A gradient-based optimisation scheme for optical tomography
Optics Express, Vol. 2, Issue 6, pp. 213-226 (1998)
http://dx.doi.org/10.1364/OE.2.000213
Acrobat PDF (545 KB)
Abstract
Optical tomography schemes using non-linear optimisation are usually based on a Newton-like method involving the construction and inversion of a large Jacobian matrix. Although such matrices can be efficiently constructed using a reciprocity principle, their inversion is still computationally difficult. In this paper we demonstrate a simple means to obtain the gradient of the objective function directly, leading to straightforward application of gradient-based optimisation methods.
© Optical Society of America
1. Introduction
1. A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988). [CrossRef] [PubMed]
2. J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990). [PubMed]
4. J. C. Hebden, R. A. Kruger, and K. S. Wong, “Time resolved imaging through a highly scattering medium,” Appl. Opt. 30, 788–794 (1991). [CrossRef] [PubMed]
6. J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997). [CrossRef] [PubMed]
7. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997). [CrossRef] [PubMed]
8. S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.
9. S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.
10. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]
11. S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992). [CrossRef]
2. Problem definition
3. Transport model
6. J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997). [CrossRef] [PubMed]
7. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997). [CrossRef] [PubMed]
18. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995). [CrossRef] [PubMed]
4. Finite element approach
17. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993). [CrossRef] [PubMed]
18. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995). [CrossRef] [PubMed]
5. Measurement types
22. S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995). [CrossRef] [PubMed]
23. M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997). [CrossRef]
6. Reconstruction methods
19. S. R. Arridge, “Photon measurement density functions. Part 1: Analytic forms,” Appl. Opt. 34, 7395–7409 (1995). [CrossRef] [PubMed]
20. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed]
24. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993). [CrossRef]
25. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995). [CrossRef] [PubMed]
29. S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).
7. Gradient calculation
26. S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997). [CrossRef]
29. S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).
20. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed]
20. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed]
20. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed]
8. Results
31. M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci. , 1230, 71–84 (1997). [CrossRef]
8.1 Circular test object
30. S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995). [CrossRef] [PubMed]
31. M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci. , 1230, 71–84 (1997). [CrossRef]
29. S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).
8.2 Neonatal head model
9. Conclusions
Acknowledgements
Footnotes
1 | In biomedical optics the absorption parameter is usually denoted μ
_{a}, with the scattering parameter denoted μ
_{s}′, and the diffusion parameter given by |
References
1. | A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988). [CrossRef] [PubMed] |
2. | J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990). [PubMed] |
3. | M. Tamura, “Multichannel near-infrared optical imaging of human brain activity,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 8–10. |
4. | J. C. Hebden, R. A. Kruger, and K. S. Wong, “Time resolved imaging through a highly scattering medium,” Appl. Opt. 30, 788–794 (1991). [CrossRef] [PubMed] |
5. | Near-infrared spectroscopy and imaging of living systems, special issue of Philos. Trans. R. Soc. London Ser. B, Vol. 352 (1997). |
6. | J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997). [CrossRef] [PubMed] |
7. | S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997). [CrossRef] [PubMed] |
8. | S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298. |
9. | S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141. |
10. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef] |
11. | S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992). [CrossRef] |
12. | H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995). [CrossRef] |
13. | B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995). [CrossRef] [PubMed] |
14. | D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988). [CrossRef] [PubMed] |
15. | B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990). [CrossRef] |
16. | J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990). |
17. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993). [CrossRef] [PubMed] |
18. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995). [CrossRef] [PubMed] |
19. | S. R. Arridge, “Photon measurement density functions. Part 1: Analytic forms,” Appl. Opt. 34, 7395–7409 (1995). [CrossRef] [PubMed] |
20. | S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed] |
21. | M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993). |
22. | S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995). [CrossRef] [PubMed] |
23. | M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997). [CrossRef] |
24. | M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993). [CrossRef] |
25. | K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995). [CrossRef] [PubMed] |
26. | S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997). [CrossRef] |
27. | O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of Münster, 1997. |
28. | R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, 1997. |
29. | S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press). |
30. | S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995). [CrossRef] [PubMed] |
31. | M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci. , 1230, 71–84 (1997). [CrossRef] |
OCIS Codes
(100.3190) Image processing : Inverse problems
(110.6960) Imaging systems : Tomography
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.6920) Medical optics and biotechnology : Time-resolved imaging
(170.6960) Medical optics and biotechnology : Tomography
ToC Category:
Focus Issue: Tomographic image reconstruction
History
Original Manuscript: January 20, 1998
Published: March 16, 1998
Citation
Simon Arridge and Martin Schweiger, "A gradient-based optimisation scheme foroptical tomography," Opt. Express 2, 213-226 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-6-213
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References
- A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope and E. O. R. Reynolds, \Cotside measurement of cerebral blood ow in ill newborn infants by near infrared spec- troscopy," Lancet 2, 770-771 (1988). [CrossRef] [PubMed]
- S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray and E. O. R. Reynolds, \Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy," J. Appl. Physiol. 68, 1086-1091 (1990). [PubMed]
- M. Tamura, \Multichannel near-infrared optical imaging of human brain activity," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 8-10.
- J. C. Hebden, R. A. Kruger and K. S. Wong, \Time resolved imaging through a highly scattering medium," Appl. Opt. 30, 788-794 (1991). [CrossRef] [PubMed]
- Near-infrared spectroscopy and imaging of living systems, special issue of Philos. Trans. R. Soc. London Ser. B, Vol. 352 (1997).
- J. C. Hebden, S. R. Arridge and D. T. Delpy, \Optical imaging in medicine: I. Experimental techniques," Phys. Med. Biol. 42, 825-840 (1997). [CrossRef] [PubMed]
- S. R. Arridge and J. C. Hebden, \Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997). [CrossRef] [PubMed]
- S. B. Colak, G. W. Hooft, D. G. Papaioannou and M. B. van der Mark, \3D backprojection tomography for medical optical imaging," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 294-298.
- S. A. Walker, S. Fantini and E. Gratton, \Back-projection reconstructions of cylindrical inho- mogeneities from frequency domain optical measurements in turbid media," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 137-141.
- M. A. O Leary, D. A. Boas, B. Chance and A. G. Yodh, \Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995). [CrossRef]
- S. R. Arridge, M. Schweiger and D. T. Delpy, \Iterative reconstruction of near-infrared absorption images," in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372-383 (1992). [CrossRef]
- H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue and M. S. Patterson, \Optical image reconstruction using frequency-domain data: Simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1995). [CrossRef]
- B. W. Pogue, M. S. Patterson, H. Jiang and K. D. Paulsen, \Initial assessment of a simple system for frequency domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995). [CrossRef] [PubMed]
- D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray and J. Wyatt, \Estimation of optical pathlength through tissue from direct time of ight measurement," Phys. Med. Biol. 33, 1433-1442 (1988). [CrossRef] [PubMed]
- B. Chance, M. Maris, J. Sorge and M. Z. Zhang, \A phase modulation system for dual wave- length difference spectroscopy of haemoglobin deoxygenation in tissue," in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481-491 (1990). [CrossRef]
- J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).
- S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, \A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
- M. Schweiger, S. R. Arridge, M. Hiraoka and D. T. Delpy, \The finite element model for the propagation of light in scattering media: Boundary and source conditions," Med. Phys. 22, 1779-1792 (1995). [CrossRef] [PubMed]
- S. R. Arridge, \Photon measurement density functions. Part 1: Analytic forms," Appl. Opt. 34, 7395-7409 (1995). [CrossRef] [PubMed]
- S. R. Arridge and M. Schweiger, \Photon measurement density functions. Part 2: Finite element calculations," Appl. Opt. 34, 8026-8037 (1995). [CrossRef] [PubMed]
- M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).
- S. R. Arridge and M. Schweiger, \Direct calculation of the moments of the distribution of photon time of ight in tissues with a finite-element method," Appl. Opt. 34, 2683-2687 (1995). [CrossRef] [PubMed]
- M. Schweiger and S. R. Arridge, \Direct calculation of the Laplace transform of the distribution of photon time of ight in tissue with a finite-element method," Appl. Opt. 36, 9042-9049 (1997). [CrossRef]
- M. Schweiger, S. R. Arridge and D. T. Delpy, \Application of the finite-element method for the forward and inverse models in optical tomography," J. Math Imag. Vision 3, 263-283 (1993). [CrossRef]
- K. D. Paulsen and H. Jiang, \Spatially-varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995). [CrossRef] [PubMed]
- S. S. Saquib, K. M. Hanson and G. S. Cunningham, \Model-based image reconstruction from time-resolved diffusion data," in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369-380 (1997). [CrossRef]
- O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of M"unster, 1997.
- R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, Uni- versity of Hertfordshire, 1997.
- S. R. Arridge and M. Schweiger, \A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis C. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer 1998, in press).
- S. R. Arridge, M. Hiraoka and M. Schweiger, \Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1558 (1995). [CrossRef] [PubMed]
- M. Schweiger and S. R. Arridge, \Optimal data types in optical tomography," in Information Processing in Medical Imaging, Lect. Notes Comput. Sci., 1230, 71-84 (1997). [CrossRef]
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