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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics


  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 7 — Jul. 27, 2011

Feasibility of U-curve method to select the regularization parameter for fluorescence diffuse optical tomography in phantom and small animal studies

Judit Chamorro-Servent, Juan Aguirre, Jorge Ripoll, Juan José Vaquero, and Manuel Desco  »View Author Affiliations

Optics Express, Vol. 19, Issue 12, pp. 11490-11506 (2011)

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When dealing with ill-posed problems such as fluorescence diffuse optical tomography (fDOT) the choice of the regularization parameter is extremely important for computing a reliable reconstruction. Several automatic methods for the selection of the regularization parameter have been introduced over the years and their performance depends on the particular inverse problem. Herein a U-curve-based algorithm for the selection of regularization parameter has been applied for the first time to fDOT. To increase the computational efficiency for large systems an interval of the regularization parameter is desirable. The U-curve provided a suitable selection of the regularization parameter in terms of Picard’s condition, image resolution and image noise. Results are shown both on phantom and mouse data.

© 2011 OSA

OCIS Codes
(100.3190) Image processing : Inverse problems
(110.6960) Imaging systems : Tomography
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(260.2510) Physical optics : Fluorescence

ToC Category:
Medical Optics and Biotechnology

Original Manuscript: February 11, 2011
Revised Manuscript: April 24, 2011
Manuscript Accepted: April 25, 2011
Published: May 31, 2011

Virtual Issues
Vol. 6, Iss. 7 Virtual Journal for Biomedical Optics

Judit Chamorro-Servent, Juan Aguirre, Jorge Ripoll, Juan José Vaquero, and Manuel Desco, "Feasibility of U-curve method to select the regularization parameter for fluorescence diffuse optical tomography in phantom and small animal studies," Opt. Express 19, 11490-11506 (2011)

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  1. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007). [CrossRef] [PubMed]
  2. X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003). [CrossRef] [PubMed]
  3. A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt. 42(25), 5181–5190 (2003). [CrossRef] [PubMed]
  4. Y. Xu, X. J. Gu, L. L. Fajardo, and H. B. Jiang, “In vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations,” Appl. Opt. 42(16), 3163–3169 (2003). [CrossRef] [PubMed]
  5. A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).
  6. V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002). [CrossRef]
  7. 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(3), 313–320 (2005). [CrossRef] [PubMed]
  8. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]
  9. J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin, 49–52 (1902).
  10. Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).
  11. C. R. Vogel, ed., Computational Methods for Inverse Problems (SIAM, 2002).
  12. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004). [CrossRef]
  13. N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009). [CrossRef] [PubMed]
  14. A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009). [CrossRef] [PubMed]
  15. J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30(2), 235–247 (2003). [CrossRef] [PubMed]
  16. A. Serdaroglu, B. Yazici, and V. Ntziachristos, "Fluorescence molecular tomography based on a priori information," in Biomedical Optics, Technical Digest (CD) (Optical Society of America, 2006), paper SH46, http://www.opticsinfobase.org/abstract.cfm?URI=BIO-2006-SH46 .
  17. Z. Xu, J. Yan, and J. Bai, “Determining the regularization parameter: a hybrid reconstruction technique in fluorescence molecular tomography,” in Communications and Photonics Conference and Exhibition (ACP),2009Asia 2009), 1 - 2.
  18. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34(4), 561–580 (1992). [CrossRef]
  19. P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993). [CrossRef]
  20. J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008). [CrossRef] [PubMed]
  21. T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009). [CrossRef] [PubMed]
  22. H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997). [CrossRef]
  23. D. Krawczyk-Stańdo and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems-the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007). [CrossRef]
  24. D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008). [CrossRef]
  25. L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010). [CrossRef]
  26. P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT 30(4), 658–672 (1990). [CrossRef]
  27. J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Co-planar FMT-CT,” in 2008 World Molecular Imaging Congress (WMIC), (2008).
  28. J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” 2008 IEEE Nuclear Science Symposium Conference Record, 5412–5413 (2008).
  29. J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003). [CrossRef] [PubMed]
  30. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978) vol. 1.
  31. I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988). [CrossRef] [PubMed]
  32. S. R. Arridge, “Photon-measurement density functions. part I: analytical forms,” Appl. Opt. 34(31), 7395–7409 (1995). [CrossRef] [PubMed]
  33. V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. role of boundary conditions,” J. Opt. Soc. Am. A 19(3), 558–566 (2002). [CrossRef]
  34. A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005). [CrossRef] [PubMed]
  35. M. Born and E. Wolf, Principles of Optics (University Press, 1999).
  36. G. H. Golub and C. F. Van Loan, Matrix computations (Johns Hopkins University Press, 1996), p. 694.
  37. P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. 27(4), 534–553 (1987). [CrossRef]
  38. P. C. Hansen, Regularization Tools Version 4.0 for MATLAB 7.3 (Springer, 2007), pp. 189–194.
  39. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997). [CrossRef] [PubMed]
  40. J. Chamorro, J. Aguirre, J. Ripoll, J. J. Vaquero, and M. Desco, “FDOT setting optimization and reconstruction using singular value analysis with automatic thresholding,” in Nuclear Science Symposium and Medical Imaging Conference (IEEE), (2009).
  41. J. Chamorro-Servent, J. Aguirre, J. Ripoll, J. J. Vaquero, and M. Desco, “Maximizing the information content in acquired measurements of a parallel plate non-contact FDOT while minimizing the computational cost: singular value analysis,” in 4th European Molecular Imaging Meeting (ESMI), (2009).
  42. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. 26(10), 701–703 (2001). [CrossRef]
  43. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007). [CrossRef] [PubMed]
  44. M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT 36(2), 287–301 (1996). [CrossRef]
  45. C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12(4), 535–548 (1996). [CrossRef]
  46. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38(13), 2950–2961 (1999). [CrossRef]

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