Coded aperture imaging for fluorescent x-rays A. Haboub, A. A. MacDowell, S. Marchesini, and D. Y. Parkinson Citation: Review of Scientific Instruments 85, 063704 (2014); doi: 10.1063/1.4882337 View online: http://dx.doi.org/10.1063/1.4882337 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Combined Imaging System for Xray Fluorescence and Transmission Xray Microtomography AIP Conf. Proc. 705, 1352 (2004); 10.1063/1.1758052 Fast Differential PhaseContrast Imaging and Total Fluorescence Yield Mapping in a Hard Xray Fluorescence Microprobe AIP Conf. Proc. 705, 1348 (2004); 10.1063/1.1758051 Performance of laminar-type holographic grating for a soft x-ray flat-field spectrograph in the 0.7–6 nm region Rev. Sci. Instrum. 74, 1156 (2003); 10.1063/1.1533097 2D imaging by X-ray fluorescence microtomography AIP Conf. Proc. 507, 539 (2000); 10.1063/1.1291207 A hard x-ray scanning microprobe for fluorescence imaging and microdiffraction at the advanced photon source AIP Conf. Proc. 507, 472 (2000); 10.1063/1.1291193
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 063704 (2014)
Coded aperture imaging for fluorescent x-rays A. Haboub,1,2 A. A. MacDowell,1 S. Marchesini,1 and D. Y. Parkinson1 1 2
Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA Lincoln University, Life and Physical Sciences Department, Jefferson City, Missouri 65101, USA
(Received 20 November 2013; accepted 28 May 2014; published online 19 June 2014) We employ a coded aperture pattern in front of a pixilated charge couple device detector to image fluorescent x-rays (6–25 KeV) from samples irradiated with synchrotron radiation. Coded apertures encode the angular direction of x-rays, and given a known source plane, allow for a large numerical aperture x-ray imaging system. The algorithm to develop and fabricate the free standing No-Two-Holes-Touching aperture pattern was developed. The algorithms to reconstruct the x-ray image from the recorded encoded pattern were developed by means of a ray tracing technique and confirmed by experiments on standard samples. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4882337] I. INTRODUCTION
In recent years, with the advances in computer speed, image processing, and charge couple device (CCD) detector technology, increasing interest has been shown in coded aperture imaging in x-ray and γ -ray fields, namely astronomical applications,1, 2 medical physics,3 plasma imaging,4 neutron imaging,5 and synchrotron beam imaging.6 Coded aperture imaging is an imaging technique without conventional lenses and was first suggested in 1961 for use in x-ray astronomy cameras.1, 7 This imaging technique involves the use of a coded aperture pattern or mask that consists of a known arrangement of transparent and opaque elements. The mask is placed between the x-ray source and a suitable area sensitive detector. Transparent elements in the mask act as individual pinhole cameras that together produce multiple overlapping images of the source on the detector. Given knowledge of the coded aperture pattern, the object can be reconstructed from the recorded multiple overlapped images. This imaging technique improves the signal noise ratio (SNR) efficiency √ of the system over a single pinhole imaging camera by ∼ N where N is the number of pinholes. Numerous coded aperture patterns have been used in x-ray and γ -ray imaging, namely Fresnel zone plate, random arrays, uniform redundant arrays (URA), modified uniform redundant arrays (MURA), and No-Two-Hole-Touching (NTHT) based on MURA.7, 8 To show the fundamental idea of the coded aperture imaging, a simulation illustration is shown in Figure 1, which compares the imaging of an object with both pinhole camera and coded aperture pattern. The single small pinhole camera shown in Fig. 1(a) generates a faithful image; however, it only allows a small amount of light to be transmitted and so the image is of low intensity. A larger pinhole shown in Fig. 1(b) increases the transmitted light but leads to degradation in the image resolution. Fig. 1(c) is the case of several discrete pinholes and results in identifiable overlapping images. For an appropriate coded aperture pattern shown in Fig. 1(d), the overlapping images become more of a continuum requiring a computational reconstruction method to decode the image while retaining the image quality of the single pinhole but with the high light gather0034-6748/2014/85(6)/063704/8/$30.00
ing power of the multiple pinhole arrays. As such the unique advantages of coded the aperture imaging technique is its simplicity (a mask + CCD) and its potential high angular acceptance of x-rays for imaging. This technique can have a high numerical aperture (NA) ∼0.1 as limited by geometry. This is
FIG. 1. Pinhole camera and coded aperture imaging simulation for a small pinhole, large pinhole, multiple small pinholes, and a coded aperture pattern.
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in contrast to the more customary-ray imaging systems such as grazing incidence Kirkpatrick-Baez Mirrors, Fresnel zone plates, and refractive lenses that are limited to small NA typically ∼0.001. In this paper, we present the development of the algorithm and the fabrication of the NTHT self-supporting coded aperture pattern – based on the Modified Uniformly Redundant Array (MURA) as well as the algorithm to reconstruct x-ray images recorded by a NTHT MURA coded aperture with CCD camera. The imaging fidelity and spatial resolution is established. The paper reports initial experimental results as a verification of the model and describes future work for the next development stage whereby an energy resolving pixilated CCD can be deployed that will allow elemental imaging.
II. THEORETICAL REVIEW AND SIMULATION
Coded aperture imaging is a two-step process. The first step is the generation of overlapping images or a coded image when photons are being emitted from a source, pass through the transparent elements of the coded aperture pattern and projected onto the detector. The second step is the decoding or the reconstruction of the image through a suitable reconstruction method. If we denote O(x, y) as a two dimen-
sional object parallel to the coded aperture pattern whose transmission function is A(x, y) being a binary array that contains “1” for each transparent element and “0” for each opaque element, then the coded image I(x, y) of the object that is projected onto the detector is given by the convolution operation, I(x, y) = O(x, y) ⊗ A(x, y),
(1)
where ⊗ is the convolution operator. The reconstructed image O (x, y) can be obtained through the convolution operation of the coded image I(x, y) by a suitable decoding function or anti-mask D(x, y),7, 8 O (x, y) = I(x, y) ⊗ D(x, y).
(2)
For an ideal imaging reconstruction, the convolution of the two functions D(x, y) and A(x, y) is the system point spread function (SPSF) that is ideally a delta function,7–9 D(x, y) ⊗ A(x, y) = δ.
(3)
The MURA10 shown in Fig. 2(a) is one of the most efficient coded aperture patterns, having 50% transparent area, and its SPSF is a perfect delta function. The MURA pattern is constructed on a square lattice whose side length p is prime; each element of the MURA A(x, y) where x and y run from 0 to p − 1, is
⎧ 0 if x = 0 ⎪ ⎪ ⎪ ⎪ if x = 0, x = 0 ⎨1 if x and y are quadratic residues modulo p . A(x, y) = 1 ⎪ ⎪ 1 if neither x nor y are quadratic residues modulo p ⎪ ⎪ ⎩ 0 otherwise
However for the 6–25 KeV x-rays that we are interested in, the transparent holes are preferred to be actual holes which makes the MURA pattern to be not self-supporting pattern and difficult to fabricate. An alternative pattern, based on the MURA is a NTHT pattern shown in Fig. 2(b) which is a selfsupporting pattern and can thus be easily fabricated. This pat-
tern consists of inserting rows and columns of closed position “0” around each open position “1” of the original MURA. Fig. 2(b) shows the 58 × 58 NTHT pattern where only one row and one column are inserted around each open position of the original 29 × 29 MURA, and therefore it presents an open fraction area of 12.5%. We have carried out computer simulations using Matlab code to study the performance of coded aperture imaging using the NTHT pattern. Fig. 3 illustrates a coded aperture imaging simulation for the 58 × 58 NTHT. The simulated object O(x, y) is a “happy face,” which is a 58 × 58 binary array. The circular convolution of the two dimensional object O(x, y) and the array transmission function A(x, y) (58 × 58 NTHT), is the coded image I(x, y) of 58 × 58 in size. The decoding array D(x, y), which is of 58 × 58 in size, is derived through 2D Fast Fourier Transforms (FFT2),7 D(x, y) = iFFT2(1/FFT2)(A(x, y)),
FIG. 2. Coded aperture patterns: (a) 29 × 29 Modified Uniformly Redundant Array (MURA) and (b) 58 × 58 No Two Holes Touching (NTHT) based on MURA pattern.
(4)
where FFT2 and iFFT2 are, respectively, the fast 2D Fourier transform and its inverse. Circularly convolving A(x, y) with D(x, y) yields the SPSF that is a delta function. Circularly convolving the coded image I(x, y) with the decoding array
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FIG. 4. (a) Schematic diagram illustrating the Ray Tracing Technique (RTT) used in the coded aperture imaging simulation. Rays are traced from the middle of each object pixel though the center of each open coded aperture pixel and projected onto the detector plane. (b) The plot of the object coordinates, (c) the plot of the mask coordinates, and (d) the plot of the coded image coordinates.
FIG. 3. Coded aperture imaging simulation for the 58 × 58 NTHT MURA. The Object O(x, y) is circularly convoluted with the NTHT MURA, A(x, y) to yield the coded image I(x, y) which when circularly convoluted with the decoding pattern D(x, y) yields the reconstructed image O (x, y) The circular convolution of NTHT MURA A(x, y) with the decoding pattern D(x, y) is required to yield a delta function.
D(x, y) yields the reconstructed image O (x, y) by means of 2D Fast Fourier Transforms: O (x, y) = iFFT2(FFT2(I(x, y) ∗ FFT2(D(x, y))).
(5)
The image reconstruction turns out to be mathematically simple for this case where the object, coded aperture, and de-
tector are all of identical sized arrays. This is the situation for x-ray astronomical applications as the object is at infinity so magnification of the coded aperture onto the area detector is unity allowing the mask and detector to be the same size. When the object is closer than infinity, the magnification of the coded aperture onto the detector is >1 and the correlation of matrices is more complicated requiring new reconstruction formulations to be developed. For this reason, we have modeled the behavior of coded aperture imaging by means of a Ray Tracing Technique (RTT) to simulate coded images from nearby samples which involve magnification. We then tested the deconvolution procedures to reproduce the original object. This ray tracing consists of Matlab code which simulates the arrangement of the experimental setup to include the source, the coded aperture pattern and detector. The system is raytraced with rays from the middle of each object pixel passing through the middle of each pinhole of the mask and projected onto the detector, as illustrated in Fig. 4. The RTT yields a realistic coded image on the detector as shown in Fig. 4(d). Note that the image size in the detector plane is larger than the object and coded aperture. Using the simulated image, the reconstruction procedure described below was developed that deals with the different array sizes. The adopted deconvolution solution involves padding and 3 × 3 replication as shown in Fig. 5. For the case of the object O(x, y) of 60 × 60 in size, which consists of several numerals, the rays are projected by ray tracing through a 58 × 58 NTHT onto a 1024 × 1024 detector plane. This 1k × 1k image is padded with zeros to yield the 4096 × 4096 coded image I(x, y) where the coded image data occupies the central region. The mask also undergoes padding by replication and tiling of the basic 58 × 58 NTHT pattern to form a 3 × 3 pattern – A3(x, y) of size 174 × 174. The derivation of
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y). To rescale A3(x, y), to the 4096 × 4096 antimask B3(x, a point p(x, y) in the object plane center is raytraced through y) and projected onto a 4096 the middle of each pixel of A3(x, × 4096 array. All pixels are zero unless raytracing indicates y) contains only 174 × 174 pixels made up otherwise. A3(x, y) of +1, 0, and −1 values, so the vast majority of B3(x, pixels are zero as indicate in Fig. 5. The reconstructed image O (x, y) is then the circular convolution of this decod y) and the coded image I(x, y) ing function B3(x, y)))). O (x, y) = iFFT2(FFT2(I(x, y). ∗ FFT2(B3(x,
(7)
The reconstructed images do suffer from some high frequency artifacts typical of Fourier reconstruction techniques; however, the central image shows the lowest amount of artifacts with the object faithfully reproduced. These artifacts do not depend on the statistical quality of the data; they are caused mainly by the system point spread function (SPSF) which is derived through the fast Fourier transformation technique.
III. GEOMETRY OF CODED APERTURE IMAGING
From the simple geometry of the coded aperture camera setup as shown in Fig. 6, various geometrical relationships exist that determine the field of view (FoV) and resolution
FIG. 5. Coded aperture imaging simulation using ray tracing technique (RTT) followed by the newly developed reconstruction technique that pads y) and and replicates the mask to yield A3(x, y), derives its anti-mask A3(x, y). through RTT magnifies and rescales it to yield the decoding array B3(x, This decoding array is of size (3k × 3k) obtained by ray tracing the object y) and projected onto center p(x, y) through the middle of each pixel of A3(x, the detector plane.
y) is done through the 2D FFT technique the anti-mask, A3(x, noted earlier, y) = iFFT2(1/FFT2(A3(x, y)). A3(x,
(6)
The circular convolution of A3(x, y) with the anti-mask y) yields a 3 × 3 array of delta SPSFs. The A3(x, y) A3(x, antimask array of size 174 × 174 is required to be rescaled
FIG. 6. (a)A schematic layout of a coded aperture experimental setup that determines the FoV. (b) A schematic layout that determines the spatial resolution Ro of the coded aperture system.
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d1 d1 + PD , Ro = c 1+ d2 d2 FoV =
D.d1 − M(d1 +d2 ) , d2
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(8)
(9)
where D is the width of the CCD, M is the coded aperture pattern size, c is the hole size used in the coded aperture, PD is the CCD pixel size, d1 is the object to coded aperture distance, and d2 is the coded aperture to CCD distance. The resolution Ro is determined by the sum of the angle subtended by the aperture hole and CCD pixel with respect to the object. Given the CCD pixel size to be smaller than the coded aperture hole size c, the resolution is mainly determined from the mask hole size and the magnification (d1 /d2 ) used.
IV. EXPERIMENTS AND RESULTS
We have conducted coded aperture imaging by developing an experimental setup that allows imaging of fluorescent Cu K alpha X-rays (8048 eV) emitted uniformly from various shaped samples. The coded aperture consists of 421–20-μm circular holes on a 58 × 58 NTHT MURA pattern of size 5.8 mm square laser drilled in a 50-μm thick tantalum sheet, see Fig. 7. The detector used, is an Andor Ikon-M CCD detector of array size 1k × 1k and with a pixel size of 13 μm. Fig. 8 shows the schematic of the experimental coded aperture setup. The sample in this case consists of a simple shaped aperture in 50 μm tantalum foil that is illuminated from behind by a large area Cu Kα source. The fluorescent Cu Kα was generated by illuminating a copper sheet with 12 KeV x-rays from beamline 8.3.2 of the Advanced Light Source.11 The large area Cu Kα source (∼40 mm2 ) was used to ensure that the hole shaped object in the tantalum foil was uniformly illuminated from a wide range of angles to ensure the coded aperture was in turn uniformly illuminated. The detector dimensions (13 × 13 mm) were larger than the mask, so for the appropriate distances (object-aperture d1 = 70 mm and
FIG. 7. A magnified photograph of the 58 × 58 NTHT used in the coded aperture imaging of fluorescent x-rays. It is a 5.8 mm sided square pattern of 421–20-μm laser drilled holes in 50 μm thick tantalum sheet.
FIG. 8. Schematic of the experimental setup used for the imaging of shaped apertures when illuminated with from behind with Cu Kα radiation.
aperture-detector d2 = 40 mm) we obtain a numerical aperture NA of ∼0.04, a magnification M of 0.57 over a field of view FoV of 6.8 × 6.8 mm. A spatial resolution of 77.75 μm was calculated given the simple geometry of distances, coded aperture hole size and CCD pixel size as depicted by the geometric layout of Fig. 6(b), which indicates that the spatial resolution is significantly affected by the hole size used. We used 20 μm holes on the 100 μm pixel spacing of the NTHT pattern. Note that using smaller holes than the NTHT pixel size improves spatial resolution at the expense of throughput. For comparison the 20 μm holes yield an estimated 78 μm spatial resolution with 0.4% open area, whereas if 100 μm square holes were used the resolution would be 298 μm with an open area of 12.5%. When the transparent elements of the coded aperture were laser milled, defects can occur, which can lead to distortion and reconstruction artifacts in the decoded image. To compensate for errors in the fabricated mask compared to the theoretical mask, a uniformly rear illuminated 200 μm pinhole object was used to project an image of the mask onto the CCD. As such the positional values of the holes in the actual mask were used in the reconstruction algorithms. Fig. 9 shows the reconstructed images for various shaped objects of increasing complexity. These objects consist of a single 200 μm pinhole, and a series of increasingly larger triangular shapes of dimensions 0.3 mm × 0.7 mm × 1.17 mm, 1 mm × 1.4 mm × 1.7 mm, and 2.1 mm × 2.2 mm × 3 mm, respectively. Coded images were collected in the single photon mode, where a detected x-ray is treated as a binary event. Dependent on the flux from the sample, typically 10–500 single photon binary events were accumulated in a typical 10 s exposure. With 106 detector pixels most of the frames are zeros ensuring that detector pile up was not an issue. Typically 100 separate 10 s frames were recorded and
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FIG. 10. Using the slanted edge method, the oversampled Modulation transfer function (MTF) as a function of cycle/pixel measured from the reconstructed images for (a) the triangle of dimensions 0.3 mm × 0.7 mm × 1.17 mm and (b) the triangle of dimensions 1 mm × 1.4 mm × 1.7 mm.
FIG. 9. Reconstructed images when Cu Ka radiation uniformly illuminate various shaped objects that were drilled in a 50 μm thick tantalum sheet: (a) a single 200-μm pinhole, (b) a small triangle of dimensions 0.3 mm × 0.7 mm × 1.2 mm, (c) medium triangle of dimensions 1 mm × 1.4 mm × 1.6 mm, and (d) a large triangle of dimensions 2.1 mm × 2.2 mm × 2 mm, (e) summed images on the CCD for pinhole object (a), (f) summed images on CCD for large triangle object (d).
summed afterwards to yield the coded images shown in Figures 9(e) and 9(f) for the pinhole and large triangle object, respectively. Peak counts in the summed coded images were typically ∼10–20/pixel. The object shapes from the experimental data were faithfully reconstructed using the mask duplication, anti-mask derivation, magnification, and rescaling method described earlier for the ray traced simulated imaging. Imaging fidelity is respectable for these simple objects even as the coded images for the large triangle case overlap on the detector plane. The low count rate of the coded image of only 10–20 counts/pixel would seem to imply noisy reconstructed images, but the additive nature of the reconstruction technique by virtue essentially summing 421 separate pinhole camera images compensates for this. The spatial resolution and performance of this x-ray imaging system can be characterized by the modulation transfer function (MTF), which is a measurement of the coded aperture system’s ability to transfer contrast from the x-ray fluorescent sample to the CCD detector. MTF measurements were performed by the slanted straight edge method that was implemented with ImageJ.12, 13 Fig. 10 represents the MTF plots of the straight edges of the triangular decoded
images (b) and (c) of Fig. 9. The MTF presentation is inversely related to the spatial frequency (cycle/pixel), and the resolution is typically defined for a MTF of 50/%, which corresponds to 5.5 and 6.2 pixels for Figs. 9(b) and 9(c), respectively. Given the 13 μm detector pixel size, the spatial resolutions obtained from these reconstructed images (b) and (c) of Fig. 9 are 72 and 81 μm, respectively. These experimental spatial resolutions derived from the decoded images agreed with the geometrical spatial resolution derived from the geometry layout. Another coded aperture imaging experiment was conducted at the micro-tomography beamline to demonstrate the achromatic nature of the coded aperture imaging technique. The shaped triangular hole was replaced by shaped copper and germanium metal samples that were affixed to x-ray transparent Kapton tape. Fig. 11 shows the experimental
FIG. 11. Schematic layout of coded aperture imaging system that was used to image fluorescent x-rays emitted from Ge-Cu sample when irradiated with x-rays.
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V. CONCLUSION
FIG. 12. (a) Radiography image of a sample, which consists of 0.9 mm × 1.4 mm × 1.67 mm copper triangle and 1.1 mm × 1.5 mm × 1.86 mm germanium triangle that emitted fluorescent x-rays, when irradiated by the excitation synchrotron radiation (12 keV). (b) Reconstructed image (200 × 200) of fluorescent x-rays emitted from the Ge-Cu sample when radiated with 12 keV.
arrangement whereby the two-element sample is front irradiated with monochromatic radiation and the fluorescent x-rays imaged by the coded aperture and CCD detector. Fig. 12(a) shows the x-ray radiograph image of the two-element sample that consists of a 0 .9 mm × 1.4 mm × 1.67 mm copper triangle and a 1.1 mm × 1.5 mm × 1.86 mm germanium triangle. The distances for object-mask d1 and mask detector d2 are 50 mm and 40 mm, respectively. In this configuration, the Numerical Aperture NA is ∼ 0.058, the magnification M is 0.8 over a field of view (FoV) of 6.39 mm × 6.39 mm. A spatial resolution for this geometric configuration is about 61.25 μm. The two-elements of the sample can radiate their respective fluorescence x-rays – namely Cu Ka (8.047 keV) and Ge (11 keV) which are imaged by the coded aperture scheme independent of their energy. The decoded image is shown in Fig. 12(b) for illumination with x-ray of energy 12 keV that is chosen to be above the fluorescing K edge of the copper and germanium. To estimate the experimental spatial resolution, the MTF plot from this x-ray image is represented in Fig. 13 as a function of cycle/pixel. The 50% of the MTF corresponds to 4.5 pixels (∼59 μm) which is again agreed with the geometrical spatial resolution.
FIG. 13. Modulation transfer function (MTF) as a function of cycle/pixel measured from the reconstructed image Fig. 12(b).
The algorithms required to fabricate the self-supporting coded aperture pattern NTHT were developed. A ray tracing technique was developed that allowed for the modeling of the experimental setup involving magnification of the coded aperture onto the detector. The reconstruction code was developed and consisted of mask padding by replication, antimask derivation, followed by magnification, rescaling and 2D Fourier reconstruction techniques. A 58 × 58 NTHT coded aperture was fabricated with 421, 20 μm holes on a 5.8 mm × 5.8 mm array to yield a field of view for two experiments of 6.82 mm × 6.82 mm and 6.39 mm × 6.39 mm, respectively, with a 13 mm × 13 mm CCD detector. The experimental value of spatial resolution derived from the reconstructed images was ∼60–80 μm for the two field of views used. The numerical aperture was 0.04–0.058 with a coded aperture throughput of 0.4%. The reconstruction codes were verified experimentally for a range of sample shapes and were established as robust. An example of achromatic imaging from a sample emitting two different x-ray energies was demonstrated. VI. FUTURE DEVELOPMENTS
This initial work indicates that x-rays can be achromatically imaged with a large NA coded aperture camera. The next phase of this project would be the replacement of this NTHT (0.4% open area) by a patterned self-supporting thin film MURA with 50% open area, which may be expected to improve throughput ∼×125. The poor 50–70 μm spatial resolution can be improved ∼×10 by the use of smaller holes in the coded aperture, and a smaller CCD pixel size. For example, the geometrical considerations described earlier indicate that 5 μm mask holes and 5 μm detector pixels yield a spatial resolution of 7.5 μm with an NA = 0.3, sample to mask distance = 5 mm, and mask to detector = 20 mm, field of view = 2.25 mm. For the x-ray energies considered here (6–25 KeV) the resolution is still limited to several microns due to limitations of fabricating holes with a high aspect ratio where the mask material is expected to be thick and absorbing. In the present case the use of 50 μm thick tantalum, requires a 10:1 aspect ratio for 5 μm holes which is practical but challenging. High aspect ratio holes will also lead to vignetting which will complicate the reconstruction algorithm. This can partly be compensated by orientation of the holes toward the object center such that the holes are not perpendicular to the mask surface. Note that the MURA pattern consists mainly of larger holes so will be less likely affected by vignetting compared to the NTHT pattern. It is noteworthy that the xray energy determines the mask thickness which will determine the hole size and thus resolution. The 50 μm thick tantalum used to date achieves 99% absorbance at 20 KeV which was arbitrarily chosen as reasonable. Signal passing through the dark areas of the mask will contribute to image noise and requires a more detailed analysis. For x-rays greater than 20 KeV, thicker tantalum mask will be required with subsequent larger holes and poorer resolution. For softer x-rays, thinner masks can be used. For instance at 2500 eV, 1 μm
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thick gold with 99% absorbance would allow for 0.1 μm mask holes, but given the limiting size of typical current CCD pixels at ∼5 μm the resolution remains in the 1–2 μm range even for the soft rays. It is worthwhile determining the typical count rates one might expect from a fluorescing sample with this MURA detector. For example in the case of irradiating a 10 μm sized contaminant of uranium irradiated with x-rays of energy of 20 keV, 80% of this energy is absorbed of which 50% is reirradiated in the form of Lα fluorescent x-rays. Given a 50% open area for a 0.3 NA MURA coded aperture the camera accepts 0.15 steradian. The flux density at the synchrotron is around ∼105 hν/s/μm2 , the 10 μm thick uranium absorbs ∼8 × 106 hν, and re-irradiates ∼4 × 106 hν/s into 4π steradian. Flux on the CCD is predicted to be ∼5 × 104 hν/s. In this case it will only require a few seconds to obtain a respectable x-ray coded image. This time scale is comparable with other scanning probe imaging techniques but the latter tend to have superior resolution than the few micron resolution that the coded aperture imaging can be predicted to reach. Other aspects of coded aperture detectors are the energy resolving capabilities of cooled CCD’s that can be exploited to yield an energy resolving imaging camera. In the case of an x-ray fluorescing sample this would allow for elemental imaging. If the sample is suitably transparent to its own fluorescent x-rays a 3D image of the sample could be built up using sheet illumination. Alternatively if multiple coded aperture detectors are positioned on an arc around a sample that is flooded with wide field illumination then by correlating the separate camera images it should be possible to reconstruct the sample elemental 3D map with accommodation made for sample absorption. Such a 3D elemental imaging instrument could have its uses as a tomography camera but would still suffer from the low resolution of a few microns due to hole and detector pixel size. Still the simplicity of the device being just a mask and CCD allows for niche uses such a beam position and profiling camera.6 Finally we note that the artifacts associated with the Fourier reconstruction algorithms can be alleviated by devel-
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opment of the new iterative reconstruction techniques becoming available.14 ACKNOWLEDGMENTS
The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. The laser miller is supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences under NSF Cooperative Agreement EAR 06-49658. 1 L.
Mertz and N. O. Young, “Fresnel transformations of images,” Proceedings of International Conference on Optical Instruments and Techniques (Chapman and Hall, London 1961), p. 305. 2 S. S. Holt, “Temporal x-ray astronomy with a pinhole camera,” Astrophys. Space Sci. 42, 123 (1976). 3 Yu. P. Kazachkov, D. S. Semenov, and. N. P. Goryacheva, “Application of coded apertures in medical gamma ray cameras,” Instrum. Exp. Tech. 50, 267–274 (2007). 4 Y.-W. Chen and K. Kishimoto, “Tomography resolution of uniformly redundant arrays coded aperture,” Rev. Sci. Instrum. 74, 2232–2235 (2003). 5 Neutron Imaging and Applications, edited by I. S. Anderson, R. McGreevy, and H. Z. Bilheux (Springer, US, 2009). 6 A. Kachatkou, N. Kyele, P. Scott, and R. van Silfhout, “In situ X-ray beam imaging using an off-axis magnifying coded aperture camera system,” J. Synchrotron Radiat. 20, 596–602 (2013). 7 E. Caroli, J. B. Steven, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x-ray and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987). 8 R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res., Sect. A 474(3), 273–284 (2001). 9 C. Brown, “Multiplex imaging with multiple-pinhole cameras,” J. Appl. Phys. 45, 1806 (1974). 10 S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28(20), 4344–4352 (1989). 11 A. A. MacDowell, D. Y. Parkinson, A. Haboub, E. Schaible, J. R. Nasiatka, C. A. Yee, J. R. Jameson, J. B. Ajo-Franklin, C. R. Brodersen, and A. J. McElron, “X-ray micro-tomography at the advanced light source,” Proc. SPIE 8506, 850618 (2012). 12 M. Carles and J. Escofet, “Medida de la MTF en cameras fotograficas digitales,” 7th Reunion National de optica, Universidad de Cantabria, Santander, 2003. 13 G. D. Boreman, Modulation Transfer Function in Optical and Electrooptical Systems (SPIE Press, Bellingham, Washington, 2001). 14 C. Lim and B. Yu, “Estimation stability with cross validation (ESCV),” preprint arXiv:1303.3128v1 [stat.ME] (2013).
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 063704 (2014)
Coded aperture imaging for fluorescent x-rays A. Haboub,1,2 A. A. MacDowell,1 S. Marchesini,1 and D. Y. Parkinson1 1 2
Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA Lincoln University, Life and Physical Sciences Department, Jefferson City, Missouri 65101, USA
(Received 20 November 2013; accepted 28 May 2014; published online 19 June 2014) We employ a coded aperture pattern in front of a pixilated charge couple device detector to image fluorescent x-rays (6–25 KeV) from samples irradiated with synchrotron radiation. Coded apertures encode the angular direction of x-rays, and given a known source plane, allow for a large numerical aperture x-ray imaging system. The algorithm to develop and fabricate the free standing No-Two-Holes-Touching aperture pattern was developed. The algorithms to reconstruct the x-ray image from the recorded encoded pattern were developed by means of a ray tracing technique and confirmed by experiments on standard samples. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4882337] I. INTRODUCTION
In recent years, with the advances in computer speed, image processing, and charge couple device (CCD) detector technology, increasing interest has been shown in coded aperture imaging in x-ray and γ -ray fields, namely astronomical applications,1, 2 medical physics,3 plasma imaging,4 neutron imaging,5 and synchrotron beam imaging.6 Coded aperture imaging is an imaging technique without conventional lenses and was first suggested in 1961 for use in x-ray astronomy cameras.1, 7 This imaging technique involves the use of a coded aperture pattern or mask that consists of a known arrangement of transparent and opaque elements. The mask is placed between the x-ray source and a suitable area sensitive detector. Transparent elements in the mask act as individual pinhole cameras that together produce multiple overlapping images of the source on the detector. Given knowledge of the coded aperture pattern, the object can be reconstructed from the recorded multiple overlapped images. This imaging technique improves the signal noise ratio (SNR) efficiency √ of the system over a single pinhole imaging camera by ∼ N where N is the number of pinholes. Numerous coded aperture patterns have been used in x-ray and γ -ray imaging, namely Fresnel zone plate, random arrays, uniform redundant arrays (URA), modified uniform redundant arrays (MURA), and No-Two-Hole-Touching (NTHT) based on MURA.7, 8 To show the fundamental idea of the coded aperture imaging, a simulation illustration is shown in Figure 1, which compares the imaging of an object with both pinhole camera and coded aperture pattern. The single small pinhole camera shown in Fig. 1(a) generates a faithful image; however, it only allows a small amount of light to be transmitted and so the image is of low intensity. A larger pinhole shown in Fig. 1(b) increases the transmitted light but leads to degradation in the image resolution. Fig. 1(c) is the case of several discrete pinholes and results in identifiable overlapping images. For an appropriate coded aperture pattern shown in Fig. 1(d), the overlapping images become more of a continuum requiring a computational reconstruction method to decode the image while retaining the image quality of the single pinhole but with the high light gather0034-6748/2014/85(6)/063704/8/$30.00
ing power of the multiple pinhole arrays. As such the unique advantages of coded the aperture imaging technique is its simplicity (a mask + CCD) and its potential high angular acceptance of x-rays for imaging. This technique can have a high numerical aperture (NA) ∼0.1 as limited by geometry. This is
FIG. 1. Pinhole camera and coded aperture imaging simulation for a small pinhole, large pinhole, multiple small pinholes, and a coded aperture pattern.
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in contrast to the more customary-ray imaging systems such as grazing incidence Kirkpatrick-Baez Mirrors, Fresnel zone plates, and refractive lenses that are limited to small NA typically ∼0.001. In this paper, we present the development of the algorithm and the fabrication of the NTHT self-supporting coded aperture pattern – based on the Modified Uniformly Redundant Array (MURA) as well as the algorithm to reconstruct x-ray images recorded by a NTHT MURA coded aperture with CCD camera. The imaging fidelity and spatial resolution is established. The paper reports initial experimental results as a verification of the model and describes future work for the next development stage whereby an energy resolving pixilated CCD can be deployed that will allow elemental imaging.
II. THEORETICAL REVIEW AND SIMULATION
Coded aperture imaging is a two-step process. The first step is the generation of overlapping images or a coded image when photons are being emitted from a source, pass through the transparent elements of the coded aperture pattern and projected onto the detector. The second step is the decoding or the reconstruction of the image through a suitable reconstruction method. If we denote O(x, y) as a two dimen-
sional object parallel to the coded aperture pattern whose transmission function is A(x, y) being a binary array that contains “1” for each transparent element and “0” for each opaque element, then the coded image I(x, y) of the object that is projected onto the detector is given by the convolution operation, I(x, y) = O(x, y) ⊗ A(x, y),
(1)
where ⊗ is the convolution operator. The reconstructed image O (x, y) can be obtained through the convolution operation of the coded image I(x, y) by a suitable decoding function or anti-mask D(x, y),7, 8 O (x, y) = I(x, y) ⊗ D(x, y).
(2)
For an ideal imaging reconstruction, the convolution of the two functions D(x, y) and A(x, y) is the system point spread function (SPSF) that is ideally a delta function,7–9 D(x, y) ⊗ A(x, y) = δ.
(3)
The MURA10 shown in Fig. 2(a) is one of the most efficient coded aperture patterns, having 50% transparent area, and its SPSF is a perfect delta function. The MURA pattern is constructed on a square lattice whose side length p is prime; each element of the MURA A(x, y) where x and y run from 0 to p − 1, is
⎧ 0 if x = 0 ⎪ ⎪ ⎪ ⎪ if x = 0, x = 0 ⎨1 if x and y are quadratic residues modulo p . A(x, y) = 1 ⎪ ⎪ 1 if neither x nor y are quadratic residues modulo p ⎪ ⎪ ⎩ 0 otherwise
However for the 6–25 KeV x-rays that we are interested in, the transparent holes are preferred to be actual holes which makes the MURA pattern to be not self-supporting pattern and difficult to fabricate. An alternative pattern, based on the MURA is a NTHT pattern shown in Fig. 2(b) which is a selfsupporting pattern and can thus be easily fabricated. This pat-
tern consists of inserting rows and columns of closed position “0” around each open position “1” of the original MURA. Fig. 2(b) shows the 58 × 58 NTHT pattern where only one row and one column are inserted around each open position of the original 29 × 29 MURA, and therefore it presents an open fraction area of 12.5%. We have carried out computer simulations using Matlab code to study the performance of coded aperture imaging using the NTHT pattern. Fig. 3 illustrates a coded aperture imaging simulation for the 58 × 58 NTHT. The simulated object O(x, y) is a “happy face,” which is a 58 × 58 binary array. The circular convolution of the two dimensional object O(x, y) and the array transmission function A(x, y) (58 × 58 NTHT), is the coded image I(x, y) of 58 × 58 in size. The decoding array D(x, y), which is of 58 × 58 in size, is derived through 2D Fast Fourier Transforms (FFT2),7 D(x, y) = iFFT2(1/FFT2)(A(x, y)),
FIG. 2. Coded aperture patterns: (a) 29 × 29 Modified Uniformly Redundant Array (MURA) and (b) 58 × 58 No Two Holes Touching (NTHT) based on MURA pattern.
(4)
where FFT2 and iFFT2 are, respectively, the fast 2D Fourier transform and its inverse. Circularly convolving A(x, y) with D(x, y) yields the SPSF that is a delta function. Circularly convolving the coded image I(x, y) with the decoding array
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FIG. 4. (a) Schematic diagram illustrating the Ray Tracing Technique (RTT) used in the coded aperture imaging simulation. Rays are traced from the middle of each object pixel though the center of each open coded aperture pixel and projected onto the detector plane. (b) The plot of the object coordinates, (c) the plot of the mask coordinates, and (d) the plot of the coded image coordinates.
FIG. 3. Coded aperture imaging simulation for the 58 × 58 NTHT MURA. The Object O(x, y) is circularly convoluted with the NTHT MURA, A(x, y) to yield the coded image I(x, y) which when circularly convoluted with the decoding pattern D(x, y) yields the reconstructed image O (x, y) The circular convolution of NTHT MURA A(x, y) with the decoding pattern D(x, y) is required to yield a delta function.
D(x, y) yields the reconstructed image O (x, y) by means of 2D Fast Fourier Transforms: O (x, y) = iFFT2(FFT2(I(x, y) ∗ FFT2(D(x, y))).
(5)
The image reconstruction turns out to be mathematically simple for this case where the object, coded aperture, and de-
tector are all of identical sized arrays. This is the situation for x-ray astronomical applications as the object is at infinity so magnification of the coded aperture onto the area detector is unity allowing the mask and detector to be the same size. When the object is closer than infinity, the magnification of the coded aperture onto the detector is >1 and the correlation of matrices is more complicated requiring new reconstruction formulations to be developed. For this reason, we have modeled the behavior of coded aperture imaging by means of a Ray Tracing Technique (RTT) to simulate coded images from nearby samples which involve magnification. We then tested the deconvolution procedures to reproduce the original object. This ray tracing consists of Matlab code which simulates the arrangement of the experimental setup to include the source, the coded aperture pattern and detector. The system is raytraced with rays from the middle of each object pixel passing through the middle of each pinhole of the mask and projected onto the detector, as illustrated in Fig. 4. The RTT yields a realistic coded image on the detector as shown in Fig. 4(d). Note that the image size in the detector plane is larger than the object and coded aperture. Using the simulated image, the reconstruction procedure described below was developed that deals with the different array sizes. The adopted deconvolution solution involves padding and 3 × 3 replication as shown in Fig. 5. For the case of the object O(x, y) of 60 × 60 in size, which consists of several numerals, the rays are projected by ray tracing through a 58 × 58 NTHT onto a 1024 × 1024 detector plane. This 1k × 1k image is padded with zeros to yield the 4096 × 4096 coded image I(x, y) where the coded image data occupies the central region. The mask also undergoes padding by replication and tiling of the basic 58 × 58 NTHT pattern to form a 3 × 3 pattern – A3(x, y) of size 174 × 174. The derivation of
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y). To rescale A3(x, y), to the 4096 × 4096 antimask B3(x, a point p(x, y) in the object plane center is raytraced through y) and projected onto a 4096 the middle of each pixel of A3(x, × 4096 array. All pixels are zero unless raytracing indicates y) contains only 174 × 174 pixels made up otherwise. A3(x, y) of +1, 0, and −1 values, so the vast majority of B3(x, pixels are zero as indicate in Fig. 5. The reconstructed image O (x, y) is then the circular convolution of this decod y) and the coded image I(x, y) ing function B3(x, y)))). O (x, y) = iFFT2(FFT2(I(x, y). ∗ FFT2(B3(x,
(7)
The reconstructed images do suffer from some high frequency artifacts typical of Fourier reconstruction techniques; however, the central image shows the lowest amount of artifacts with the object faithfully reproduced. These artifacts do not depend on the statistical quality of the data; they are caused mainly by the system point spread function (SPSF) which is derived through the fast Fourier transformation technique.
III. GEOMETRY OF CODED APERTURE IMAGING
From the simple geometry of the coded aperture camera setup as shown in Fig. 6, various geometrical relationships exist that determine the field of view (FoV) and resolution
FIG. 5. Coded aperture imaging simulation using ray tracing technique (RTT) followed by the newly developed reconstruction technique that pads y) and and replicates the mask to yield A3(x, y), derives its anti-mask A3(x, y). through RTT magnifies and rescales it to yield the decoding array B3(x, This decoding array is of size (3k × 3k) obtained by ray tracing the object y) and projected onto center p(x, y) through the middle of each pixel of A3(x, the detector plane.
y) is done through the 2D FFT technique the anti-mask, A3(x, noted earlier, y) = iFFT2(1/FFT2(A3(x, y)). A3(x,
(6)
The circular convolution of A3(x, y) with the anti-mask y) yields a 3 × 3 array of delta SPSFs. The A3(x, y) A3(x, antimask array of size 174 × 174 is required to be rescaled
FIG. 6. (a)A schematic layout of a coded aperture experimental setup that determines the FoV. (b) A schematic layout that determines the spatial resolution Ro of the coded aperture system.
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d1 d1 + PD , Ro = c 1+ d2 d2 FoV =
D.d1 − M(d1 +d2 ) , d2
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(8)
(9)
where D is the width of the CCD, M is the coded aperture pattern size, c is the hole size used in the coded aperture, PD is the CCD pixel size, d1 is the object to coded aperture distance, and d2 is the coded aperture to CCD distance. The resolution Ro is determined by the sum of the angle subtended by the aperture hole and CCD pixel with respect to the object. Given the CCD pixel size to be smaller than the coded aperture hole size c, the resolution is mainly determined from the mask hole size and the magnification (d1 /d2 ) used.
IV. EXPERIMENTS AND RESULTS
We have conducted coded aperture imaging by developing an experimental setup that allows imaging of fluorescent Cu K alpha X-rays (8048 eV) emitted uniformly from various shaped samples. The coded aperture consists of 421–20-μm circular holes on a 58 × 58 NTHT MURA pattern of size 5.8 mm square laser drilled in a 50-μm thick tantalum sheet, see Fig. 7. The detector used, is an Andor Ikon-M CCD detector of array size 1k × 1k and with a pixel size of 13 μm. Fig. 8 shows the schematic of the experimental coded aperture setup. The sample in this case consists of a simple shaped aperture in 50 μm tantalum foil that is illuminated from behind by a large area Cu Kα source. The fluorescent Cu Kα was generated by illuminating a copper sheet with 12 KeV x-rays from beamline 8.3.2 of the Advanced Light Source.11 The large area Cu Kα source (∼40 mm2 ) was used to ensure that the hole shaped object in the tantalum foil was uniformly illuminated from a wide range of angles to ensure the coded aperture was in turn uniformly illuminated. The detector dimensions (13 × 13 mm) were larger than the mask, so for the appropriate distances (object-aperture d1 = 70 mm and
FIG. 7. A magnified photograph of the 58 × 58 NTHT used in the coded aperture imaging of fluorescent x-rays. It is a 5.8 mm sided square pattern of 421–20-μm laser drilled holes in 50 μm thick tantalum sheet.
FIG. 8. Schematic of the experimental setup used for the imaging of shaped apertures when illuminated with from behind with Cu Kα radiation.
aperture-detector d2 = 40 mm) we obtain a numerical aperture NA of ∼0.04, a magnification M of 0.57 over a field of view FoV of 6.8 × 6.8 mm. A spatial resolution of 77.75 μm was calculated given the simple geometry of distances, coded aperture hole size and CCD pixel size as depicted by the geometric layout of Fig. 6(b), which indicates that the spatial resolution is significantly affected by the hole size used. We used 20 μm holes on the 100 μm pixel spacing of the NTHT pattern. Note that using smaller holes than the NTHT pixel size improves spatial resolution at the expense of throughput. For comparison the 20 μm holes yield an estimated 78 μm spatial resolution with 0.4% open area, whereas if 100 μm square holes were used the resolution would be 298 μm with an open area of 12.5%. When the transparent elements of the coded aperture were laser milled, defects can occur, which can lead to distortion and reconstruction artifacts in the decoded image. To compensate for errors in the fabricated mask compared to the theoretical mask, a uniformly rear illuminated 200 μm pinhole object was used to project an image of the mask onto the CCD. As such the positional values of the holes in the actual mask were used in the reconstruction algorithms. Fig. 9 shows the reconstructed images for various shaped objects of increasing complexity. These objects consist of a single 200 μm pinhole, and a series of increasingly larger triangular shapes of dimensions 0.3 mm × 0.7 mm × 1.17 mm, 1 mm × 1.4 mm × 1.7 mm, and 2.1 mm × 2.2 mm × 3 mm, respectively. Coded images were collected in the single photon mode, where a detected x-ray is treated as a binary event. Dependent on the flux from the sample, typically 10–500 single photon binary events were accumulated in a typical 10 s exposure. With 106 detector pixels most of the frames are zeros ensuring that detector pile up was not an issue. Typically 100 separate 10 s frames were recorded and
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FIG. 10. Using the slanted edge method, the oversampled Modulation transfer function (MTF) as a function of cycle/pixel measured from the reconstructed images for (a) the triangle of dimensions 0.3 mm × 0.7 mm × 1.17 mm and (b) the triangle of dimensions 1 mm × 1.4 mm × 1.7 mm.
FIG. 9. Reconstructed images when Cu Ka radiation uniformly illuminate various shaped objects that were drilled in a 50 μm thick tantalum sheet: (a) a single 200-μm pinhole, (b) a small triangle of dimensions 0.3 mm × 0.7 mm × 1.2 mm, (c) medium triangle of dimensions 1 mm × 1.4 mm × 1.6 mm, and (d) a large triangle of dimensions 2.1 mm × 2.2 mm × 2 mm, (e) summed images on the CCD for pinhole object (a), (f) summed images on CCD for large triangle object (d).
summed afterwards to yield the coded images shown in Figures 9(e) and 9(f) for the pinhole and large triangle object, respectively. Peak counts in the summed coded images were typically ∼10–20/pixel. The object shapes from the experimental data were faithfully reconstructed using the mask duplication, anti-mask derivation, magnification, and rescaling method described earlier for the ray traced simulated imaging. Imaging fidelity is respectable for these simple objects even as the coded images for the large triangle case overlap on the detector plane. The low count rate of the coded image of only 10–20 counts/pixel would seem to imply noisy reconstructed images, but the additive nature of the reconstruction technique by virtue essentially summing 421 separate pinhole camera images compensates for this. The spatial resolution and performance of this x-ray imaging system can be characterized by the modulation transfer function (MTF), which is a measurement of the coded aperture system’s ability to transfer contrast from the x-ray fluorescent sample to the CCD detector. MTF measurements were performed by the slanted straight edge method that was implemented with ImageJ.12, 13 Fig. 10 represents the MTF plots of the straight edges of the triangular decoded
images (b) and (c) of Fig. 9. The MTF presentation is inversely related to the spatial frequency (cycle/pixel), and the resolution is typically defined for a MTF of 50/%, which corresponds to 5.5 and 6.2 pixels for Figs. 9(b) and 9(c), respectively. Given the 13 μm detector pixel size, the spatial resolutions obtained from these reconstructed images (b) and (c) of Fig. 9 are 72 and 81 μm, respectively. These experimental spatial resolutions derived from the decoded images agreed with the geometrical spatial resolution derived from the geometry layout. Another coded aperture imaging experiment was conducted at the micro-tomography beamline to demonstrate the achromatic nature of the coded aperture imaging technique. The shaped triangular hole was replaced by shaped copper and germanium metal samples that were affixed to x-ray transparent Kapton tape. Fig. 11 shows the experimental
FIG. 11. Schematic layout of coded aperture imaging system that was used to image fluorescent x-rays emitted from Ge-Cu sample when irradiated with x-rays.
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V. CONCLUSION
FIG. 12. (a) Radiography image of a sample, which consists of 0.9 mm × 1.4 mm × 1.67 mm copper triangle and 1.1 mm × 1.5 mm × 1.86 mm germanium triangle that emitted fluorescent x-rays, when irradiated by the excitation synchrotron radiation (12 keV). (b) Reconstructed image (200 × 200) of fluorescent x-rays emitted from the Ge-Cu sample when radiated with 12 keV.
arrangement whereby the two-element sample is front irradiated with monochromatic radiation and the fluorescent x-rays imaged by the coded aperture and CCD detector. Fig. 12(a) shows the x-ray radiograph image of the two-element sample that consists of a 0 .9 mm × 1.4 mm × 1.67 mm copper triangle and a 1.1 mm × 1.5 mm × 1.86 mm germanium triangle. The distances for object-mask d1 and mask detector d2 are 50 mm and 40 mm, respectively. In this configuration, the Numerical Aperture NA is ∼ 0.058, the magnification M is 0.8 over a field of view (FoV) of 6.39 mm × 6.39 mm. A spatial resolution for this geometric configuration is about 61.25 μm. The two-elements of the sample can radiate their respective fluorescence x-rays – namely Cu Ka (8.047 keV) and Ge (11 keV) which are imaged by the coded aperture scheme independent of their energy. The decoded image is shown in Fig. 12(b) for illumination with x-ray of energy 12 keV that is chosen to be above the fluorescing K edge of the copper and germanium. To estimate the experimental spatial resolution, the MTF plot from this x-ray image is represented in Fig. 13 as a function of cycle/pixel. The 50% of the MTF corresponds to 4.5 pixels (∼59 μm) which is again agreed with the geometrical spatial resolution.
FIG. 13. Modulation transfer function (MTF) as a function of cycle/pixel measured from the reconstructed image Fig. 12(b).
The algorithms required to fabricate the self-supporting coded aperture pattern NTHT were developed. A ray tracing technique was developed that allowed for the modeling of the experimental setup involving magnification of the coded aperture onto the detector. The reconstruction code was developed and consisted of mask padding by replication, antimask derivation, followed by magnification, rescaling and 2D Fourier reconstruction techniques. A 58 × 58 NTHT coded aperture was fabricated with 421, 20 μm holes on a 5.8 mm × 5.8 mm array to yield a field of view for two experiments of 6.82 mm × 6.82 mm and 6.39 mm × 6.39 mm, respectively, with a 13 mm × 13 mm CCD detector. The experimental value of spatial resolution derived from the reconstructed images was ∼60–80 μm for the two field of views used. The numerical aperture was 0.04–0.058 with a coded aperture throughput of 0.4%. The reconstruction codes were verified experimentally for a range of sample shapes and were established as robust. An example of achromatic imaging from a sample emitting two different x-ray energies was demonstrated. VI. FUTURE DEVELOPMENTS
This initial work indicates that x-rays can be achromatically imaged with a large NA coded aperture camera. The next phase of this project would be the replacement of this NTHT (0.4% open area) by a patterned self-supporting thin film MURA with 50% open area, which may be expected to improve throughput ∼×125. The poor 50–70 μm spatial resolution can be improved ∼×10 by the use of smaller holes in the coded aperture, and a smaller CCD pixel size. For example, the geometrical considerations described earlier indicate that 5 μm mask holes and 5 μm detector pixels yield a spatial resolution of 7.5 μm with an NA = 0.3, sample to mask distance = 5 mm, and mask to detector = 20 mm, field of view = 2.25 mm. For the x-ray energies considered here (6–25 KeV) the resolution is still limited to several microns due to limitations of fabricating holes with a high aspect ratio where the mask material is expected to be thick and absorbing. In the present case the use of 50 μm thick tantalum, requires a 10:1 aspect ratio for 5 μm holes which is practical but challenging. High aspect ratio holes will also lead to vignetting which will complicate the reconstruction algorithm. This can partly be compensated by orientation of the holes toward the object center such that the holes are not perpendicular to the mask surface. Note that the MURA pattern consists mainly of larger holes so will be less likely affected by vignetting compared to the NTHT pattern. It is noteworthy that the xray energy determines the mask thickness which will determine the hole size and thus resolution. The 50 μm thick tantalum used to date achieves 99% absorbance at 20 KeV which was arbitrarily chosen as reasonable. Signal passing through the dark areas of the mask will contribute to image noise and requires a more detailed analysis. For x-rays greater than 20 KeV, thicker tantalum mask will be required with subsequent larger holes and poorer resolution. For softer x-rays, thinner masks can be used. For instance at 2500 eV, 1 μm
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thick gold with 99% absorbance would allow for 0.1 μm mask holes, but given the limiting size of typical current CCD pixels at ∼5 μm the resolution remains in the 1–2 μm range even for the soft rays. It is worthwhile determining the typical count rates one might expect from a fluorescing sample with this MURA detector. For example in the case of irradiating a 10 μm sized contaminant of uranium irradiated with x-rays of energy of 20 keV, 80% of this energy is absorbed of which 50% is reirradiated in the form of Lα fluorescent x-rays. Given a 50% open area for a 0.3 NA MURA coded aperture the camera accepts 0.15 steradian. The flux density at the synchrotron is around ∼105 hν/s/μm2 , the 10 μm thick uranium absorbs ∼8 × 106 hν, and re-irradiates ∼4 × 106 hν/s into 4π steradian. Flux on the CCD is predicted to be ∼5 × 104 hν/s. In this case it will only require a few seconds to obtain a respectable x-ray coded image. This time scale is comparable with other scanning probe imaging techniques but the latter tend to have superior resolution than the few micron resolution that the coded aperture imaging can be predicted to reach. Other aspects of coded aperture detectors are the energy resolving capabilities of cooled CCD’s that can be exploited to yield an energy resolving imaging camera. In the case of an x-ray fluorescing sample this would allow for elemental imaging. If the sample is suitably transparent to its own fluorescent x-rays a 3D image of the sample could be built up using sheet illumination. Alternatively if multiple coded aperture detectors are positioned on an arc around a sample that is flooded with wide field illumination then by correlating the separate camera images it should be possible to reconstruct the sample elemental 3D map with accommodation made for sample absorption. Such a 3D elemental imaging instrument could have its uses as a tomography camera but would still suffer from the low resolution of a few microns due to hole and detector pixel size. Still the simplicity of the device being just a mask and CCD allows for niche uses such a beam position and profiling camera.6 Finally we note that the artifacts associated with the Fourier reconstruction algorithms can be alleviated by devel-
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opment of the new iterative reconstruction techniques becoming available.14 ACKNOWLEDGMENTS
The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. The laser miller is supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences under NSF Cooperative Agreement EAR 06-49658. 1 L.
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