Abstract

This paper proposes an integral method to achieve a more accurate weighting matrix that makes very positive contributions to the image reconstruction in inertial confinement fusion research. Standard algebraic reconstruction techniques with a positivity constraint included are utilized. The final normalized mean-square error between the simulated and reconstructed projection images is 0.000365%, which is a nearly perfect result, indicating that the weighting matrix is very important. Compared with the error between the simulated and reconstructed phantoms, which is 2.35%, it seems that the improvement of the accuracy of the projection image does not mean the improvement of the phantom. The proposed method can reconstruct a simulated laser-imploded target consisting of 100×100×100 voxels.

© 2012 Optical Society of America

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  1. Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
    [CrossRef]
  2. T. Zhuang, CT Principles and Algorithms (Shanghai Jiao Tong University, 1992).
  3. J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6, 513–537 (1977).
    [CrossRef]
  4. G. N. Minerbo, J. G. Sanderson, D. B. van Hulsteyn, and P. Lee, “Three-dimensional reconstruction of the x-ray emission in laser imploded targets,” Appl. Opt. 19, 1723–1728 (1980).
    [CrossRef]
  5. A. Holland, E. T. Powell, and R. J. Fonck, “Image reconstruction methods for the PBX-M pinhole camera,” Appl. Opt. 30, 3740–3751 (1991).
    [CrossRef]
  6. G. T. Herman and A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
    [CrossRef]
  7. G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
    [CrossRef]
  8. A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
    [CrossRef]
  9. D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
    [CrossRef]
  10. G. Gillman and I. Macleod, “Reconstruction of x-ray sources from penumbral images,” Comput. Graph. Image Process. 11, 227–241 (1979).
    [CrossRef]
  11. D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
    [CrossRef]
  12. T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
    [CrossRef]
  13. 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. Control. Fusion 49, 1145–1150 (2007).
    [CrossRef]
  14. W. Yao and K. Leszczynski, “Analytically derived weighting factors for transmission tomography cone beam projections,” Phys. Med. Biol. 54, 513–533 (2009).
    [CrossRef]
  15. H. Zhao and A. J. Reader, “Fast ray-tracing technique to calculate line integral paths in voxel arrays,” IEEE Nucl. Sci. Symp. Conf. Record 4, 2808–2812 (2003).
    [CrossRef]

2010 (1)

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

2009 (1)

W. Yao and K. Leszczynski, “Analytically derived weighting factors for transmission tomography cone beam projections,” Phys. Med. Biol. 54, 513–533 (2009).
[CrossRef]

2007 (1)

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

2003 (2)

H. Zhao and A. J. Reader, “Fast ray-tracing technique to calculate line integral paths in voxel arrays,” IEEE Nucl. Sci. Symp. Conf. Record 4, 2808–2812 (2003).
[CrossRef]

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

1992 (1)

D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
[CrossRef]

1991 (1)

1988 (1)

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[CrossRef]

1980 (1)

1979 (2)

G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
[CrossRef]

G. Gillman and I. Macleod, “Reconstruction of x-ray sources from penumbral images,” Comput. Graph. Image Process. 11, 227–241 (1979).
[CrossRef]

1977 (1)

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6, 513–537 (1977).
[CrossRef]

1976 (1)

G. T. Herman and A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[CrossRef]

Azuma, R.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Cao, L.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Chen, Y.-W.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

Colsher, J. G.

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6, 513–537 (1977).
[CrossRef]

Da Silva, L.

D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
[CrossRef]

Ding, Y.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Dong, J.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Ellis, R. J.

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Fonck, R. J.

Fujioka, S.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Gillman, G.

G. Gillman and I. Macleod, “Reconstruction of x-ray sources from penumbral images,” Comput. Graph. Image Process. 11, 227–241 (1979).
[CrossRef]

Hao, Y.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Herman, G. T.

G. T. Herman and A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef]

Holland, A.

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[CrossRef]

Kodama, R.

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

Kohatsu, T.

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

Lane, S. M.

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Lee, P.

Lent, A.

G. T. Herman and A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef]

Lerche, R. A.

D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
[CrossRef]

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Leszczynski, K.

W. Yao and K. Leszczynski, “Analytically derived weighting factors for transmission tomography cone beam projections,” Phys. Med. Biol. 54, 513–533 (2009).
[CrossRef]

Macleod, I.

G. Gillman and I. Macleod, “Reconstruction of x-ray sources from penumbral images,” Comput. Graph. Image Process. 11, 227–241 (1979).
[CrossRef]

Minerbo, G.

G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
[CrossRef]

Minerbo, G. N.

Nishimura, H.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Nozaki, S.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

Nugent, K. A.

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Powell, E. T.

Pu, Y.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Reader, A. J.

H. Zhao and A. J. Reader, “Fast ray-tracing technique to calculate line integral paths in voxel arrays,” IEEE Nucl. Sci. Symp. Conf. Record 4, 2808–2812 (2003).
[CrossRef]

Ress, D.

D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
[CrossRef]

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Sanderson, J. G.

Ueda, T.

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

van Hulsteyn, D. B.

Wu, S.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Yao, W.

W. Yao and K. Leszczynski, “Analytically derived weighting factors for transmission tomography cone beam projections,” Phys. Med. Biol. 54, 513–533 (2009).
[CrossRef]

Zhao, H.

H. Zhao and A. J. Reader, “Fast ray-tracing technique to calculate line integral paths in voxel arrays,” IEEE Nucl. Sci. Symp. Conf. Record 4, 2808–2812 (2003).
[CrossRef]

Zhao, Z.

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Zhuang, T.

T. Zhuang, CT Principles and Algorithms (Shanghai Jiao Tong University, 1992).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Ress, R. A. Lerche, and L. Da Silva, “Demonstration of an x-ray ring-aperture microscope for inertial-confinement fusion experiments,” Appl. Phys. Lett. 60, 410–412 (1992).
[CrossRef]

Comput. Biol. Med. (1)

G. T. Herman and A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef]

Comput. Graph. Image Process. (3)

G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
[CrossRef]

G. Gillman and I. Macleod, “Reconstruction of x-ray sources from penumbral images,” Comput. Graph. Image Process. 11, 227–241 (1979).
[CrossRef]

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6, 513–537 (1977).
[CrossRef]

IEEE Nucl. Sci. Symp. Conf. Record (1)

H. Zhao and A. J. Reader, “Fast ray-tracing technique to calculate line integral paths in voxel arrays,” IEEE Nucl. Sci. Symp. Conf. Record 4, 2808–2812 (2003).
[CrossRef]

Phys. Med. Biol. (1)

W. Yao and K. Leszczynski, “Analytically derived weighting factors for transmission tomography cone beam projections,” Phys. Med. Biol. 54, 513–533 (2009).
[CrossRef]

Plasma Phys. Control. Fusion (1)

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. Control. Fusion 49, 1145–1150 (2007).
[CrossRef]

Rev. Sci. Instrum. (2)

T. Ueda, S. Fujioka, S. Nozaki, R. Azuma, Y.-W. Chen, and H. Nishimura, “A uniformly redundant imaging array of penumbral apertures coupled with a heuristic reconstruction for hard x-ray and neutron imaging,” Rev. Sci. Instrum. 81, 073505 (2010).
[CrossRef]

Y.-W. Chen, T. Kohatsu, S. Nozaki, and R. Kodama, “Heuristic reconstruction of three-dimensional laser-imploded targets from x-ray pinhole images,” Rev. Sci. Instrum. 74, 2236–2239(2003).
[CrossRef]

Science (1)

D. Ress, R. A. Lerche, R. J. Ellis, S. M. Lane, and K. A. Nugent, “Neutron imaging of laser fusion targets,” Science 241, 956–958 (1988).
[CrossRef]

Ultrason. Imag. (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[CrossRef]

Other (1)

T. Zhuang, CT Principles and Algorithms (Shanghai Jiao Tong University, 1992).

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Figures (9)

Fig. 1.
Fig. 1.

Geometric relationship between the source and projections.

Fig. 2.
Fig. 2.

Geometric analysis of the weighting factors: (a) projection to single pixel and (b) projection to subpixel. The shadowed volume in the right part of (a) corresponds to a detector pixel, while the cuboids in the right part of (b) correspond to a subpixel.

Fig. 3.
Fig. 3.

Calculation of hq. Regarding the geometric symmetry relationship between the ray and the voxel array, define a voxel of x[0,av], y[0,av], z[0,av]. Assume that the start point, or the left intersection point of the line and the voxel, has been worked out; here is the end point (xr,yr,zr), or the right intersection point of the line and the voxel at the plane y=av: (a) xr(0,av), zr(0,av), which is in the surface of the voxel, and (b) xr(0,av), zr>av, which is not in the surface as zr is beyond the boundary. This will be remedied by further calculation of the intersection point of the line and the boundary plane z=av, (c) xr(0,av), zr=av, which is on the edge, and (d) xr=0, zr=av, which is in the vertex. The bold cubic in this figure is the next voxel the ray going through is decided by the current voxel and its end point (xr,yr,zr), which is also the start point of the next voxel.

Fig. 4.
Fig. 4.

Comparison of the original and reconstructed phantom: (a) original 3D phantom and (b) reconstructed 3D phantom.

Fig. 5.
Fig. 5.

Layers of the original and reconstructed phantom: (a) six selected typical layers of the original phantom and (b) six corresponding layers of the reconstructed image by the proposed method. The label “x=” denotes the layer sequence.

Fig. 6.
Fig. 6.

NMSEs of the 100 layers, which are reconstructed by using weighting factor Ait, along axes X, Y, Z, respectively.

Fig. 7.
Fig. 7.

Relative errors compared with the maximum intensity of the reconstructed phantom.

Fig. 8.
Fig. 8.

Comparison between the reconstructed and simulated projection images: (a) reconstructed projection images and (b) simulated projection images calculated based on the projection relationship with the phantom xorg.

Fig. 9.
Fig. 9.

Difference between the originally simulated and reconstructed projection images. Inset, detail of the six originally simulated projection images.

Tables (1)

Tables Icon

Table 1. NMSEs of b and x Using Two Different Weighting Factorsa

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

bmnp=i,j,kamnijkpxijk,
amnijkp=VmnijkpVvoxel,
aijk=q((ac/(f+1))(ldq/lsq))2hqav3,
x(g)=x(g1)+λ(g)bmnp(amnp)Tx(g1)amnpamnp,
b=Axorg,
AAit.
Ne(y,z)=yz2y2.

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