## A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization

Optics Express, Vol. 18, Issue 8, pp. 8630-8646 (2010)

http://dx.doi.org/10.1364/OE.18.008630

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### Abstract

Through the reconstruction of the fluorescent probe distributions, fluorescence molecular tomography (FMT) can three-dimensionally resolve the molecular processes in small animals *in vivo*. In this paper, we propose an FMT reconstruction algorithm based on the iterated shrinkage method. By incorporating a surrogate function, the original optimization problem can be decoupled, which enables us to use the general sparsity regularization. Due to the sparsity characteristic of the fluorescent sources, the performance of this method can be greatly enhanced, which leads to a fast reconstruction algorithm. Numerical simulations and physical experiments were conducted. Compared to Newton method with Tikhonov regularization, the iterated shrinkage based algorithm can obtain more accurate results, even with very limited measurement data.

© 2010 OSA

## 1. Introduction

*In vivo*small animal optical molecular imaging has become an important and rapidly developing method for biomedical research, and has been widely utilized for cancer detection, drug discovery and gene expression visualization, etc [1

1. 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]

4. J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. **27**(5), 48–57 (2008). [CrossRef] [PubMed]

5. C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. **4**(1), 235–260 (2002). [CrossRef] [PubMed]

6. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. **8**(1), 1–33 (2006). [CrossRef] [PubMed]

7. X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express **15**(26), 18300–18317 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=OE-15-26-18300. [CrossRef] [PubMed]

*a priori*information should be incorporated to regularize the FMT problem [8

8. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse a priori Information,” Opt. Express **17**(10), 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062. [CrossRef] [PubMed]

10. D. Wang, X. Song, and J. Bai, “Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution,” Opt. Express **15**(15), 9722–9730 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9722. [CrossRef] [PubMed]

11. W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. **24**(3), 034011 (2008). [CrossRef]

12. N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express **15**(21), 13695–13708 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13695. [CrossRef] [PubMed]

8. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse a priori Information,” Opt. Express **17**(10), 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062. [CrossRef] [PubMed]

9. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. **46**(10), 1679–1685 (2007). [CrossRef] [PubMed]

12. N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express **15**(21), 13695–13708 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13695. [CrossRef] [PubMed]

9. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. **46**(10), 1679–1685 (2007). [CrossRef] [PubMed]

*a priori*information for FMT. A straightforward way to integrate the sparsity constraint is to replace the Tikhonov method with L0-norm regularization. However, the optimization problem becomes NP-hard in this case, and cannot be solved efficiently [13

13. I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. **45**(3), 600–616 (1997). [CrossRef]

*p*is within this range, large values in the solution are penalized less severely compared with Tikhonov regularization. Therefore, the Lp regularization (

8. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse a priori Information,” Opt. Express **17**(10), 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062. [CrossRef] [PubMed]

13. I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. **45**(3), 600–616 (1997). [CrossRef]

14. E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. **59**(8), 1207–1223 (2006). [CrossRef]

15. I. Daubechies, M. Defrise, and C. DeMol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. **57**(11), 1413–1457 (2004). [CrossRef]

## 2. Method

### 2.1 Photon propagation model

16. F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express **16**(17), 13104–13121 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-17-13104. [CrossRef] [PubMed]

20. C. Qin, J. Tian, X. Yang, K. Liu, G. Yan, J. Feng, Y. Lv, and M. Xu, “Galerkin-based meshless methods for photon transport in the biological tissue,” Opt. Express **16**(25), 20317–20333 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20317. [CrossRef] [PubMed]

*x*and

*m*denote the excitation and emission wavelengths, respectively;

*Ω*denotes the domain of the problem;

*g*is the anisotropy parameter;

*Θ*denotes the amplitude of the point sources. The coupled equations are complemented by Robin-type boundary conditions which depict the refractive index mismatch between the external medium and Ω:where

*A*is a constant depending on the optical reflective index mismatch at the boundary [21

21. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. **22**(11), 1779–1792 (1995). [CrossRef] [PubMed]

### 2.2 Linear relationship establishment

*F*is obtained by discretizing the unknown fluorescent yield distribution. Vector

*X*denotes the fluorescent yield to be reconstructed. For each excitation point source at

*inverse crime*problem,

*X*with linear elements. As matrix

7. X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express **15**(26), 18300–18317 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=OE-15-26-18300. [CrossRef] [PubMed]

### 2.3 Reconstruction based on iterated shrinkage method

*A*is not zero. Therefore, the solution is not unique in this case. Even if the FMT problem becomes less ill-posed when more fluorescence measurement data can be captured by the CCD camera, it can still remain ill-conditioned (with a large condition number). Therefore, errors in the FMT problem can be magnified, which will affect the reconstruction results. Errors are inevitable and can be introduced in several ways, e.g. the fluorescence measurement errors caused by CCD camera noise and the approximation errors caused by data interpolation. A standard way to regularize the problem is to incorporate additional constraints on the solution, which can be considered as a kind of

*a priori*information:where

*λ*is a positive real number called the regularization parameter which is used to balance the two terms. In this paper,

*X*is considered to be non-negative. When L2-norm penalty function is used, this becomes the popular Tikhonov regularization method. Here, we only consider the case when

*N*is the dimension of

*X*. Here, we define

*X*is equivalent to solving each

15. I. Daubechies, M. Defrise, and C. DeMol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. **57**(11), 1413–1457 (2004). [CrossRef]

*c*is a constant scalar and

*c*should be chosen so that

*X*and setting the result to zero, we have:Let

*Shrink*:This function maps the values smaller than the threshold to zero; values larger than the threshold are “shrinked”, thus the name of the function. An important feature of the

*Shrink*function (not limited to

*Shrink*function, the optimal solution to the optimization problem can be represented as:

*c*satisfying

*c*can be set to the largest Eigen value plus

*ε*, where

*ε*is a small positive value. However, this could be rather time-consuming. Therefore, we provide two alternative ways to determine

*c*. For the first one, we estimate the upper bound of the maximum Eigen value instead of actually calculating it, and

*c*can be set to the upper bound plus

*ε*. Several estimation algorithms can be adopted, e.g. the popular Minc method [24]. The second way is to calculate the maximum Eigen value directly using e.g. the Power method [25

25. M. T. Chu and J. L. Watterson, “On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method,” SIAM J. Sci. Comput. **14**(5), 1089–1106 (1993). [CrossRef]

*c*can be set similarly. Sometimes, the estimated upper bound of maximum Eigen value may be too large compared with the true value. This will increase the influence of the surrogate function on the optimization problem and will slow down the convergence rate. Therefore, for the above two ways, we prefer the latter one, because it is more accurate.

*A*and Φ. For a certain iteration

*k*,

*A*, and

*Shrink*operations) and the calculation of

*X*is considered, the performance of both strategies can be greatly improved. Suppose

*n*. When multiplying

## 3. Results

### 3.1 Simulation verifications

**17**(10), 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062. [CrossRef] [PubMed]

### 3.2 Physical experiments

## 4. Conclusion

*In vivo*mouse studies using the proposed method will be reported in the future.

## Acknowledgments

## References and links

1. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

2. | R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology |

3. | J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. |

4. | J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. |

5. | C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. |

6. | V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. |

7. | X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express |

8. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse a priori Information,” Opt. Express |

9. | P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. |

10. | D. Wang, X. Song, and J. Bai, “Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution,” Opt. Express |

11. | W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. |

12. | N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express |

13. | I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. |

14. | E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. |

15. | I. Daubechies, M. Defrise, and C. DeMol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. |

16. | F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express |

17. | Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. |

18. | A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express |

19. | J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express |

20. | C. Qin, J. Tian, X. Yang, K. Liu, G. Yan, J. Feng, Y. Lv, and M. Xu, “Galerkin-based meshless methods for photon transport in the biological tissue,” Opt. Express |

21. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. |

22. | R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. R. Stat. Soc., B |

23. | M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” in |

24. | H. Minc, Nonnegative matrices, (Wiley, New York, 1988). |

25. | M. T. Chu and J. L. Watterson, “On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method,” SIAM J. Sci. Comput. |

26. | A. X. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: January 13, 2010

Revised Manuscript: March 25, 2010

Manuscript Accepted: April 1, 2010

Published: April 9, 2010

**Virtual Issues**

Vol. 5, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Dong Han, Jie Tian, Shouping Zhu, Jinchao Feng, Chenghu Qin, Bo Zhang, and Xin Yang, "A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization," Opt. Express **18**, 8630-8646 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8630

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### References

- 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]
- R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001). [PubMed]
- J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008). [CrossRef] [PubMed]
- J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008). [CrossRef] [PubMed]
- C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4(1), 235–260 (2002). [CrossRef] [PubMed]
- V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006). [CrossRef] [PubMed]
- X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express 15(26), 18300–18317 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=OE-15-26-18300 . [CrossRef] [PubMed]
- Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse a priori Information,” Opt. Express 17(10), 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062 . [CrossRef] [PubMed]
- P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. 46(10), 1679–1685 (2007). [CrossRef] [PubMed]
- D. Wang, X. Song, and J. Bai, “Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution,” Opt. Express 15(15), 9722–9730 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9722 . [CrossRef] [PubMed]
- W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24(3), 034011 (2008). [CrossRef]
- N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express 15(21), 13695–13708 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13695 . [CrossRef] [PubMed]
- I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45(3), 600–616 (1997). [CrossRef]
- E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006). [CrossRef]
- I. Daubechies, M. Defrise, and C. DeMol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004). [CrossRef]
- F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express 16(17), 13104–13121 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-17-13104 . [CrossRef] [PubMed]
- Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35(12), 5703–5707 (2008). [CrossRef]
- A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5402 . [CrossRef] [PubMed]
- J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15640 . [CrossRef] [PubMed]
- C. Qin, J. Tian, X. Yang, K. Liu, G. Yan, J. Feng, Y. Lv, and M. Xu, “Galerkin-based meshless methods for photon transport in the biological tissue,” Opt. Express 16(25), 20317–20333 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20317 . [CrossRef] [PubMed]
- M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995). [CrossRef] [PubMed]
- R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. R. Stat. Soc., B 58, 267–288 (1996).
- M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York2, 1924–1931 (2006).
- H. Minc, Nonnegative matrices, (Wiley, New York, 1988).
- M. T. Chu and J. L. Watterson, “On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method,” SIAM J. Sci. Comput. 14(5), 1089–1106 (1993). [CrossRef]
- A. X. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13(24), 9847–9857 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-24-9847 . [CrossRef] [PubMed]

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