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

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

http://dx.doi.org/10.1364/OE.19.011490

Acrobat PDF (1766 KB)

### Abstract

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

## 1. Introduction

*in vivo*have been developed. These techniques are able to retrieve the 3D distribution of a wide set of intrinsic and extrinsic optical contrasts, and have many preclinical and clinical applications, such as breast imaging [1

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]

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).

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]

*d*is a vector that contains the measurements,

*f*represents the unknown contrast concentration at each voxel, and

*W*is a weight matrix that relates the measurements and the unknowns.

8. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41–R93 (1999). [CrossRef]

*α*, is manually selected, using a sequence of regularization parameters and selecting the parameter value that leads to the best results, as judged by the operator . However, this procedure is subjective and slow. To avoid this, several automatic methods for selecting regularization parameters have been suggested over the years, for instance: the Unbiased predictive risk estimator method (UPRE), the Discrepancy principle (DP) and strategies based on the calculation of the variance of the solution that require prior knowledge of the noise, or others such as Generalized cross-validation (GCV) and L-curve (LC) that do not need a-priori information.

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]

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]

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.

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]

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]

- - The U-curve method provides an interval where the optimal regularization parameter exists. Given that any value out of this interval is not a candidate for being a regularization parameter, the computation of U-curve for values out of this interval is not necessary. Thus, the use of this interval can greatly increase the computational efficiency in selecting the regularization parameter.
- - The U-curve performed better than the L-curve, obtaining a better reconstruction result that can retain its robustness and efficacy in blurred or noisy situations.

26. P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT **30**(4), 658–672 (1990). [CrossRef]

## 2. Materials and methods

### 2.1 fDOT experimental set up

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]

### 2.2 Weight matrix formulation. Forward problem

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]

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]

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]

*r*by a source at position

*r*to the detector at location

*r*multiplied by the fluorescence yield and

*h*is the voxel volumen.

### 2.3 Tikhonov solution based on SVD

*U*and

*V*are orthonormal matrices MxM and NxN, respectively, containing the left and right singular vectors of

*W*. ∑ is an MxN diagonal matrix that contains the singular values (σ

_{i}) of

*W*.

37. P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. **27**(4), 534–553 (1987). [CrossRef]

*ith*column of the matrices

*U*and

*V*respectively.

37. P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. **27**(4), 534–553 (1987). [CrossRef]

### 2.4 L-curve method

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]

### 2.5 Selection of the noise threshold by the U-curve method (regularization parameter)

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]

**17**(2), 157–164 (2007). [CrossRef]

*α*,

*α*took 200 values equally spaced from

### 2.6 Experiments

#### 2.6.1 Phantom experiments

_{a}=0.3 cm

^{−1}and a reduced scattering coefficient of μ

_{s}=10 cm

^{−1}[39

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]

#### 2.6.2 Ex-vivo mouse data

### 2.7 Validation of the regularization parameter obtained by the U-curve method

*α*parameters in the 10

^{−1}to 10

^{−6}range, which included the U-curve-based regularization parameter,

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]

*σ*. The Discrete Picard’s condition (DPC) [26

_{i}26. P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT **30**(4), 658–672 (1990). [CrossRef]

*d*is said to satisfy the DPC if the data space coefficients

*σ*[26

_{i}26. P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT **30**(4), 658–672 (1990). [CrossRef]

**30**(4), 658–672 (1990). [CrossRef]

## 3. Results

### 3.1 Phantom

^{−2}.

*α*parameters in the 10

^{−1}to 10

^{−6}range and with

^{−2}.

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]

*α*values of 10

^{−1}, 4.38*10

^{−2}(

^{−2}, the FWHM of the profiles decreases. For

*α*= 10

^{−3}the trend is truncated because image noise starts to prevail, while for

*α*<10

^{−3}the reconstruction is noise only, and the object is no longer visible in the reconstructed images. Taking these considerations into account, we observe a heuristic range of

^{−1}and 10

^{−3}).

^{−3}, on average, decay to zero faster than those of the respective

^{−1}and 10

^{−3}, particularly the singular values above the U-curve cut-off (4.38*10

^{−2}), fulfill Picard’s condition.

### 3.2 Ex-vivo experiments

**=**5.72*10

^{−2}.

*α*parameters in the 10

^{−1}to 10

^{−6}range and with

**=**5.72*10

^{−2}.

*α*values that produces reconstructed images with a reasonable amount of noise and resolution. The U-curve-based value fall within this range, which is

^{−1}and 10

^{−2}).

^{−1}and 10

^{−2}, particularly the singular values above the U-curve cut-off (5.72*10

^{−2}), fulfill Picard’s condition. The singular values decay faster than the singular values of Figs. 7(c)–7(d) (for phantom experiment), therefore the problem for the ex-vivo experiment is more ill-posed than the problem for the phantom experiment.

## 4. Discussion and conclusion

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]

43. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**(4), 389–399 (2007). [CrossRef] [PubMed]

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]

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.

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]

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]

22. H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. **63**(2), 243–248 (1997). [CrossRef]

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 *4 ^{th} European Molecular Imaging Meeting (ESMI)*, (2009).

*α*parameters out of this interval). Thus, this is a case that shows the robustness and the computation effectiveness of the U-curve method.

*α*parameters) and Figs. 6 and 11 (profiles and FWHM versus noise plot), we can choose values for the regularization parameter that are lower than the value at which the reconstructed image started to be noisy. Figures 7(d) and 12 demonstrated that, for these lower values, Picard’s condition is satisfied, as the U-curve parameter is in this range.

**30**(2), 235–247 (2003). [CrossRef] [PubMed]

*ex-vivo*experiment, the resolution is better (Fig. 11(b) versus Fig. 6(b)), due to the fact that the tip of the capillary is closer to the surface than in the phantom experiment. As the resolution of DOT systems are depth-dependent, resolution improves the closer the object is to either side of the slab [46

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]

**17**(2), 157–164 (2007). [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]

## Acknowledgments

## References and links

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 |

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. |

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. |

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. |

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 |

6. | V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging |

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. |

8. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

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. |

11. | C. R. Vogel, ed., |

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 |

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. |

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. |

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. |

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 |

18. | P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. |

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. |

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. |

21. | T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. |

22. | H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. |

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. |

24. | D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. |

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 |

27. | J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Co-planar FMT-CT,” in |

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. |

30. | A. Ishimaru, |

31. | I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. |

32. | S. R. Arridge, “Photon-measurement density functions. part I: analytical forms,” Appl. Opt. |

33. | V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. role of boundary conditions,” J. Opt. Soc. Am. A |

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 |

35. | M. Born and E. Wolf, |

36. | G. H. Golub and C. F. Van Loan, |

37. | P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. |

38. | P. C. Hansen, |

39. | R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. |

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 |

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 |

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. |

43. | T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. |

44. | M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT |

45. | C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. |

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. |

**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

**History**

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*

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11490

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### References

- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]
- J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin, 49–52 (1902).
- Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).
- C. R. Vogel, ed., Computational Methods for Inverse Problems (SIAM, 2002).
- 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]
- 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]
- 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]
- 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]
- 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 .
- 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.
- P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34(4), 561–580 (1992). [CrossRef]
- 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]
- 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]
- 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]
- H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997). [CrossRef]
- 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]
- 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]
- 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]
- P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT 30(4), 658–672 (1990). [CrossRef]
- J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Co-planar FMT-CT,” in 2008 World Molecular Imaging Congress (WMIC), (2008).
- 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).
- 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]
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978) vol. 1.
- 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]
- S. R. Arridge, “Photon-measurement density functions. part I: analytical forms,” Appl. Opt. 34(31), 7395–7409 (1995). [CrossRef] [PubMed]
- 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]
- 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]
- M. Born and E. Wolf, Principles of Optics (University Press, 1999).
- G. H. Golub and C. F. Van Loan, Matrix computations (Johns Hopkins University Press, 1996), p. 694.
- P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. 27(4), 534–553 (1987). [CrossRef]
- P. C. Hansen, Regularization Tools Version 4.0 for MATLAB 7.3 (Springer, 2007), pp. 189–194.
- 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]
- 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).
- 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).
- 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]
- T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007). [CrossRef] [PubMed]
- M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT 36(2), 287–301 (1996). [CrossRef]
- C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12(4), 535–548 (1996). [CrossRef]
- 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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