Abstract

We present a useful strategy for imaging perturbations of the macroscopic absorption cross section of dense-scattering media using steady-state light sources. A perturbation model based on transport theory is derived, and the inverse problem is simplified to a system of linear equations, WΔμ = ΔR, where W is the weight matrix, Δμ is a vector of the unknown perturbations, and ΔR is the vector of detector readings. Monte Carlo simulations compute the photon flux across the surfaces of phantoms containing simple or complex inhomogeneities. Calculation of the weight matrix is also based on the results of Monte Carlo simulations. Three reconstruction algorithms—conjugate gradient descent, projection onto convex sets, and the simultaneous algebraic reconstruction technique, with or without imposed positivity constraints—are used for image reconstruction. A rescaling technique that improves the conditioning of the weight matrix is also developed. Results show that the analysis of time-independent data by a perturbation model is capable of resolving the internal structure of a dense-scattering medium. Imposition of positivity constraints improves image quality at the cost of a reduced convergence rate. Use of the rescaling technique increases the initial rate of convergence, resulting in accurate images in a smaller number of iterations.

© 1996 Optical Society of America

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1995 (2)

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

1994 (1)

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J.2501–2510 (1994).
[CrossRef] [PubMed]

1993 (2)

F. Liu, K. M. Yoo, R. R. Alfano, “Ultrafast laser-pulse transmission and imaging through biological tissues,” Appl. Opt. 32, 554–558 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vis. 3, 263–283 (1993).
[CrossRef]

1984 (1)

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

1972 (2)

G. A. Deschamps, H. S. Garayan, “Antenna synthesis and solution of inverse problems by regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[CrossRef]

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

1970 (1)

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Alfano, R. R.

Anderson, A. H.

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

Andersson-Engels, S.

R. Berg, S. Andersson-Engels, S. Svanberg, “Time-resolved transillumination imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 397–424.

Aronson, R.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

J. Chang, R. Aronson, H. L. Graber, R. L. Barbour, “Imaging diffusive media using time-independent and time-harmonic sources: dependence of image quality on imaging algorithms, target volume, weight matrix, and view angles,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 448–464 (1995).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

Arridge, S. R.

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vis. 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 378–388 (1995).
[CrossRef]

Barbour, R. L.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, R. Aronson, H. L. Graber, R. L. Barbour, “Imaging diffusive media using time-independent and time-harmonic sources: dependence of image quality on imaging algorithms, target volume, weight matrix, and view angles,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 448–464 (1995).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Progress toward optical mammography: imaging in dense scattering media using time-independent optical sources,” in Proceedings of 1994 IEEE Medical Imaging Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1484–1488.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

Beauvoit, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J.2501–2510 (1994).
[CrossRef] [PubMed]

Bender, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Berg, R.

R. Berg, S. Andersson-Engels, S. Svanberg, “Time-resolved transillumination imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 397–424.

Boas, D. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive wave within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J.2501–2510 (1994).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive wave within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Chang, J.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, H. L. Graber, R. L. Barbour, “Progress toward optical mammography: imaging in dense scattering media using time-independent optical sources,” in Proceedings of 1994 IEEE Medical Imaging Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1484–1488.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, R. Aronson, H. L. Graber, R. L. Barbour, “Imaging diffusive media using time-independent and time-harmonic sources: dependence of image quality on imaging algorithms, target volume, weight matrix, and view angles,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 448–464 (1995).
[CrossRef]

Cunningham, J. R.

H. E. Johns, J. R. Cunningham, The Physics of Radiology, 4th ed. (Thomas, Springfield, Ill., 1983).

Delpy, D. T.

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vis. 3, 263–283 (1993).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, H. S. Garayan, “Antenna synthesis and solution of inverse problems by regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[CrossRef]

Feng, S.

S. Feng, F.-A. Zeng, “Perturbation theory of photon migration in the presence of a single defect,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 217–228.

Garayan, H. S.

G. A. Deschamps, H. S. Garayan, “Antenna synthesis and solution of inverse problems by regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[CrossRef]

Gilbert, P.

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Gill, P. E.

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

Gordon, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Graber, H. L.

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, R. Aronson, H. L. Graber, R. L. Barbour, “Imaging diffusive media using time-independent and time-harmonic sources: dependence of image quality on imaging algorithms, target volume, weight matrix, and view angles,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 448–464 (1995).
[CrossRef]

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Progress toward optical mammography: imaging in dense scattering media using time-independent optical sources,” in Proceedings of 1994 IEEE Medical Imaging Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1484–1488.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

Herman, G. T.

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

Hünlich, R.

R. Model, R. Hünlich, M. Orlt, M. Walzel, “Image reconstruction for random media by diffusion tomography,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 400–410 (1995).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jacques, S. L.

M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for optical diffusion theory: a general approach for absorbing and scattering objects in tissue,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 98–102 (1995).
[CrossRef]

Jiang, H.

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Johns, H. E.

H. E. Johns, J. R. Cunningham, The Physics of Radiology, 4th ed. (Thomas, Springfield, Ill., 1983).

Kak, A. C.

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

Kak, A. V.

A. V. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineering, New York, 1988).

Kalender, W. A.

W. A. Kalender, “X-ray computed tomography—state of the art,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. I511 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 10–27.

Kitai, T.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J.2501–2510 (1994).
[CrossRef] [PubMed]

Lamarsh, J.

J. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966).

Liu, F.

Lubowsky, J.

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

Model, R.

R. Model, R. Hünlich, M. Orlt, M. Walzel, “Image reconstruction for random media by diffusion tomography,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 400–410 (1995).
[CrossRef]

Murray, W.

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

O'Leary, M. A.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive wave within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Orlt, M.

R. Model, R. Hünlich, M. Orlt, M. Walzel, “Image reconstruction for random media by diffusion tomography,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 400–410 (1995).
[CrossRef]

Ostermeyer, M. R.

M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for optical diffusion theory: a general approach for absorbing and scattering objects in tissue,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 98–102 (1995).
[CrossRef]

Patterson, M. S.

M. S. Patterson, B. W. Pougue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 513–533.

Paulsen, K. D.

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Pougue, B. W.

M. S. Patterson, B. W. Pougue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 513–533.

Schweiger, M.

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vis. 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 378–388 (1995).
[CrossRef]

Slaney, M.

A. V. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineering, New York, 1988).

Strang, G.

G. Strang, Introduction to Applied Mathematics (Wellesley-Cambridge Press, Wellesley, Mass., 1986).

Svanberg, S.

R. Berg, S. Andersson-Engels, S. Svanberg, “Time-resolved transillumination imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 397–424.

van Assendelft, O. W.

O. W. van Assendelft, Spectrophotometry of Haemoglobin Derivatives (Thomas, Springfield, Ill., 1970).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Chap. 3.

Walzel, M.

R. Model, R. Hünlich, M. Orlt, M. Walzel, “Image reconstruction for random media by diffusion tomography,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 400–410 (1995).
[CrossRef]

Wang, Y.

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

Wilson, B. C.

M. S. Patterson, B. W. Pougue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 513–533.

Wright, M. H.

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

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

Yodh, A. G.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive wave within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Yoo, K. M.

Youla, D. C.

D. C. Youla, “Mathematical theory of image reconstruction by the method of convex projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987).

Zeng, F.-A.

S. Feng, F.-A. Zeng, “Perturbation theory of photon migration in the presence of a single defect,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 217–228.

Zhu, W.

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

Zhu, W. Z.

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (1)

Biophys. J. (1)

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J.2501–2510 (1994).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

G. A. Deschamps, H. S. Garayan, “Antenna synthesis and solution of inverse problems by regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[CrossRef]

J. Math. Imag. Vis. (1)

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vis. 3, 263–283 (1993).
[CrossRef]

J. Theor. Biol. (2)

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Med. Phys. (1)

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

Ultrason. Imag. (1)

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

Other (34)

S. R. Arridge, M. Schweiger, “Sensitivity to prior knowledge in optical tomographic reconstruction,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 378–388 (1995).
[CrossRef]

S. Feng, F.-A. Zeng, “Perturbation theory of photon migration in the presence of a single defect,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 217–228.

M. A. O'Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive wave within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for optical diffusion theory: a general approach for absorbing and scattering objects in tissue,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 98–102 (1995).
[CrossRef]

R. Model, R. Hünlich, M. Orlt, M. Walzel, “Image reconstruction for random media by diffusion tomography,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 400–410 (1995).
[CrossRef]

M. S. Patterson, B. W. Pougue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 513–533.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

J. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

J. Chang, H. L. Graber, R. L. Barbour, “Progress toward optical mammography: imaging in dense scattering media using time-independent optical sources,” in Proceedings of 1994 IEEE Medical Imaging Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1484–1488.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Chap. 3.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. Graber, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer stripping approach for recovery of scattering medium using time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “Time-resolved imaging in dense scattering media,” in Physiological Imaging, Spectroscopy, and Early Detection Diagnostic Methods, R. L. Barbour, M. J. Carvlin, eds., Proc. SPIE1887, 108–119 (1993).
[CrossRef]

H. L. Graber, J. Chang, J. Lubowsky, R. Aronson, R. L. Barbour, “Near infrared absorption imaging of dense scattering media by steady-state diffusion tomography,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 372–386 (1993).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 193–201.

J. Chang, R. Aronson, H. L. Graber, R. L. Barbour, “Imaging diffusive media using time-independent and time-harmonic sources: dependence of image quality on imaging algorithms, target volume, weight matrix, and view angles,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 448–464 (1995).
[CrossRef]

W. Z. Zhu, Y. Wang, J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction in scattering media from time-independent data: a total least squares approach,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 420–430 (1995).
[CrossRef]

D. C. Youla, “Mathematical theory of image reconstruction by the method of convex projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987).

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

G. Strang, Introduction to Applied Mathematics (Wellesley-Cambridge Press, Wellesley, Mass., 1986).

H. E. Johns, J. R. Cunningham, The Physics of Radiology, 4th ed. (Thomas, Springfield, Ill., 1983).

D. D. Stark, W. G. Bradley, eds., Magnetic Resonance Imaging, 2nd ed. (Mosby, St. Louis, Mo., 1992).

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

A. V. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineering, New York, 1988).

H. Stark, ed., Image Recovery: Theory and Application (Academic, New York, 1987).

W. A. Kalender, “X-ray computed tomography—state of the art,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. I511 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 10–27.

G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. I511 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

R. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994).

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389, (1995).

O. W. van Assendelft, Spectrophotometry of Haemoglobin Derivatives (Thomas, Springfield, Ill., 1970).

R. Berg, S. Andersson-Engels, S. Svanberg, “Time-resolved transillumination imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 397–424.

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Figures (17)

Fig. 1
Fig. 1

(a) Longitudinal view of a portion of the cylindrical reference medium with coordinates z and ρ explicitly indicated, illustrating the division of the volume into layers by planes, (Z1 and Z2) perpendicular to the cylinder axis, i.e., by surfaces of constant z, and the division of the layers into rings by a set of surfaces of constant ρ (P1 and P2). (b) A cross-sectional view, showing the voxel geometry and the cylindrical coordinate system used in this study; the z direction is perpendicular to the plane of the sketch.

Fig. 2
Fig. 2

(a) Longitudinal view of a portion of the cylindrical reference medium with coordinates z and ρ explicitly indicated, illustrating the division of the surface into bands by planes (H1 and H2) perpendicular to the cylinder axis and the bands into sectors by a second set of planes (V1 and V2) intersecting along the axis. (b) Simulation source and detector configurations.

Fig. 3
Fig. 3

Phantoms modeled in the Monte Carlo simulations: (a) centered rod, (b) off-axis rod, and (c) 13 rods.

Fig. 4
Fig. 4

Monte Carlo simulation results: polar plot of the logarithms of the absolute intensities (R 0) and the intensity differences (ΔR = R 0R).

Fig. 5
Fig. 5

(a) Monte Carlo simulation results: intensity differences (ΔR = R 0R) and their second-order least-squares fit for the centered absorber. The correlation coefficient of the fit is 0.73. (b) Monte Carlo simulation results: intensity differences (ΔR = R 0R) for the off-center absorber; the 270° data are the mirror image (with respect to 180°) of the 90° data.

Fig. 6
Fig. 6

Two-dimensional reconstructed images of the centered-rod phantom after 10,000 iterations: (a) The target: the white disk represents a cross-sectional cut through the cylinder, and the black area within it indicates the location, the size, and the shape of the heterogeneity (Δμa = ∞). Reconstructions by the (b) POCS, (c) CGD, and (d) SART algorithms. The maximum value of the reconstructed Δμa is explicitly indicated on the linear scale, shading gradually from white to black, under each image.

Fig. 7
Fig. 7

2-D reconstruction results of the off-center rod phantom after 10,000 iterations: (a) The target, (b) POCS reconstruction, (c) CGD reconstruction, and (d) SART reconstruction.

Fig. 8
Fig. 8

2-D reconstructed images of the 13-rod phantom after 10,000 iterations: (a) The target. Reconstructions by the (b) POCS, (c) CGD, (d) SART, (10,000 iterations), and (e) SART (100,000 iterations) algorithms.

Fig. 9
Fig. 9

3-D reconstructed images of the centered-rod phantom after 1,000 iterations: (a) The target. Reconstructions by the (b) POCS, (c) CGD, and (d) SART algorithms.

Fig. 10
Fig. 10

3-D reconstruction results of the off-center rod phantom after 10,000 iterations: (a) The target. Reconstructions by the (b) POCS, (c) CGD, and (d) SART algorithms.

Fig. 11
Fig. 11

3-D reconstructed images of the 13-rod phantom after 1,000 iterations: (a) The target. Reconstructions by the (b) POCS, (c), CGD, and (d) SART algorithms.

Fig. 12
Fig. 12

2-D reconstruction results of the off-center rod phantom with the weights on only the plane z = 0 (i.e., 2-D limited reconstruction) after 10,000 iterations: (a) The target. Reconstructions by the (b) POCS, (c) CGD, and (d) SART algorithms.

Fig. 13
Fig. 13

Mean-squared error versus the number of iterations for different algorithms: (a) unconstrained and (b) constrained reconstructions.

Fig. 14
Fig. 14

Image reconstructed without positivity constraints (left), image reconstructed with positivity constraints (center), and target (right) by the CGD algorithm after 10,000 iterations: (a) centered and (b) off-center rod phantom.

Fig. 15
Fig. 15

Reconstructed images obtained by the POCS algorithm and (a) no rescaling (W), (b) rescaling the maximum of each column to 1 (W′), or (c) rescaling the average of each column to 1 (W″). Results are plotted after 100 (left), 1000 (center), and 10,000 (right) iterations.

Fig. 16
Fig. 16

Same as Fig. 15 except that the reconstructed images are obtained by the CGD algorithm.

Fig. 17
Fig. 17

Same as Fig. 15 except that the reconstructed images are obtained by the SART algorithm.

Equations (18)

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Ω ϕ ( r , Ω ) + μ T ( r ) ϕ ( r , Ω ) 4 π μ s ( r , Ω Ω ) × ϕ ( r , Ω ) d Ω = s ( r , Ω ) ,
Δ R = V w a ( r ) Δ μ a ( r ) d 3 r ,
Δ R = V 4 π Δ μ a ( r ) ϕ ( r , Ω ) ϕ + ( r , Ω ) d Ω d 3 r .
w a ( r ) = 4 π ϕ ( r , Ω ) ϕ + ( r , Ω ) d Ω .
w a ( r ) = 1 4 π ϕ ( r ) ϕ + ( r ) ,
ϕ ( r ) = 4 π ϕ ( r , Ω ) d Ω , ϕ + ( r ) = 4 π ϕ + ( r , Ω ) d Ω .
Δ R i = j w a , i j Δ μ a , j ,
W a Δ μ a = Δ R ,
W a = [ w a , 11 w a , 12 w a , 1 J w a , 21 w a , 22 w a , 2 J w a , I 1 w a , I 2 w a , I J ] , Δ μ a = [ Δ μ a , 1 Δ μ a , 2 Δ μ a , J ] , Δ R = [ Δ R 1 Δ R 2 Δ R I ] .
E = ( W Δ μ a Δ R ) T ( W Δ μ a Δ R ) = Δ μ a T A Δ μ a b T Δ μ a + Δ R T Δ R ,
g ( Δ μ a ) = E Δ μ a = A Δ μ a b = 0 ,
Δ μ n + 1 = P L P L 1 P 1 Δ μ n .
Δ μ a , j n = Δ μ a , j n 1 + i = 1 I w i j Δ R i j = 1 J w i j Δ μ a , j n 1 j = 1 J w i j i = 1 I w i j .
Δ μ a n = Δ μ a n 1 α n d n ,
α n = g n 1 2 Wd n 2 , d n = g n 1 + β n d n 1 , β n = g n 1 2 g n 2 2 , g n 1 = A Δ μ a n b = g n 2 α n 1 Ad n 1 .
Δ μ a , j n = 0 if Δ μ a , j n < 0 .
r E n E n 1 = E ( Δ μ a n ) E ( Δ μ a n 1 ) ,
μ tr = ( 1 g ) μ s + μ a ,

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