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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 9, Iss. 5 — Apr. 29, 2014
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Blind deconvolution for spatial distribution of Kα emission from ultraintense laser-plasma interaction

Weihua He, Zongqing Zhao, Jian Wang, Bo Zhang, Feng Qian, Zuhua Yang, Min Shui, Feng Lu, Jian Teng, Leifeng Cao, and Yuqiu Gu  »View Author Affiliations


Optics Express, Vol. 22, Issue 5, pp. 5875-5882 (2014)
http://dx.doi.org/10.1364/OE.22.005875


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Abstract

The spatial distributions of the Kα emission from foil targets irradiated with ultra-intensity laser pulses have been studied using the x-ray coded imaging technique. Due to the effect of hard x-ray background contamination, noise as well as imperfection of imaging system, it is hard to determine the PSF analytically or measure it experimentally. Therefore, we propose a blind deconvolution method to restore both the spatial distributions of the Kα emission and the system's PSF from the coded images based on the maximum-likelihood scheme. Experimental restoration results from penumbral imaging and ring coded imaging demonstrated that both the structure integrity and the rich detail information can be well preserved.

© 2014 Optical Society of America

1. Introduction

The interaction of an ultra-intense laser pulse with a solid-target surface rapidly creates plasma, as electrons near the critical density of plasma are accelerated and then penetrate into the solid. These electrons will knock out electrons from inner electronic shells of atoms there, which lead to emission of a short burst of characteristic Kα x-ray [1

1. W. Lu, M. Nicoul, U. Shymanovich, A. Tarasevitch, P. Zhou, K. Sokolowski-Tinten, D. von der Linde, M. Mašek, P. Gibbon, and U. Teubner, “Optimized Kalpha x-ray flashes from femtosecond-laser-irradiated foils,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(2), 026404 (2009). [CrossRef] [PubMed]

]. Such x-ray sources have great potential applications in many fields such as fast ignition for inertial confinement fusion [2

2. S. Le Pape, P. Neumayer, C. Fortmann, T. Döppner, P. Davis, A. Kritcher, O. Landen, and S. Glenzer, “X-ray radiography and scattering diagnosis of dense shock-compressed matter,” Phys. Plasmas 17(5), 056309 (2010). [CrossRef]

5

5. T. Ma, H. Sawada, P. K. Patel, C. D. Chen, L. Divol, D. P. Higginson, A. J. Kemp, M. H. Key, D. J. Larson, S. Le Pape, A. Link, A. G. MacPhee, H. S. McLean, Y. Ping, R. B. Stephens, S. C. Wilks, and F. N. Beg, “Hot electron temperature and coupling efficiency scaling with prepulse for cone-guided fast ignition,” Phys. Rev. Lett. 108(11), 115004 (2012). [CrossRef] [PubMed]

], high energy density physics [6

6. H. S. Park, D. M. Chambers, H. K. Chung, R. J. Clarke, R. Eagleton, E. Giraldez, T. Goldsack, R. Heathcote, N. Izumi, M. H. Key, J. A. King, J. A. Koch, O. L. Landen, A. Nikroo, P. K. Patel, D. F. Price, B. A. Remington, H. F. Robey, R. A. Snavely, D. A. Steinman, R. B. Stephens, C. Stoeckl, M. Storm, M. Tabak, W. Theobald, R. P. J. Town, J. E. Wickersham, and B. B. Zhang, “High-energy Kα radiography using high-intensity, short-pulse lasers,” Phys. Plasmas 13(5), 056309 (2006). [CrossRef]

], laser induced radiation sources [7

7. N. L. Kugland, C. G. Constantin, P. Neumayer, H. K. Chung, A. Collette, E. L. Dewald, D. H. Froula, S. H. Glenzer, A. Kemp, A. L. Kritcher, J. S. Ross, and C. Niemann, “High Kα x-ray conversion efficiency from extended source gas jet targets irradiated by ultra short laser pulses,” Appl. Phys. Lett. 92(24), 241504 (2008). [CrossRef]

], and particle acceleration [8

8. M. N. Quinn, X. H. Yuan, X. X. Lin, D. C. Carroll, O. Tresca, R. J. Gray, M. Coury, C. Li, Y. T. Li, C. M. Brenner, A. P. L. Robinson, D. Neely, B. Zielbauer, B. Aurand, J. Fils, T. Kuehl, and P. McKenna, “Refluxing of fast electrons in solid targets irradiated by intense, picosecond laser pulses,” Plasma Phys. Contr. Fusion 53(2), 025007 (2011). [CrossRef]

], all of which critically depend on the source characteristics to achieve better physical performance.

In past years, much effort has been devoted to investigate the size and distribution of Kα source. For instance, Eder et al. [9

9. D. C. Eder, G. Pretzler, E. Fill, K. Eidmann, and A. Saemann, “Spatial characteristics of Kα radiation from weakly relativistic laser plasmas,” Appl. Phys. B 70(2), 211–217 (2000). [CrossRef]

] employed one-dimensional x-ray shadowgraphy at a knife edge to estimate the size and the total yield of a Cu Kα source. With the help of Monte Carlo electron transport software, they reproduced the total Cu Kα yield under different laser intensity, but they found the predicted source size was smaller than the measured. Reich et al. [10

10. Ch. Reich, I. Uschmann, F. Ewald, S. Düsterer, A. Lübcke, H. Schwoerer, R. Sauerbrey, E. Förster, and P. Gibbon, “Spatial characteristics of Kalpha x-ray emission from relativistic femtosecond laser plasmas,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 68(5), 056408 (2003). [CrossRef] [PubMed]

] studied the spatial structure of the Kα emission from Ti target using a two dimensional monochromatic imaging technique. Conventional single pinhole imaging can capture the interested source image, however its low photon collection efficiency due to the small aperture limits the application of this technique [11

11. W. He, Z. Zhao, F. Li, J. Teng, M. Shui, L. Cao, and Y. Gu, “A parallel deconvolution algorithm in coded imaging,” ICIC EL 7, 2407–2410 (2013).

]. The collection efficiency of Kα source can be enhanced by adopting Wölter microscope [12

12. J. Hu, L. Cheng, X. Wu, Y. Sun, and Y. Bai, “A new method of image reconstruction with high resolution in x-ray coded aperture imaging,” Proc. SPIE 5918, 591818 (2005).

,13

13. P. Troussel, P. Munsch, and J. J. Ferme, “Microfocusing between 1 and 5 keV with Wölter-type optics,” Proc. SPIE 3773, 60–69 (1999). [CrossRef]

] or Fresnel zone-plate imaging [14

14. H. H. Barrett and F. A. Horrigan, “Fresnel zone plate imaging of gamma rays; Theory,” Appl. Opt. 12(11), 2686–2702 (1973). [CrossRef] [PubMed]

], but the former one would add complexity to the imaging system, and for the latter case it is expensive to implement the required “thick” zone plate with about 10μm resolution. At present, Bent Bragg crystal coupled either to x-ray films or to CCD detectors are among the mostly used x-ray diagnostics for Kα emission [15

15. R. B. Stephens, R. A. Snavely, Y. Aglitskiy, F. Amiranoff, C. Andersen, D. Batani, S. D. Baton, T. Cowan, R. R. Freeman, T. Hall, S. P. Hatchett, J. M. Hill, M. H. Key, J. A. King, J. A. Koch, M. Koenig, A. J. MacKinnon, K. L. Lancaster, E. Martinolli, P. Norreys, E. Perelli-Cippo, M. Rabec Le Gloahec, C. Rousseaux, J. J. Santos, and F. Scianitti, “Kα fluorescence measurement of relativistic electron transport in the context of fast ignition,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(6), 066414 (2004). [CrossRef] [PubMed]

,16

16. K. U. Akli, M. H. Key, H. K. Chung, S. B. Hansen, R. R. Freeman, M. H. Chen, G. Gregori, S. Hatchett, D. Hey, N. Izumi, J. King, J. Kuba, P. Norreys, A. J. Mackinnon, C. D. Murphy, R. Snavely, R. B. Stephens, C. Stoeckel, W. Theobald, and B. Zhang, “Temperature sensitivity of Cu Kα imaging efficiency using a spherical Bragg reflecting crystal,” Phys. Plasmas 14(2), 023102 (2007). [CrossRef]

]. It can provide 2D x-ray monochromatic image with micrometer-scale spatial resolution. However, bent crystals in the Bragg configuration suffer from the important limitation in the small area of the crystal reflecting at a constant angle of incidence [16

16. K. U. Akli, M. H. Key, H. K. Chung, S. B. Hansen, R. R. Freeman, M. H. Chen, G. Gregori, S. Hatchett, D. Hey, N. Izumi, J. King, J. Kuba, P. Norreys, A. J. Mackinnon, C. D. Murphy, R. Snavely, R. B. Stephens, C. Stoeckel, W. Theobald, and B. Zhang, “Temperature sensitivity of Cu Kα imaging efficiency using a spherical Bragg reflecting crystal,” Phys. Plasmas 14(2), 023102 (2007). [CrossRef]

]. More recently, the Energy-encoded Pinhole Camera (EPiC) scheme [17

17. L. Labate, P. Köster, T. Levato, and L. A. Gizzi, “A novel technique for single-shot energy-resolved 2D x-ray imaging of plasmas relevant for the inertial confinement fusion,” Rev. Sci. Instrum. 83(10), 103504 (2012). [CrossRef] [PubMed]

] is proposed for x-ray diagnostic of laser-fusion plasma, based upon the use of an array of many pinholes coupled to a large area CCD camera operating in the single-photon mode. It can obtain 2D monochromatic images at any x-ray photon energy over a large spectral domain, while the available spatial resolution is merely comparable to the one of a conventional pinhole camera scheme. In contrast, the coded aperture imaging is another simpler but effective means for x-ray diagnostics [18

18. Z. Zhao, Y. Ding, J. Dong, Y. Hao, S. Wu, L. Cao, and Y. Pu, “Richardson-Lucy method for decoding x-ray ring code image,” Plasma Phys. Contr. Fusion 49(8), 1145–1150 (2007). [CrossRef]

]. It owns the following advantages [19

19. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17(3), 337–347 (1978). [CrossRef] [PubMed]

]: a) the number of collected photons will be increased due to its large open area. Compared with the single pinhole imaging, the photon collection efficiency will be greatly improved, which is very helpful for producing the image with higher signal-to-noise ratio, b) it has high angular resolution commensurate with the conventional pinhole imaging, and c) the alignment of the coded-imaging experimental setup is much simpler than that of the crystal imaging, meanwhile, the fabrication process of the coded aperture is much easier than the preparation of the bent crystal. Therefore we would investigate the spatial distributions of the Kα emission with the coded aperture imaging scheme in this paper.

Accurate restoration of the spatial distribution of Kα source from the coded image remains a great challenge. A coded image can be approximated as the convolution of the ground truth source image with the system point spread function (PSF). A direct deconvolution operation, which assumes PSF to be known, can only be employed under ideal conditions [20

20. W. Stefan, “Image restoration by blind deconvolution,” Diploma Thesis, Technische Universität München and Arizona State University, 2003.

]. However, due to the effect of deep penetration of high energy particles, noise as well as imperfection of imaging system, it is hard to determine the PSF analytically or measure it experimentally. In order to accurately restore the Kα source, we propose an iteration blind deconvolution method based on maximum likelihood scheme. In this paper, we experimentally captured the coded imaging of Kα source on ultra-intensity laser and provided the restoration results.

The rest of the paper is organized as follows: Section 2 describes the experimental setup for coded imaging on SILEX-I laser facility. Section 3 presents the model and the principle of the proposed blind deconvolution method. Section 4 reports the experimental restoration results, and some discussions are also made in the same section, finally conclusions are made in Section 5.

2. Experimental setup

From a series of tests, two specific coded imaging experiments performed on the Terawatt SILEX-I laser facility at Laser Fusion Research Center in MianYang, China, are chosen here for their pertinence in the present work.

2.1 Experiment I: Penumbral coded imaging

Schematic of the experimental setup is shown in Fig. 1(a)
Fig. 1 (a) Experimental arrangement for the spatial distribution measurement of Kα source, (b) The 12μm × 7μm elliptic focal spot.
. After reflected from an f/3 off-axis-parabolic mirror, the p-polarized 800 nm laser pulse delivered 8J in 30 fs (FWHM) on target. The laser was focused to a 12μm × 7μm elliptic focal spot and the peak intensity is W/cm2, while the intensity contrast ratio of the nanosecond level pre-pulse to the main pulse is better than 10−7 [Fig. 1(b)]. A target was mounted on an electronic adjustable frame and the incident angle of the laser is 24°.

A target with a 10μm thick copper foil was employed in this work. To observe propagation properties of fast electron beam, a penumbral imaging system, with a 20μm thick copper filter, was installed to obtain the Kα coded image from the rear side of the target with a bird’s-eye view of 22° and 15° to the target normal, as shown in Fig. 1(a). This imaging system consisted of an 800μm diameter penumbral aperture and a calibrated 1340 × 1300 Princeton single photon counting x-ray CCD, and the penumbral aperture was generated by a laser drilling machine punching an 800μm hole on a 50μm thick tantalum.

2.2 Experiment II: Ring coded imaging

The Schematic of the experimental setup for ring aperture imaging is very similar to that of penumbral aperture imaging. The laser pulse duration is 700ps. A ring aperture, coated with 10μm thick Au, was placed behind the Ti solid target. The inner and external diameters of the ring aperture were 990μm and 1000μm, respectively. It should be noted that the ring aperture was not intact, which has been fabricated with small gaps. There also was a 5μm diameter pinhole at the center of the ring aperture to measure theKαprofile directly, which gave the property of the fast electrons. The x-ray CCD detector can record the ring coded image and the pinhole image at the same time.

3. The proposed iterative blind deconvolution method

The coded imaging technique mainly includes two steps. The first is to generate the coded image, in which a geometrical shadow of the source is cast on the CCD detector through a coded aperture. Mathematically, the measured coded imageY can be modeled as
Y=XH+η.
(1)
in which X, Hand η represent the source image, system’s PSF and noise, respectively, and * denotes the convolution operation.

The second step is to reconstruct the original image from coded image by restoration technique. Traditional restoration methods such as the direct deconvolution operation [15

15. R. B. Stephens, R. A. Snavely, Y. Aglitskiy, F. Amiranoff, C. Andersen, D. Batani, S. D. Baton, T. Cowan, R. R. Freeman, T. Hall, S. P. Hatchett, J. M. Hill, M. H. Key, J. A. King, J. A. Koch, M. Koenig, A. J. MacKinnon, K. L. Lancaster, E. Martinolli, P. Norreys, E. Perelli-Cippo, M. Rabec Le Gloahec, C. Rousseaux, J. J. Santos, and F. Scianitti, “Kα fluorescence measurement of relativistic electron transport in the context of fast ignition,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(6), 066414 (2004). [CrossRef] [PubMed]

] or Richardson-Lucy method [18

18. Z. Zhao, Y. Ding, J. Dong, Y. Hao, S. Wu, L. Cao, and Y. Pu, “Richardson-Lucy method for decoding x-ray ring code image,” Plasma Phys. Contr. Fusion 49(8), 1145–1150 (2007). [CrossRef]

], which assume the PSF to be known, can only be employed under ideal conditions. In this paper, we developed the blind deconvolution method based on the maximum-likelihood scheme to restore the coded images. The maximum-likelihood scheme was firstly introduced for application in positron emission tomography by Shepp and Vardi in 1982 [21

21. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1(2), 113–122 (1982). [CrossRef] [PubMed]

]. Since then numerous researchers have further studied on it for restoring Hubble space telescope imagery [22

22. J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002). [CrossRef]

], three-dimensional fluorescence microscopy imagery [23

23. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef] [PubMed]

], single photon emission computed tomography (SPECT) transmission imagery in medical science [24

24. M. V. Narayanan, C. L. Byrne, and M. A. King, “An interior point iterative maximum-likelihood reconstruction algorithm incorporating upper and lower bounds with application to SPECT transmission imaging,” IEEE Trans. Med. Imaging 20(4), 342–353 (2001). [CrossRef] [PubMed]

] and so on.

The maximum-likelihood scheme is derived from the Bayesian framework. In our case we want to find the most probable Xgiven an observation ofY, i.e., we want to maximizep(X/Y). With the Bayesian rule,
p(X/Y)=p(Y/X)p(X)p(Y),
(2)
where p()denotes the probability density function. Given p(Y)and p(X)are constant, this optimization problem is equivalent to maximizing the conditional probability term on the right, named maximum likelihood scheme.

X^=argmaxp(X/Y)=argmaxp(Y/X),
(3)

Our proposed method is an alternating variable optimization method to estimate both the source image and PSF. The optimization technique employed here is an iterative deconvolution method based on the maximum-likelihood scheme. Table 1

Table 1. Procedure of the proposed blind deconvolution method

table-icon
View This Table
outlines the procedure of the proposed method.

To estimate X^n and H^n (see the step 4 and step 5 in Table 1), iterative deconvolution based on maximum-likelihood scheme is employed. Considering the observed image data as a set of independent Gaussian distribution, the conditional probability of Yk(k=1,,N) can be expressed as
p(Yk|Xk)=12πσexp{12[Yk(HX)kσ]2},
(4)
in whichH=Hn1andX=Xn1.Then the corresponding joint log-likelihood function is

L(X)=lnk=1Np(Yk|Xk)=12σ2k=1N[(Yk(HX)k)2]N2lnσ2N2ln(2π),
(5)

The maximum likelihood solution is located at the point where all the partial derivatives of L(X)with respect to Xj are equal to zero

LXj=1σ2[k=1N(Hkj(i=1NHkiXi)HkjYk)],
(6)

According to Eq. (6) we can derive the iterative scheme
X^j(t+1)=X^j(t)k=1NHkjYkk=1NHkj(i=1NHkiX^it).
(7)
AfterTiterations, the restored image is denoted asX^T, and we setX^n=X^T.

Similarly, the estimated PSF also can be derived as
H^j(t+1)=H^j(t)k=1NXkjYkk=1NXkj(i=1NXkiH^it).
(8)
in an iterative way. After Titerations, the restored PSF is denoted asH^T, and we setH^n=H^T.

4. Results and analysis

This section presents the restoration results of coded images for Kα source by the proposed blind deconvolution method. Figure 2(a)
Fig. 2 The penumbral coded image and the restoration results. (a) Penumbral coded image, (b) An electron microscope image of the real penumbral aperture fabricated by a laser drilling machine, (c) Initial estimated PSF, (d) The restored PSF, (e) The restored spatial distribution of Kα emission.
shows the penumbral image obtained in our penumbral imaging experiment. Herein, an obvious bright spot is observed in the central of the image. The spot is mainly contributed by Kα photons passing through the penumbral hole directly. Around this spot, there exists a partially illuminated penumbral ring due to the vignetting effect. Some high energy electrons and bremsstrahlung photons also penetrate the tantalum substrate to form the background noise. Figure 2(b) shows an electron microscope image of the penumbral aperture generated by a laser drilling machine. An ideal circle is employed as the initial estimated PSF whose diameter is 300 pixels [Fig. 2(c)]. The restored PSF and the distribution of the Kα source with our method are given in Figs. 2(d) and 2(e), respectively. From Fig. 2(d), it can be observed that while the shape of the initial estimated PSF and restored PSF are similar, their intensity distributions are quite different. Specifically, the intensity for the former is uniform, while that of the latter is non-uniform due to the edge imperfection, the non-ideal laser focal spot, and the high penetration of electrons and the bremsstrahlung photons emitted by the target. From Fig. 2(e), it can be seen that main Kα peak is surrounded by several holes of weak emission. The spatial half-width of the restored Kα source is measured about 68μm, which is about 6 times larger than the half-width of the laser focus spot. The theoretical spatial resolution of the image system can be calculated following the way given in [25

25. L. Disdier, A. Rouyer, I. Lantuéjoul, O. Landoas, J. L. Bourgade, T. C. Sangster, V. Y. Glebov, and R. A. Lerche, “Inertial confinement fusion neutron images,” Phys. Plasmas 13(5), 056317 (2006). [CrossRef]

]. As our penumbral aperture is a thin cylindrical hole, the penumbral coded imaging system’s theoretical spatial resolution of the restored source could be approximately expressed asΔScodeΔd/M, where M and Δdrepresent the magnification parameter and the detector resolution, respectively. In contrast, the theoretical spatial resolution for the pinhole imaging system can be expressed asΔShole=(1+1/M)2D2+(Δd/M)2, where D represents the diameter of the pinhole. It is noticeable that we will haveΔShole>ΔScode, when keeping the values of the magnification and detector resolution parameters unchanged. Specifically, lettingM=6, Δd=20μm(as in our experiments) and D=5μm, the spatial resolutions of the penumbral coded imaging system and the pinhole imaging system are about 4μm and 6μm, respectively.

Figure 3(a)
Fig. 3 The ring coded image and the restoration results. (a) Ring coded image, (b) The pinhole image at the center of the ring aperture, (c) An electron microscope image of the real ring aperture fabricated by a laser drilling machine, (d) Initial estimated PSF, (e) restored PSF, (f) The restored spatial distribution of Kα emission.
shows the ring coded image obtained in the ring coded imaging experiment. The detail information of the pinhole image centered at the ring coded image is shown in Fig. 3(b). Figure 3(c) shows an electron microscope image of the ring aperture fabricated by a laser drilling machine. Therein, the green region indicates the ring aperture while the surrounding black region shows the uninterested laser-ablation region. From this figure, it can be found that the fabricated ring aperture has some imperfections, such as the discrepancy in ring width, the burr in ring edge as well as the lack of integrity (the ring even includes two gaps). During the blind deconvolution process, an ideal ring, whose inner diameter and external diameter are 400 and 500 pixels, is employed as the initial estimated PSF. After about 20 iterations, the restored results of PSF and the distribution of the Kα source are demonstrated in Figs. 3(e) and 3(f), respectively. From Fig. 3(e), it can be observed that the shape of the restored PSF is not an intact ring but with small gaps, which is consistent with the imperfectly fabricated ring aperture. Moreover, the non-uniform intensity in the restored PSF could be attributed to the discrepancy in ring width and the burr edge of the fabricated ring aperture. From Fig. 3(f), it can be observed that the restored image has a good agreement with the pinhole image in structure, but the quality of the restored image in clarity, detail and structure integrity has been greatly improved compared with the pinhole image. It should be mentioned that, Fig. 3(f) shows multiple spots instead of a single emission due to the aberrations in the laser focal spot.

5. Conclusions

In this paper, we have experimentally studied the Kα emission on ultra-intensity laser with the coded imaging technique. As precision knowledge of PSF information is always unavailable, we proposed a blind deconvolution method based on maximum-likelihood scheme in this paper. The experimental results for penumbral imaging and ring coded imaging show that the structure integrity and rich detail information can be well preserved in the restored image. We also found that the two restored Kα emissions have quite different spatial distributions which mainly be affected by many factors including laser intensities, properties of target materials, and so on.

Acknowledgments

The authors acknowledge helpful discussions with Y.H. Yan, T.K. Zhang, J.L. Jiao and Y.C. Wu. We would also like to thank the staff of the SILEX-I Group for their support in laser system operation and G. Niu for target fabrication. This work is supported by the National Science Foundation of China (Grant Nos.1174259, 11175030, 109050512, and 11375161) and China Academy of Engineering Physics Foundation (Grant No. 2011B0102021).

References and links

1.

W. Lu, M. Nicoul, U. Shymanovich, A. Tarasevitch, P. Zhou, K. Sokolowski-Tinten, D. von der Linde, M. Mašek, P. Gibbon, and U. Teubner, “Optimized Kalpha x-ray flashes from femtosecond-laser-irradiated foils,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(2), 026404 (2009). [CrossRef] [PubMed]

2.

S. Le Pape, P. Neumayer, C. Fortmann, T. Döppner, P. Davis, A. Kritcher, O. Landen, and S. Glenzer, “X-ray radiography and scattering diagnosis of dense shock-compressed matter,” Phys. Plasmas 17(5), 056309 (2010). [CrossRef]

3.

S. Atzeni, A. Schiavi, and J. R. Davies, “Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition,” Plasma Phys. Contr. Fusion 51(1), 015016 (2009). [CrossRef]

4.

S. Chawla, M. S. Wei, R. Mishra, K. U. Akli, C. D. Chen, H. S. McLean, A. Morace, P. K. Patel, H. Sawada, Y. Sentoku, R. B. Stephens, and F. N. Beg, “Effect of target material on fast-electron transport and resistive collimation,” Phys. Rev. Lett. 110(2), 025001 (2013). [CrossRef] [PubMed]

5.

T. Ma, H. Sawada, P. K. Patel, C. D. Chen, L. Divol, D. P. Higginson, A. J. Kemp, M. H. Key, D. J. Larson, S. Le Pape, A. Link, A. G. MacPhee, H. S. McLean, Y. Ping, R. B. Stephens, S. C. Wilks, and F. N. Beg, “Hot electron temperature and coupling efficiency scaling with prepulse for cone-guided fast ignition,” Phys. Rev. Lett. 108(11), 115004 (2012). [CrossRef] [PubMed]

6.

H. S. Park, D. M. Chambers, H. K. Chung, R. J. Clarke, R. Eagleton, E. Giraldez, T. Goldsack, R. Heathcote, N. Izumi, M. H. Key, J. A. King, J. A. Koch, O. L. Landen, A. Nikroo, P. K. Patel, D. F. Price, B. A. Remington, H. F. Robey, R. A. Snavely, D. A. Steinman, R. B. Stephens, C. Stoeckl, M. Storm, M. Tabak, W. Theobald, R. P. J. Town, J. E. Wickersham, and B. B. Zhang, “High-energy Kα radiography using high-intensity, short-pulse lasers,” Phys. Plasmas 13(5), 056309 (2006). [CrossRef]

7.

N. L. Kugland, C. G. Constantin, P. Neumayer, H. K. Chung, A. Collette, E. L. Dewald, D. H. Froula, S. H. Glenzer, A. Kemp, A. L. Kritcher, J. S. Ross, and C. Niemann, “High Kα x-ray conversion efficiency from extended source gas jet targets irradiated by ultra short laser pulses,” Appl. Phys. Lett. 92(24), 241504 (2008). [CrossRef]

8.

M. N. Quinn, X. H. Yuan, X. X. Lin, D. C. Carroll, O. Tresca, R. J. Gray, M. Coury, C. Li, Y. T. Li, C. M. Brenner, A. P. L. Robinson, D. Neely, B. Zielbauer, B. Aurand, J. Fils, T. Kuehl, and P. McKenna, “Refluxing of fast electrons in solid targets irradiated by intense, picosecond laser pulses,” Plasma Phys. Contr. Fusion 53(2), 025007 (2011). [CrossRef]

9.

D. C. Eder, G. Pretzler, E. Fill, K. Eidmann, and A. Saemann, “Spatial characteristics of Kα radiation from weakly relativistic laser plasmas,” Appl. Phys. B 70(2), 211–217 (2000). [CrossRef]

10.

Ch. Reich, I. Uschmann, F. Ewald, S. Düsterer, A. Lübcke, H. Schwoerer, R. Sauerbrey, E. Förster, and P. Gibbon, “Spatial characteristics of Kalpha x-ray emission from relativistic femtosecond laser plasmas,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 68(5), 056408 (2003). [CrossRef] [PubMed]

11.

W. He, Z. Zhao, F. Li, J. Teng, M. Shui, L. Cao, and Y. Gu, “A parallel deconvolution algorithm in coded imaging,” ICIC EL 7, 2407–2410 (2013).

12.

J. Hu, L. Cheng, X. Wu, Y. Sun, and Y. Bai, “A new method of image reconstruction with high resolution in x-ray coded aperture imaging,” Proc. SPIE 5918, 591818 (2005).

13.

P. Troussel, P. Munsch, and J. J. Ferme, “Microfocusing between 1 and 5 keV with Wölter-type optics,” Proc. SPIE 3773, 60–69 (1999). [CrossRef]

14.

H. H. Barrett and F. A. Horrigan, “Fresnel zone plate imaging of gamma rays; Theory,” Appl. Opt. 12(11), 2686–2702 (1973). [CrossRef] [PubMed]

15.

R. B. Stephens, R. A. Snavely, Y. Aglitskiy, F. Amiranoff, C. Andersen, D. Batani, S. D. Baton, T. Cowan, R. R. Freeman, T. Hall, S. P. Hatchett, J. M. Hill, M. H. Key, J. A. King, J. A. Koch, M. Koenig, A. J. MacKinnon, K. L. Lancaster, E. Martinolli, P. Norreys, E. Perelli-Cippo, M. Rabec Le Gloahec, C. Rousseaux, J. J. Santos, and F. Scianitti, “Kα fluorescence measurement of relativistic electron transport in the context of fast ignition,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(6), 066414 (2004). [CrossRef] [PubMed]

16.

K. U. Akli, M. H. Key, H. K. Chung, S. B. Hansen, R. R. Freeman, M. H. Chen, G. Gregori, S. Hatchett, D. Hey, N. Izumi, J. King, J. Kuba, P. Norreys, A. J. Mackinnon, C. D. Murphy, R. Snavely, R. B. Stephens, C. Stoeckel, W. Theobald, and B. Zhang, “Temperature sensitivity of Cu Kα imaging efficiency using a spherical Bragg reflecting crystal,” Phys. Plasmas 14(2), 023102 (2007). [CrossRef]

17.

L. Labate, P. Köster, T. Levato, and L. A. Gizzi, “A novel technique for single-shot energy-resolved 2D x-ray imaging of plasmas relevant for the inertial confinement fusion,” Rev. Sci. Instrum. 83(10), 103504 (2012). [CrossRef] [PubMed]

18.

Z. Zhao, Y. Ding, J. Dong, Y. Hao, S. Wu, L. Cao, and Y. Pu, “Richardson-Lucy method for decoding x-ray ring code image,” Plasma Phys. Contr. Fusion 49(8), 1145–1150 (2007). [CrossRef]

19.

E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17(3), 337–347 (1978). [CrossRef] [PubMed]

20.

W. Stefan, “Image restoration by blind deconvolution,” Diploma Thesis, Technische Universität München and Arizona State University, 2003.

21.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1(2), 113–122 (1982). [CrossRef] [PubMed]

22.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002). [CrossRef]

23.

C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef] [PubMed]

24.

M. V. Narayanan, C. L. Byrne, and M. A. King, “An interior point iterative maximum-likelihood reconstruction algorithm incorporating upper and lower bounds with application to SPECT transmission imaging,” IEEE Trans. Med. Imaging 20(4), 342–353 (2001). [CrossRef] [PubMed]

25.

L. Disdier, A. Rouyer, I. Lantuéjoul, O. Landoas, J. L. Bourgade, T. C. Sangster, V. Y. Glebov, and R. A. Lerche, “Inertial confinement fusion neutron images,” Phys. Plasmas 13(5), 056317 (2006). [CrossRef]

OCIS Codes
(140.7090) Lasers and laser optics : Ultrafast lasers
(300.2140) Spectroscopy : Emission
(340.7430) X-ray optics : X-ray coded apertures
(340.7440) X-ray optics : X-ray imaging
(100.1455) Image processing : Blind deconvolution

ToC Category:
Image Processing

History
Original Manuscript: December 5, 2013
Revised Manuscript: January 17, 2014
Manuscript Accepted: February 9, 2014
Published: March 6, 2014

Virtual Issues
Vol. 9, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Weihua He, Zongqing Zhao, Jian Wang, Bo Zhang, Feng Qian, Zuhua Yang, Min Shui, Feng Lu, Jian Teng, Leifeng Cao, and Yuqiu Gu, "Blind deconvolution for spatial distribution of Kα emission from ultraintense laser-plasma interaction," Opt. Express 22, 5875-5882 (2014)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-5875


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References

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  11. W. He, Z. Zhao, F. Li, J. Teng, M. Shui, L. Cao, Y. Gu, “A parallel deconvolution algorithm in coded imaging,” ICIC EL 7, 2407–2410 (2013).
  12. J. Hu, L. Cheng, X. Wu, Y. Sun, Y. Bai, “A new method of image reconstruction with high resolution in x-ray coded aperture imaging,” Proc. SPIE 5918, 591818 (2005).
  13. P. Troussel, P. Munsch, J. J. Ferme, “Microfocusing between 1 and 5 keV with Wölter-type optics,” Proc. SPIE 3773, 60–69 (1999). [CrossRef]
  14. H. H. Barrett, F. A. Horrigan, “Fresnel zone plate imaging of gamma rays; Theory,” Appl. Opt. 12(11), 2686–2702 (1973). [CrossRef] [PubMed]
  15. R. B. Stephens, R. A. Snavely, Y. Aglitskiy, F. Amiranoff, C. Andersen, D. Batani, S. D. Baton, T. Cowan, R. R. Freeman, T. Hall, S. P. Hatchett, J. M. Hill, M. H. Key, J. A. King, J. A. Koch, M. Koenig, A. J. MacKinnon, K. L. Lancaster, E. Martinolli, P. Norreys, E. Perelli-Cippo, M. Rabec Le Gloahec, C. Rousseaux, J. J. Santos, F. Scianitti, “Kα fluorescence measurement of relativistic electron transport in the context of fast ignition,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(6), 066414 (2004). [CrossRef] [PubMed]
  16. K. U. Akli, M. H. Key, H. K. Chung, S. B. Hansen, R. R. Freeman, M. H. Chen, G. Gregori, S. Hatchett, D. Hey, N. Izumi, J. King, J. Kuba, P. Norreys, A. J. Mackinnon, C. D. Murphy, R. Snavely, R. B. Stephens, C. Stoeckel, W. Theobald, B. Zhang, “Temperature sensitivity of Cu Kα imaging efficiency using a spherical Bragg reflecting crystal,” Phys. Plasmas 14(2), 023102 (2007). [CrossRef]
  17. L. Labate, P. Köster, T. Levato, L. A. Gizzi, “A novel technique for single-shot energy-resolved 2D x-ray imaging of plasmas relevant for the inertial confinement fusion,” Rev. Sci. Instrum. 83(10), 103504 (2012). [CrossRef] [PubMed]
  18. Z. Zhao, Y. Ding, J. Dong, Y. Hao, S. Wu, L. Cao, Y. Pu, “Richardson-Lucy method for decoding x-ray ring code image,” Plasma Phys. Contr. Fusion 49(8), 1145–1150 (2007). [CrossRef]
  19. E. E. Fenimore, T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17(3), 337–347 (1978). [CrossRef] [PubMed]
  20. W. Stefan, “Image restoration by blind deconvolution,” Diploma Thesis, Technische Universität München and Arizona State University, 2003.
  21. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1(2), 113–122 (1982). [CrossRef] [PubMed]
  22. J. L. Starck, E. Pantin, F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002). [CrossRef]
  23. C. Preza, J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef] [PubMed]
  24. M. V. Narayanan, C. L. Byrne, M. A. King, “An interior point iterative maximum-likelihood reconstruction algorithm incorporating upper and lower bounds with application to SPECT transmission imaging,” IEEE Trans. Med. Imaging 20(4), 342–353 (2001). [CrossRef] [PubMed]
  25. L. Disdier, A. Rouyer, I. Lantuéjoul, O. Landoas, J. L. Bourgade, T. C. Sangster, V. Y. Glebov, R. A. Lerche, “Inertial confinement fusion neutron images,” Phys. Plasmas 13(5), 056317 (2006). [CrossRef]

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