## Active constrained truncated Newton method for simple-bound optical tomography

JOSA A, Vol. 17, Issue 9, pp. 1627-1641 (2000)

http://dx.doi.org/10.1364/JOSAA.17.001627

Enhanced HTML Acrobat PDF (464 KB)

### Abstract

In the past, nonlinear unconstrained optimization of the optical imaging problem has focused on Newton–Raphson techniques. Besides requiring expensive computation of the Jacobian, the unconstrained minimization with Tikhonov regularization can pose significant storage problems for large-scale reconstructions, involving a large number of unknowns necessary for realization of optical imaging. We formulate the inverse optical imaging problem as both simple-bound constrained and unconstrained minimization problems in order to illustrate the reduction in computational time and storage associated with constrained image reconstructions. The forward simulator of excitation and generated fluorescence, consisting of the Galerkin finite-element formulation, is used in an inverse algorithm to find the spatial distribution of absorption and lifetime that minimizes the difference between predicted and synthetic frequency-domain measurements. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The reconstruction results confirm that the physically based, constrained minimization with efficient optimization schemes may offer a more logical approach to the large-scale optical imaging problem than unconstrained minimization with regularization.

© 2000 Optical Society of America

**OCIS Codes**

(110.6960) Imaging systems : Tomography

**History**

Original Manuscript: May 9, 2000

Revised Manuscript: January 23, 2001

Manuscript Accepted: December 20, 2000

Published: September 1, 2000

**Citation**

Ranadhir Roy and Eva M. Sevick-Muraca, "Active constrained truncated Newton method for simple-bound optical
tomography," J. Opt. Soc. Am. A **17**, 1627-1641 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-9-1627

Sort: Year | Journal | Reset

### References

- J. S. Reynolds, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997). [CrossRef] [PubMed]
- E. M. Sevick, J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, M. L. Johnson, “Frequency domain imaging of absorbers obscured by scattering,” J. Photochem. Photobiol., B 16, 169–185 (1992). [CrossRef]
- J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991). [CrossRef]
- A. M. Siegel, J. J. A. Marota, D. A. Boas, “Design and evaluation of a continuous-wave diffuse optical tomography system,” Opt. Express 4, 287–298 (1999). [CrossRef] [PubMed]
- V. Ntziachristos, “Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998). [CrossRef]
- R. L. Barbour, R. Andronics, Q. Sha, H. L. Graber, I. Soller, “Development and evaluation of the IRIS-OPI scanner, a general-purpose optical tomography imaging system,” in Advances in Optical Imaging and Photon Migration, J. G. Fujimoto, M. S. Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 251–255.
- K. Wells, J. C. Hebden, F. E. W. Schmidt, D. T. Delpy, “The UCL multichannel time-resolved system for optical tomography,” in Optical Tomography and Spectroscopy of Tissue, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 590–607 (1997).
- M. Miwa, Y. Ueda, “Development of time-resolved spectroscopy system for quantitative noninvasive tissue measurement,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 142–149 (1995).
- S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. (Bellingham) 34, 32–42 (1995). [CrossRef]
- S. J. Madsen, E. R. Anderson, R. C. Haskell, B. J. Tromberg, “Portable, high-bandwidth frequency-domain photon migration instrument for tissue spectroscopy,” Opt. Lett. 19, 1934–1936 (1994). [CrossRef] [PubMed]
- E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photonics News, July1996, pp. 25–28.
- M. A. O’Leary, D. A. Boas, D. X. L. B. Chance, A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996). [CrossRef] [PubMed]
- D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997). [CrossRef] [PubMed]
- J. Chang, H. L. Graberm, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997). [CrossRef]
- E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997). [CrossRef] [PubMed]
- D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical heterogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994). [CrossRef]
- M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]
- T. M. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999). [CrossRef] [PubMed]
- S. A. Walker, S. Fantini, E. Gratton, “Back-projection reconstructions in cylindrical inhomogeneities from frequency-domain optical measurements in turbid media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C.), 1992, pp. 137–141.
- S. B. Colak, G. W. Hooft, D. G. Papaioannou, M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1992), pp. 294–298.
- R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.
- Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997). [CrossRef]
- W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1995). [CrossRef]
- J. C. Ye, K. J. Webb, R. P. Millane, T. J. Downar, “Modified distorted Born iterative method with an approximate Frechet derivative for optical diffusion tomography,” J. Opt. Soc. Am. A 16, 1814–1830 (1999). [CrossRef]
- S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992). [CrossRef]
- H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996). [CrossRef]
- K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–710 (1995). [CrossRef] [PubMed]
- R. Roy, “Image reconstruction from light measurements on biological tissue,” Ph.D. thesis (in Mathematics) (University of Hertfordshire, Hatfield, England, 1996).
- S. R. Arridge, M. Schweiger, “A gradient-based optimization scheme for optical tomography,” Opt. Express 2, 213–226 (1997). [CrossRef]
- A. H. Hielscher, A. D. Klose, K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271 (1999). [CrossRef] [PubMed]
- A. D. Klose, A. H. Hielscher, “Iterative reconstructions scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999). [CrossRef] [PubMed]
- R. Roy, E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence opticaltomography: Part I: Theory and formulation,” Opt. Express 4, 353–371 (1999). [CrossRef] [PubMed]
- R. Roy, E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part II: Reconstruction from synthetic measurements,” Opt. Express 4, 372–382 (1999). [CrossRef] [PubMed]
- A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).
- V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417 (1966).
- G. Wahba, Spline Models of Observational Data (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
- P. C. Hansen, “Analysis of discrete ill-posed problem by means of the L-curve,” SIAM Rev. 34, 561–580 (1992). [CrossRef]
- A. H. Hielscher, A. D. Klose, “Use of a priori information and penalty terms in gradient-based iterative reconstruction schemes,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. Tromberg, ed., Proc. SPIE3597, 36–44 (1999).
- B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999). [CrossRef]
- M. J. Eppstein, D. E. Dougherty, T. L. Troy, E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parameterization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38, 2138–2150 (1999). [CrossRef]
- B. A. Murtagh, M. A. Saunders“Large-scale linearly constrained optimization,” Math. Program. 14, 41–72 (1978). [CrossRef]
- B. A. Murtagh, M. A. Saunders, “A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints,” Math. Program. 16, 4–117 (1982).
- R. Fletcher, M. P. Jackson, “Minimization of a quadratic function on many variables subject only to upper and lower bounds,” J. Inst. Math. Appl. 14, 159–174 (1974). [CrossRef]
- P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
- A. R. Conn, I. M. Gould, Ph. L. Toint, LANCELOT: a Fortran Package for Large-Scale Nonlinear Optimization (release A), Vol. 17 of Computational Mathematics Series (Springer-Verlag, New York, 1992). [CrossRef]
- A. R. Conn, I. M. Gould, Ph. L. Toint, “Testing a class of methods for solving minimization problems with simply bounds on the variables,” Math. Comput. 50, 399–430 (1988). [CrossRef]
- F. Facchine, J. Judice, J. Soares, “An active set Newton algorithm for large-scale nonlinear programs with box constraints,” SIAM J. Optim. 8, 158–186 (1998). [CrossRef]
- R. Pytlak, “An efficient algorithm for large-scale nonlinear programming problems with simple bounds on the variables,” SIAM J. Optim. 8, 532–560 (1998). [CrossRef]
- M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 97–105 (1999). [CrossRef]
- B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992). [CrossRef]
- S. R. Arridge, M. R. Schweiger, “Image reconstruction in optical tomography,” Philos. Trans. R. Soc. London, Ser. B 352, 717–726 (1997). [CrossRef] [PubMed]
- J. C. Hebden, S. Arridge, D. T. Delpy, “Optical imagingeb in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997). [CrossRef] [PubMed]
- S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997). [CrossRef] [PubMed]
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
- O. C. Zienkiewcz, R. L. Taylor, The Finite Element Methods in Engineering Science (McGraw-Hill, New York, 1989).
- D. P. Bertsekas, “Projected Newton method for optimization problems with simple contraints,” SIAM J. Control Optim. 20, 221–246 (1982). [CrossRef]
- Q. Ni, Y. Yuan, “A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization,” Math. Comput. 66, 1509–1520 (1997). [CrossRef]
- L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math. 16, 1–3 (1966). [CrossRef]
- R. S. Dembo, T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimization,” Math. Program. 26, 190–212 (1983). [CrossRef]
- L. C. W. Dixon, R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. Optim. Theory Appl. 56, 245–255 (1988). [CrossRef]
- P. Wolfe, “Convergence condition for ascent method,” SIAM Rev. 11, 226–253 (1969). [CrossRef]
- A. Griewank, “On automatic differentiation,” in Mathematical Programming: Recent Developments and Applications, M. Iri, K. Tanaka, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 83–108.
- B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. Software 8, 53–67 (1997). [CrossRef]
- A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. Eng. Softw. 28, 217–221 (1997). [CrossRef]
- J. Lee, E. M. Sevick-Muraca, “Lifetime and absorption imaging with fluorescence FDPM,” in Time-Resolved Fluorescence Spectroscopy and Imaging in Tissues, E. M. Sevick-Muraca, ed. Proc. SPIE3600, 246–254 (1999).
- R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in the matoporhyrin derivate–sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997). [CrossRef] [PubMed]
- H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998). [CrossRef]
- M. Schweiger, S. R. Arridge, “Comparison of two- and three-dimensional reconstruction methods in optical tomography,” Appl. Opt. 37, 7419–7428 (1998). [CrossRef]
- Y. Yao, Y. Pei, Y. Wang, R. L. Barbour, “Born-iterative methods for imaging of heterogeneous scattering media and its application to simulated breast tissue,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 231–239 (1997).
- W. Cai, B. B. Das, F. Liu, F. A. Feng, M. Lax, R. R. Alfano, “Three-dimensional image reconstruction in highly scattering turbid media,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, B. Chance, R. R. Alfano, eds. Proc. SPIE2979, 241–249 (1997).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.