Abstract

Bioluminescence tomography (BLT) is a promising optical molecular imaging technique on the frontier of biomedical optics. In this paper, a generalized hybrid algorithm has been proposed based on the graph cuts algorithm and gradient-based algorithms. The graph cuts algorithm is adopted to estimate a reliable source support without prior knowledge, and different gradient-based algorithms are sequentially used to acquire an accurate and fine source distribution according to the reconstruction status. Furthermore, multilevel meshes for the internal sources are used to speed up the computation and improve the accuracy of reconstruction. Numerical simulations have been performed to validate this proposed algorithm and demonstrate its high performance in the multi-source situation even if the detection noises, optical property errors and phantom structure errors are involved in the forward imaging.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng.4, 235–260 (2002).
    [CrossRef] [PubMed]
  2. C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging16, 378–387 (2002).
    [CrossRef] [PubMed]
  3. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
    [CrossRef] [PubMed]
  4. G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).
  5. A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
    [CrossRef]
  6. M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery2, 123–131 (2003).
    [CrossRef]
  7. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
    [CrossRef] [PubMed]
  8. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol.51, 2045–2053 (2006).
    [CrossRef] [PubMed]
  9. G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express16, 1719–1732 (2008).
    [CrossRef] [PubMed]
  10. G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express14, 7801–7809 (2006).
    [CrossRef] [PubMed]
  11. X. Ma, J. Tian, C. Qin, X. Yang, B. Zhang, Z. Xue, X. Zhang, D. Han, D. Dong, and X. Liu, “Early detection of liver cancer based on bioluminescence tomography,” Appl. Opt.50, 1389–1395 (2011).
    [CrossRef] [PubMed]
  12. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE5535, 335–351 (2004).
    [CrossRef]
  13. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
    [CrossRef] [PubMed]
  14. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
    [CrossRef] [PubMed]
  15. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express15, 11095–11116 (2007).
    [CrossRef] [PubMed]
  16. S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
    [CrossRef] [PubMed]
  17. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17, 8062–8080 (2009).
    [CrossRef] [PubMed]
  18. B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express18, 6477–6491 (2010).
    [CrossRef] [PubMed]
  19. K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express18, 3732–3745 (2010).
    [CrossRef] [PubMed]
  20. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
    [CrossRef] [PubMed]
  21. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
    [CrossRef] [PubMed]
  22. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16, 15640–15654 (2008).
    [CrossRef] [PubMed]
  23. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express1, 512–526 (2010).
    [CrossRef]
  24. M. A. Naser and M. S. Patterson, “Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region,” Biomed. Opt. Express2, 3179–3193 (2011).
    [CrossRef] [PubMed]
  25. H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
    [CrossRef] [PubMed]
  26. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express14, 8211–8223 (2006).
    [CrossRef] [PubMed]
  27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob.15, R41–R93 (1999).
    [CrossRef]
  28. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, 2001).
    [CrossRef]
  29. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt.34, 8026–8037 (1995).
    [CrossRef] [PubMed]
  30. M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob.13, 441–463 (1997).
    [CrossRef]
  31. M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging22, 569–579 (2003).
    [CrossRef] [PubMed]
  32. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob.18, 441–453 (2002).
    [CrossRef]
  33. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).
  34. Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell.26, 1124–1137 (2004).
    [CrossRef]
  35. V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell.26, 147–159 (2004).
    [CrossRef] [PubMed]
  36. P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
    [CrossRef]
  37. M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision (Thompson Learning, 2008).

2011

2010

2009

2008

G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express16, 1719–1732 (2008).
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16, 15640–15654 (2008).
[CrossRef] [PubMed]

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
[CrossRef] [PubMed]

2007

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express15, 11095–11116 (2007).
[CrossRef] [PubMed]

2006

2005

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
[CrossRef] [PubMed]

2004

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell.26, 1124–1137 (2004).
[CrossRef]

V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell.26, 147–159 (2004).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
[CrossRef] [PubMed]

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE5535, 335–351 (2004).
[CrossRef]

2003

M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery2, 123–131 (2003).
[CrossRef]

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging22, 569–579 (2003).
[CrossRef] [PubMed]

2002

C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob.18, 441–453 (2002).
[CrossRef]

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng.4, 235–260 (2002).
[CrossRef] [PubMed]

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging16, 378–387 (2002).
[CrossRef] [PubMed]

2000

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

1999

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob.15, R41–R93 (1999).
[CrossRef]

1997

M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob.13, 441–463 (1997).
[CrossRef]

1995

1984

P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
[CrossRef]

Ahn, S.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol.51, 2045–2053 (2006).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Arridge, S. R.

Bachmann, M. H.

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng.4, 235–260 (2002).
[CrossRef] [PubMed]

Bading, J. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Bertero, M.

M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob.13, 441–463 (1997).
[CrossRef]

Bouman, C. A.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

Boykov, Y.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell.26, 1124–1137 (2004).
[CrossRef]

Boyle, R.

M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision (Thompson Learning, 2008).

Byrne, C.

C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob.18, 441–453 (2002).
[CrossRef]

Cardozo, S. J.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Chan, T. F.

Chatziioannou, A. F.

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17, 8062–8080 (2009).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol.51, 2045–2053 (2006).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Chaudhari, A. J.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Cheng, J.

Cherry, S. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Cong, A.

Cong, W.

Contag, C. H.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging16, 378–387 (2002).
[CrossRef] [PubMed]

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng.4, 235–260 (2002).
[CrossRef] [PubMed]

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Conti, P. S.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Darvas, F.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Davis, S. C.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
[CrossRef] [PubMed]

Dehghani, H.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
[CrossRef] [PubMed]

Dong, D.

Douraghy, A.

Dubey, P.

Durairaj, K.

Feng, J.

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Gupta, S.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Hall, D. E.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Hammer, P. L.

P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
[CrossRef]

Han, D.

Hansen, P.

P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
[CrossRef]

Henry, M.

Hlavac, V.

M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision (Thompson Learning, 2008).

Hoffman, E.

Hoffman, E. A.

Jia, K.

Jiang, M.

M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express15, 11095–11116 (2007).
[CrossRef] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
[CrossRef] [PubMed]

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE5535, 335–351 (2004).
[CrossRef]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
[CrossRef] [PubMed]

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging22, 569–579 (2003).
[CrossRef] [PubMed]

Kolmogorov, V.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell.26, 1124–1137 (2004).
[CrossRef]

V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell.26, 147–159 (2004).
[CrossRef] [PubMed]

Kumar, D.

Leahy, R. M.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Li, H.

Li, X.

Li, Y.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
[CrossRef] [PubMed]

Liu, D.

Liu, K.

Liu, X.

Liu, Y.

Lu, Y.

Luo, J.

Lv, Y.

Ma, X.

McCray, P. B.

McLennan, G.

Meinel, J.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

Moats, R. A.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Naser, M. A.

Natterer, F.

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, 2001).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

Patterson, M. S.

Piana, M.

M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob.13, 441–463 (1997).
[CrossRef]

Pogue, B. W.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
[CrossRef] [PubMed]

Qian, X.

Qin, C.

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol.51, 2045–2053 (2006).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Rehemtulla, A.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Ripoll, J.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

Ross, B. D.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging16, 378–387 (2002).
[CrossRef] [PubMed]

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Rudin, M.

M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery2, 123–131 (2003).
[CrossRef]

Schweiger, M.

Shen, H.

Simeone, B.

P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
[CrossRef]

Sinn, P.

Smith, D. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Sonka, M.

M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision (Thompson Learning, 2008).

Stegman, L. D.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Stout, D.

Sun, L.

Suter, M.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

Tian, J.

X. Ma, J. Tian, C. Qin, X. Yang, B. Zhang, Z. Xue, X. Zhang, D. Han, D. Dong, and X. Liu, “Early detection of liver cancer based on bioluminescence tomography,” Appl. Opt.50, 1389–1395 (2011).
[CrossRef] [PubMed]

B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express18, 6477–6491 (2010).
[CrossRef] [PubMed]

K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express18, 3732–3745 (2010).
[CrossRef] [PubMed]

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17, 8062–8080 (2009).
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16, 15640–15654 (2008).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express14, 8211–8223 (2006).
[CrossRef] [PubMed]

Wang, G.

G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express16, 1719–1732 (2008).
[CrossRef] [PubMed]

M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express15, 11095–11116 (2007).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express14, 8211–8223 (2006).
[CrossRef] [PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express14, 7801–7809 (2006).
[CrossRef] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
[CrossRef] [PubMed]

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE5535, 335–351 (2004).
[CrossRef]

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging22, 569–579 (2003).
[CrossRef] [PubMed]

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
[CrossRef] [PubMed]

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

Wang, Y.

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery2, 123–131 (2003).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Wübbeling, F.

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, 2001).
[CrossRef]

Xu, M.

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

Xue, Z.

Yan, G.

Yang, W.

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express14, 8211–8223 (2006).
[CrossRef] [PubMed]

Yang, X.

Zabin, R.

V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell.26, 147–159 (2004).
[CrossRef] [PubMed]

Zabner, J.

Zhang, B.

Zhang, X.

Zhong, J.

Zhou, T.

Zhu, S.

Annu. Rev. Biomed. Eng.

C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng.4, 235–260 (2002).
[CrossRef] [PubMed]

Appl. Opt.

Biomed. Opt. Express

IEEE Trans. Med. Imaging

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging22, 569–579 (2003).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell.26, 1124–1137 (2004).
[CrossRef]

V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell.26, 147–159 (2004).
[CrossRef] [PubMed]

Inverse Prob.

C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob.18, 441–453 (2002).
[CrossRef]

M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob.13, 441–463 (1997).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob.15, R41–R93 (1999).
[CrossRef]

J. Magn. Reson. Imaging

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging16, 378–387 (2002).
[CrossRef] [PubMed]

Math. Program.

P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program.28, 121–155 (1984).
[CrossRef]

Med. Phys.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.35, 4863–4871 (2008).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31, 2289–2299 (2004).
[CrossRef] [PubMed]

Nat. Biotechnol.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23, 313–320 (2005).
[CrossRef] [PubMed]

Nat. Rev. Drug Discovery

M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery2, 123–131 (2003).
[CrossRef]

Neoplasia

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia2, 491–495 (2000).
[CrossRef]

Opt. Express

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13, 6756–6771 (2005).
[CrossRef] [PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express14, 7801–7809 (2006).
[CrossRef] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express14, 8211–8223 (2006).
[CrossRef] [PubMed]

M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express15, 11095–11116 (2007).
[CrossRef] [PubMed]

G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express16, 1719–1732 (2008).
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16, 15640–15654 (2008).
[CrossRef] [PubMed]

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17, 8062–8080 (2009).
[CrossRef] [PubMed]

K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express18, 3732–3745 (2010).
[CrossRef] [PubMed]

B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express18, 6477–6491 (2010).
[CrossRef] [PubMed]

Phys. Med. Biol.

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol.52, 4497–4512 (2007).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53, 3921–3942 (2008).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol.51, 2045–2053 (2006).
[CrossRef] [PubMed]

Proc. SPIE

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE5535, 335–351 (2004).
[CrossRef]

Radiology

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology229(P), 566 (2003).

Other

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, 2001).
[CrossRef]

M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision (Thompson Learning, 2008).

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic Press, 1981).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Flowchart of the proposed hybrid algorithm for BLT problem.

Fig. 2
Fig. 2

The heterogeneous mouse phantom and its internal sources.

Fig. 3
Fig. 3

Reconstructed results of the hybrid algorithm, EM, Landweber and modified Newton algorithm. (a), (c), (e) and (g) are the cross-section at z = 0mm of the reconstructed results corresponding to the four algorithms, and the white circles with actual source radius (1mm) are marked to show the true source location. (b), (d), (f) and (h) are the 3D view of the reconstructed results.

Fig. 4
Fig. 4

Reconstructed result obtained by the hybrid algorithm with +50% optical property error.

Fig. 5
Fig. 5

Reconstructed result obtained by the hybrid algorithm with the lungs being shifted 1mm outward to the phantom edge.

Fig. 6
Fig. 6

Reconstructed result obtained by the hybrid algorithm with the influence of the mixed factors.

Tables (6)

Tables Icon

Algorithm 1 (Generalized hybrid algorithm for BLT)

Tables Icon

Table 1 Optical Properties for the Numerical Phantom

Tables Icon

Table 2 Setting up of the Internal Sources

Tables Icon

Table 3 Finite Element Information for the Simulations

Tables Icon

Table 4 Quantitative Comparison of the Reconstructed Results Between the Hybrid Algorithm and the Single EM, Landweber, and Modified Newton Algorithm

Tables Icon

Table 5 Quantitative Analysis of the Reconstructed Results Obtained by the Hybrid Algorithm with Some Specific Influence Factors*

Equations (10)

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

BLT { ( D ( x ) u 0 ( x ) ) + μ a ( x ) u 0 ( x ) = q 0 ( x ) , in Ω , u 0 ( x ) + 2 D ( x ) u 0 ν ( x ) = 0 , on Γ , D ( x ) u 0 ν ( x ) = g ( x ) , on Γ .
q = b .
E ( q ) = q b L 2 2 + λ G ( q ) .
q ( n + 1 ) = 1 T 1 q ( n ) T [ b q ( n ) ] .
q ( n + 1 ) = q ( n ) + γ n T ( b q ( n ) ) .
q ( n + 1 ) = q ( n ) + ( T + α I ) 1 T ( b q ( n ) ) .
E ( q ) = q b L 2 2 + λ G ( q ) = θ const + i θ i ( q i ) + i < j θ i j ( q i , q j ) .
{ θ const = b T b , θ i ( q i ) = ( m i T m i ) q i 2 + ( λ 2 b T m i ) q i , θ i j ( q i , q j ) = 2 ( m i T m j ) q i q j .
ϕ i ( x ) = χ Ω i ( x ) , Ω i = { x | x x i < Δ x i / 2 } , i = 1 , 2 , , N .
Ω 0 = { ( x , y , z ) | 6 mm x 2 + y 2 12 mm , 3 mm < z < 3 mm } .

Metrics