Artifact-less Coded Aperture Imaging in the X-ray Band with Multiple Different Random Patterns
Tomoaki Kasuga, Hirokazu Odaka, Kosuke Hatauchi, Satoshi Takashima, Tsubasa Tamba, Yuki Aizawa, Soichiro Hashiba, Aya Bamba, Yuanhui Zhou, Toru Tamagawa
aa r X i v : . [ a s t r o - ph . I M ] J u l Artifact-less Coded Aperture Imaging in the X-ray Band withMultiple Different Random Patterns
Tomoaki Kasuga a,* , Hirokazu Odaka a,b,c , Kosuke Hatauchi a , Satoshi Takashima a , TsubasaTamba a , Yuki Aizawa a , Soichiro Hashiba a,b , Aya Bamba a,b , Yuanhui Zhou d,e , ToruTamagawa d,e,f a Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo113-0033, Japan b Research Center for the Early Universe, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,Tokyo 113-0033, Japan c Kavli IPMU (WPI), UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan d RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan e Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan f RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Abstract.
The coded aperture imaging technique is a useful method of X-ray imaging in observational astrophysics.However, the presence of imaging noise or so-called artifacts in a decoded image is a drawback of this method. Wepropose a new coded aperture imaging method using multiple different random patterns for significantly reducing theimage artifacts. This aperture mask contains multiple different patterns each of which generates a different artifactdistribution in its decoded image. By summing all decoded images of the different patterns, the artifact distributionsare cancelled out, and we obtain a remarkably accurate image. We demonstrate this concept with imaging experimentsof a monochromatic 16 keV hard X-ray beam at the synchrotron photon facility SPring-8, using the combination of aCMOS image sensor and an aperture mask that has four different random patterns composed of holes with a diameterof 27 µ m and a separation of 39 µ m. The entire imaging system is installed in a 25 cm-long compact size, and achievesan angular resolution of < ′′ (full width at half maximum). In addition, we show by Monte Carlo simulation thatthe artifacts can be reduced more effectively if the number of different patterns increases to 8 or 16. Keywords:
X-ray, coded aperture, CMOS. * Tomoaki Kasuga, [email protected]
The coded aperture imaging is one of the useful methods for imaging X-rays. This method is em-ployed by the
Swift -BAT
4, 5 and the
INTEGRAL -IBIS and -SPI
6, 9 for astronomical observatories,and gamma-ray imaging systems on ground for nuclear medicine and nondestructive inspection.In the imaging process, we obtain the projection of an aperture pattern by a position-sensitive de-tector, and estimate the directional distribution of photon sources based on the projected pattern.In contrast to focused imaging using a multilayer coated mirror such as the
NuSTAR and the Hitomi -HXI, the coded aperture does not need a complicated optics and a long focal length. This1implicity allows us to build a compact imaging system that can be installed in an ultra-compactsatellite like a CubeSat.There are two major categories of the coded aperture patterns. One is a category of patternsthat have a nature with the characteristics of uniform redundancy. This is an application of thedifference set and the pseudo noise generation
3, 12 to computational imaging. Several patternshave been proposed historically: e.g., M-sequence, Uniformly Redundant Arrays (URA),
14, 15 andModified URA (MURA).
14, 16
Since these patterns make use of a property that the autocorrelationis the delta function, we should use periodic repetition of these patterns for realizing a modulooperation in the convolution operation in an image decoding process. Then the reconstructedimage in the fully coded field-of-view (FC-FoV) of this category of the patterns is completelyartifact-less except for statistical fluctuations.
3, 17
On the other hand, concerning the outside of theFC-FoV, which is named partially coded FoV (PC-FoV), we have eight strong artifacts and weakerbut structural ones in the PC-FoV due to the periodicity, when a photon source is in the FC-FoV.For example, the artifacts have at most half intensity of the source peak in the IBIS detector on
INTEGRAL , which employs the MURA 53 × A worse problem is that when the source islocated in the PC-FoV, confusing artifacts appear even in the FC-FoV. Such artifacts induceserious uncertainties in the reconstructed image.An alternative way is using a category of the random patterns. This pattern is coded by a two-dimensional array of random numbers sampled from {
0, 1 } , and we can straightforwardly obtainhigher randomness with a larger number of elements. In the pattern, artifacts in the decoded imageare generally not structural and are not remarkable. This is a clear advantage of the random pat-terns, and it is apparent in images obtained with the BAT detector on Swift , which adopts a randompattern composed of ∼
4, 5
However, artifacts can be distributed in the entire FoV2ncluding the FC-FoV, in which an on-axis primary target source may come for pointing observa-tion. This weak point of the random patterns becomes significant due to insufficient randomnesswhen the number of the coded aperture elements is small. Particularly for a narrow FoV configu-ration, the elements number should be limited and thus it is important to gain high randomness ofthe pattern under the restriction of the image size.The angular resolution is another important factor of the coded aperture imaging. Since ex-isting missions such as
Swift have been aimed at the all-sky survey of hard X-ray sources, theirinstruments have large FoVs by the sacrifice of the imaging resolution. They are typically 10 timesor more inferior to the focused imaging method like
NuSTAR , whose angular resolution is ′′ (FWHM). The angular resolution of the coded aperture imaging, in principle, can be improvedby adopting even finer element sizes of the coded apertures and the detector pixels.In this paper, we propose a new method to reduce artifacts that appear in images decoded withthe random pattern coded aperture. Section 2 describes the concept of our method , which simulta-neously uses multiple images with different random patterns. We demonstrate this concept by hardX-ray beam experiments using our compact-scale imaging system for narrow FoV observations,as described in Section 3. Section 4 shows the results of these experiments. Here, we also demon-strate that our coded aperture system realizes a fine imaging resolution comparable to
NuSTAR even in dimensional limitations of a compact-scale satellite of 25 cm. In Section 5, we also con-duct Monte Carlo simulations for further discussions and discuss practical issues in observations.We give our conclusions in Section 6. 3
Coded Aperture Imaging with Multiple Different Random Patterns
Mathematically, the coded aperture imaging is described as a convolution operation of a sky image S with an aperture pattern A . In the encoding process corresponding to the measurement, thedetected image D is written by: D = A ∗ S + B , (1)where ∗ denotes the convolution operator and B is the background on the detector. The purpose ofthe imaging is to estimate S by computing a decoded image ˜ S . This decoding process is describedwith a decoding pattern ˜ A , which can be generated from A , as: ˜ S = ˜ A ∗ D ˜ S ( i − k,j − l ) = ( ˜ A ∗ D ) ( i − k,j − l ) = P ( i,j ) P ( k,l ) ˜ A ( i,j ) D ( k,l ) . (2)Substituting Equation 1 for Equation 2, ˜ S is represented as: ˜ S = ( ˜ A ∗ A ) ∗ S + ˜ A ∗ B . (3)Ignoring B , which is independent of A , ˜ S is reconstructed into the original image S if the convo-lution ˜ A ∗ A is approximately the δ function. However, the difference between S and ˜ S results inartifacts. Since artifacts are originated from and unique to the pattern structure A , they are regardedas systematic errors of the coded aperture imaging. Such artifacts cause over- and under- estima-tions of the source intensity S . In the case of random patterns, ˜ A ∗ A converges to the delta functionwith the degree of randomness. Thus, the problem is reduced to how to increase the number of thepattern elements. 4 ig 1 Example of multiple different random patterns. They are different only in terms of the configuration of apertures.This set of patterns was used for the experiments described in Section 3.
We consider a coded aperture imaging system with a high-resolution and narrow-FoV configu-ration. The diameter of each aperture and the total pattern size should be determined by scientificrequirements for the angular resolution and the FoV size. Since a general narrow FoV configura-tion does not need a large size of the entire system unit compared to the total size of its satellite,we can gain the effective area by configuring them parallelly. Random patterns have N cases ofvarieties when the pattern size is N × N . Figure 1 shows one example. These four patterns aremade randomly and independently, but the pattern size of each is equally set to be × . Thenumber of the aperture elements is 684 in order to keep the aperture ratio almost 50 %. Here, wepropose a simple idea to use this N variation. Configuring parallelly some different random pat-terns with the same size instead of exactly same patterns, they give independent decoded images ˜ S p with the same FoV size and the angular resolution, where p is a label for a pattern. They can be5ddable and a new summed image ˜ S is given by: ˜ S = X p ˜ S p . (4)Due to the different artifact distribution in each ˜ S p , their contribution should be canceled out in thesummed ˜ S . It is important that the decoding is processed independently for each decoding pattern ˜ A , and is simply implemented in parallel computing. Fig 2
The setup for X-ray beam experiments.
In order to demonstrate the concept of the coded aperture imaging with multiple different randompatterns, we performed X-ray beam experiments. Figure 2 shows the experimental setup. We con-figured the coded aperture plane in front of a CMOS imager with a distance of 250 mm. Changingthe relative direction of the beam to the detector plane, we took three datasets for different imagingtypes of photon sources, as summarized in Table 1.We made a coded aperture mask with the four different random patterns described in the pre-vious section and shown in Figure 1. This mask is made of a 100 µ m-thick SUS304 board and theapertures are created as 27 µ m-diameter holes on it. To keep the mechanical strength, the pitch6 able 1 Datasets in our experiments. The detected event counts are of unit dataset per each pattern before summingprocesses described in Section 4.
Photon Source Detected Event Counts(i) a single point-like source on-axis ∼ ′′ apart ∼ ′′ ∼ µ m. Therefore, the entire pattern size of each pattern is ∼ µ min a × configuration. Then the total size of the multiple random patterns is ∼ as a decoding method. As the geometry of theapertures does not affect the quality of imaging,
18, 19 we use the “delta decoding method”, wherewe assume that each photon comes through the center of the aperture circle. In the balancedcorrelation method, ˜ A ( i,j ) follows ˜ A ( i,j ) = A ( i,j ) is an aperture . ) − A ( i,j ) is a mask . ) , (5)for an aperture fraction of 50 %, where ( i, j ) denotes indices of an aperture element. We alsouse the detected value D ( k,l ) at a detector element ( k, l ) , where the background B ( k,l ) is alreadysubtracted. Using the central coordinates ( a x ( i ) , a y ( j ) ) of the coded aperture element ( i, j ) and7 d x ( k ) , d y ( l ) ) of the detector element ( k, l ) , the photon direction ( s x , s y ) which reaches the detectorelement ( k, l ) through the coded aperture element ( i, j ) is given by: ( s x , s y ) = (cid:18) a x ( i ) − d x ( k ) L , a y ( j ) − d y ( l ) L (cid:19) , (6)where L is the distance from the detector plane to the coded aperture plane. Finally, substitutingEquation 6 for Equation 2, we obtain the sky image from the direction ( s x , s y ) by this convolution: ˜ S ( s x ,s y ) = X ( i,j ) X ( k,l ) ˜ A ( i,j ) D ( k,l ) . (7)For these experiments, we used the synchrotron X-ray beam line BL20B2 in SPring-8 (SuperPhoton ring - 8 GeV). This is a 215 m-long beam line and the beam can be regarded as parallellyincident X-rays, emulating an infinitely distant celestial source. The 16 keV monochromatic beamcame into the coded aperture board and reached the detector. The beam size was 10 mm square,which was sufficiently larger than the size of the patterns 3.5 mm. Since the direction of the beamis fixed, we realized the directional shifts of the X-ray source by leaning the entire detector systemwith the coded aperture using rotary and goniometer stages (the right side of Figure 2) with respectto the beam axis. We generated the observation data of multiple sources (ii) and (iii) by stackingX-ray events at different relative directions. Since this stacking process is done before the decodingcalculation, the encoded images of the sources (ii) and (iii) are identical to images which wouldbe obtained by simultaneous observations of the corresponding multiple sources. Due to a verysmall misalignment between the coded aperture plane and the detector plane, we applied a rotationcorrection of ′′ in the detection image plane before the decoding process.8 ig 3 An X-ray spectrum for the monochromatic 16 keV beam detected by the CMOS imager. The emphasized areais the energy band from which X-ray events were extracted for the imaging in these experiments.
As a detector, we used a 25M pixel CMOS imager with a pixel pitch of 2.5 µ m. This imagerhas an ability of X-ray detection below 24 keV and has an energy resolution of 176 eV (Full Widthat Half Maximum; FWHM) at 5.9 keV at a room temperature of 25 ◦ C. We set a frame timeexposure of 90 ms to reduce detector backgrounds B ( k,l ) and pedestals in a frame. The pedestalsare estimated from the latest set of dark images with the X-ray beam off. To extract X-ray eventsafter subtracting pedestals, we define two kinds of threshold energies. An event threshold is usedfor determining pixels with X-ray detection. In one X-ray event, the charge generated in a pixelcan be extended to its adjacent pixels due to the ejection of a photoelectron and Auger electronsand diffusion of the charge carriers. We also apply a split threshold to the pixels surrounding theX-ray detection pixel in order to take account of pixels with the charge sharing. Pixel values largerthan the split threshold are added to the central value to determine the total energy of an event. Weset the event and split thresholds to 500 and 50 eV, respectively. Figure 3 shows an X-ray spectrumfor the 16 keV beam. We selected events within the 16 keV peak for our imaging experimentsas indicated in this spectrum. Then we regard the highest pixel position in an X-ray event as the9hoton detected position. Figure 4 shows a distribution of the event positions for each dataset bythe pattern (A). We can see a clear shadow of the aperture pattern, though there are some eventsdetected in the area eventually corresponding to the mask or the gaps due to the transmission ofthe X-ray beam through the SUS104 board ( ∼
10 % for 16 keV).
Fig 4
The position distributions of X-ray detections in the case of pattern (A) for the datasets in Table 1. These imagesare shown in linear scales. Results
Fig 5
Imaging results for the dataset (i) in Table 1.
The top-left 4 panels : The decoded image by each random patternin Figure 1. These images are binned in 5 arcsec and expanded near the source. The scale is normalized such that thepeak value of ˜ S ( s x ,s y ) is and the color at 0 is common among all images. The top-right panel : The summed imageof all decoded images. The color scale is made in the same way.
The bottom-left 4 panels : The decoded images with4 times longer exposure experiments in each pattern. Then the photon statistics is comparable to that of the summedimage. The color scale is made in the same way.
The lower-right panel : The histogram of ˜ S ( s x ,s y ) within the entireFoV excluding the source region considering the angular resolution σ , i.e. a histogram of the artifact levels. The value0 means artifact-less. The solid line is for the summed image and colored dashed lines for each long-exposure decodedimage, where the color red, blue, green, and magenta represent the pattern (A), (B), (C), and (D) respectively. Fig 6
The same as Figure 5 but for the dataset (ii). ig 7 The same as Figure 5 but for the dataset (iii). Note that these images are displayed in a wider sky area comparedto Figures 5 & 6 in order to show the artifact pattern distributed in the entire FoV.
Figures 5–7 show the results of the decoding process for the three datasets in Table 1. The top-left4 panels of each figure shows the decoded images by all patterns shown in Figure 1. Because thesedecoded values depend on the decoding process, we show only the ratio to the peak value here inorder to compare the magnitude of artifacts. The top-right panel shows the sum of the four decodedimages, which is the result of our newly proposed method of the artifact reduction. To compare theeffect of the artifact reduction under the same statistical conditions, we demonstrate the situationthat we configured 4 same patterns and did the same experiments. The bottom-left 4 panels arethe results with 4 times longer exposure time by the same setup. The bottom-right panel showshistograms of the normalized pixel values ˜ S ( s x ,s y ) of the source-free region in the decoded imagesto evaluate the effect of summing for multiple different patterns. These histograms give the degreeof the fluctuations due to the artifacts.First, we evaluated the performance for point-like sources using dataset (i) and (ii). For thedataset (ii), the detected images are separated by 50 pixels from each other as shown in Figure 4(ii). It corresponds to a source separation of ′′ . Considering the reading error of 2 pixels and the12ngular resolution σ , this is consistent with our decoded image in Figure 6. This confirms that ourdecoding method works precisely. As we clearly see in Figures 5 and 6, the shape of the decodedsources are extended although the actual beam is regarded not to be diffused in the angular space.This extension is due to the angular resolution σ , and our results imply σ (FWHM) is ∼ ′′ . It isconsistent with the calculation that: σ = p r + p L , (8)where r is the diameter of the aperture, p D is the pixel pitch of the detector, and L is the distancebetween the coded aperture and the detector. This angular resolution is comparable to that ofthe NuSTAR satellite in terms of FWHM ′′ . This means that our experimental setup achievedan excellent imaging performance with a dramatically downsized imaging system compared toexisting missions. Next, the dataset (iii) is for a demonstration of a diffuse source. The imagingsystem measured a circular source with a radius of ′′ . As shown in Figure 7, a circle at the centerof the FoV is clearly decoded with the precise size considering the angular resolution. Its shape isslightly distorted, which is due to the discrete stage control. Table 2
The standard deviation of normalized ˜ S ( s x ,s y ) of each histogram in Figures 5–7. Dataset (A) (B) (C) (D) Summed(i) 0.0166 0.0173 0.0175 0.0172 0.0143(ii) 0.0236 0.0250 0.0249 0.0247 0.0197(iii) 0.0774 0.0895 0.1002 0.0837 0.0704The decoded images in Figures 5–7 show less artifacts in the summed image. For estimatingthis effect precisely, Table 2 shows the standard deviation of each artifact level histogram in Fig-ures 5–7. This value should be close to 0 in an artifact-free image. Each pattern shows different13haracteristics. Pattern (A) shows the smallest standard deviation in all datasets. Pattern (C) looksgenerally the worst, but (B) is slightly worse for the dataset (ii). Pattern (D) shows a similarity to(B) but is good for the dataset (iii). In all the datasets, the summation of the decoded images withthe different random patterns reduces the fluctuations of the source-free regions at least by 10 %.This also quantitatively supports our proposed method of the artifact reduction.
Figures 5–7 and Table 2 show the effective reduction of imaging artifacts by using multiple differ-ent random patterns. However, the effective area per one coded aperture unit of our experimentalsetup is very small for astronomical observations in fact. We have to configure parallelly muchmore than 4 units to gain the total area because the unit size is determined by the scientific re-quirements. Our method performs more effectively in such a situation by using a different randompattern in each unit. 14 ig 8
Decoded images : Simulation results. n is the number of patterns we used. The lower right panel : The standarddeviation of ˜ S ( s x ,s y ) in the source-free regions as a function of the number of patterns. We examine the effect of increasing the number of different patterns. It would cost a lot tomake this experimentally, so we used Monte Carlo simulations instead here. We used ComptonSoft, which depends on the Geant4 toolkit library for Monte Carlo simulations. We built thesame geometric configuration of our beam experiment (iii), i.e., a 1.5 mm square coded apertureand a detector with a pixel pitch of 2.5 µ m, with a separation of 250 mm. We assumed a circularobject with a radius of ′′ on the optical axis, and it emits monochromatic 16 keV photons uni-formly inside the thin circle. Concerning the coded apertures, we prepared 16 different randompatterns. The number of multiple different random patterns was changed from 1 to 2, 4, 8, and 16,and all simulations were done so that the number of total events kept common to 500,000. Figure 8shows the decoded images together with the standard deviations of all the conditions. In those de-15oded images, the artifact structure gradually disappears as the number of patterns gets increased,which is the effect consistent with the result of our beam experiment. In particular, artifacts aresufficiently reduced in the 8- and 16- pattern cases. The standard deviation of artifacts also de-creases, i.e., the signal-to-noise ratio increases with the number of patterns. Therefore, we are ableto gain the effective area and to reduce artifacts simultaneously by using our method. Combiningthis imaging capability with the polarization detectability of the CMOS imager, our system canbe used for the imaging polarimetry in hard X-rays.For a practical use, we should consider backgrounds as well as celestial signals. We need dis-tinguish two major components: the Cosmic X-ray Background (CXB) and the Non X-ray Back-ground (NXB). The CXB is originated from point sources of active galactic nuclei, distributedalmost uniformly in all sky. Our narrow FoV configuration is designed to reduce the contamina-tion of the CXB. It can also restrict the possibility of contaminating bright sources including theappearance of transient sources in the FoV. As to the NXB, which is composed by particles in orbitand secondary X-rays from the satellite and instrument structures, we can estimate it using nightEarth occultation and/or a fully masked unit. Considering these practical issues about the effec-tive area and the backgrounds, we stress that our compact system will be promising as a CubeSatmission that targets small bright objects such as solar flares, the Crab nebula, and bright X-raybinaries.Finally, we should note three points about our experiments. First, we can also use a forwardfitting method taking account of the projection pattern for the decoding process. This methodshould work well with appropriate treatment of the statistical model of the signal and backgroundobservations, but the problem of systematic artifacts originated from a coded pattern will remain.In addition, It requires more computing costs and then an online processing would not be easy for16 limited resource mission. Second, there are other evaluation factors of the imaging performanceimproved by our method though we only show the artifact intensity spectra. In practice, it wouldbe important to evaluate the point-like source sensitivity and the resolving power of double pointsources under the effects of artifacts. These are interesting performance indices but such furtherquantitative evaluations will be done in the future using a more practical system as an observatory.Third, in our experimental setup the photon positions were highly over-sampled on the detectorplane since the pixel pitch of 2.5 µ m is significantly smaller than the requirement by the samplingtheorem for the aperture size and pitch. Such over-sampling generally improves the accuracy ofimaging and reduces image noises. But the reduction of artifacts shown in Section 4 is primarilydue to the multiple random patterns as we propose. These three points do not affect the demon-stration of out proposed concept. We propose a new imaging method of the coded aperture using multiple different random patternssimultaneously to decrease systematic artifacts. We actually made an experimental system with 4different random patterns. Using a 16 keV monochromatic X-ray beam, we achieved a fine angularresolution of < ′′ (FWHM) with a small configuration of 25 cm. For both point-like and diffusesources, artifacts decreased effectively in the summed image compared to a single decoded imageby each pattern. By Monte Carlo simulations, our concept is shown to be more effective when thenumber of patterns gets larger. 17 cknowledgments We thank the anonymous referees for comments to improve the manuscript. We also thank At-sushi Togo for his master thesis, Kentaro Uesugi, Masato Hoshino, Togo Shimozawa, Shigemi Ot-suka, Shunsaku Nagasawa, Kairi Mine, Tomoshi Takeda, Yuto Yoshida, and Keisuke Uchiyama forhelping our beam experiments, and Yuuki Wada, Hiromasa Suzuki, Tadayuki Takahashi, NoriyukiNarukage, and Kiyoshi Hayashida for helpful suggestions. T.K. is supported by the AdvancedLeading Graduate Course for Photon Science (ALPS) in the University of Tokyo. This workis partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI GrantNumber 19H01906.
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