J. Opt. Soc. Am. 62(1), 55 (1972), http://www.opticsinfobase.org/abstract.cfm?URI=josa-62-1-55. [CrossRef]

18. G. Harding and J. Kosanetzky, “Status and outlook of coherent-x-ray scatter imaging,” J. Opt. Soc. Am. A 4(5), 933–944 (1987), http://www.ncbi.nlm.nih.gov/pubmed/3598745. [CrossRef] [PubMed]

19. D. L. Batchelar, M. T. M. Davidson, W. Dabrowski, and I. A. Cunningham, “Bone-composition imaging using coherent-scatter computed tomography: Assessing bone health beyond bone mineral density,” Med. Phys. 33(4), 904–915 (2006), http://scitation.aip.org/content/aapm/journal/medphys/33/4/10.1118/1.2179151. [CrossRef] [PubMed]

Complementary coded apertures for 4-dimensional x-ray coherent scatter imaging

Abstract

1. Introduction

2. Theory

2.1 Mathematical model for the system

2.2 Complementary structured illumination combined with the detector-side coded aperture

3. Materials and methods

3.1 Experiment setup

3.2 System simulation and reconstruction

4. Results and discussions

4.1 Imaging of a “point” graphite sample and resolution analysis

4.2 Imaging extended objects with structured illumination

4.3 Imaging of 4D objects

5 Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (16)

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