## Compressive sensing based reconstruction in bioluminescence tomography improves image resolution and robustness to noise |

Biomedical Optics Express, Vol. 3, Issue 9, pp. 2131-2141 (2012)

http://dx.doi.org/10.1364/BOE.3.002131

Acrobat PDF (1502 KB)

### Abstract

Bioluminescence Tomography attempts to quantify 3-dimensional luminophore distributions from surface measurements of the light distribution. The reconstruction problem is typically severely under-determined due to the number and location of measurements, but in certain cases the molecules or cells of interest form localised clusters, resulting in a distribution of luminophores that is spatially sparse. A Conjugate Gradient-based reconstruction algorithm using Compressive Sensing was designed to take advantage of this sparsity, using a multistage sparsity reduction approach to remove the need to choose sparsity weighting a priori. Numerical simulations were used to examine the effect of noise on reconstruction accuracy. Tomographic bioluminescence measurements of a Caliper XPM-2 Phantom Mouse were acquired and reconstructions from simulation and this experimental data show that Compressive Sensing-based reconstruction is superior to standard reconstruction techniques, particularly in the presence of noise.

© 2012 OSA

## 1. Introduction

1. S. Arridge and J. Hebden, “Optical imaging in medicine: II. modelling and reconstruction,” Phys. Med. Biol. **42**, 841–853 (1997). [CrossRef] [PubMed]

**Jx**=

**y**, where

**J**is a matrix representing the physics of light propagation in tissue [2

2. C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. **12**, 024007 (2007). [CrossRef] [PubMed]

**x**is a vector representing the bioluminescent source distribution (the image), and

**y**is a vector representing the resulting light distribution (the surface measurements). The

**J**matrix is commonly called a Jacobian or weight matrix, and is constructed from prior knowledge of the optical properties of the subject.

**y**through measurement, and prior knowledge of

**J**, it is necessary to reconstruct

**x**. However, BLT problems are typically strongly underdetermined as a result of limitations on the number of measurements, and correlation between measurements. For example, in the experimental data examined in this paper it is necessary to reconstruct 10171 unknowns from 526 measurements.

3. R. G. Baraniuk, “Compressive sensing,” IEEE Signal. Process. Mag. **24**, 118–124 (2007). [CrossRef]

4. E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. **23**, 969–985 (2007). [CrossRef]

5. H. Rauhut, “Compressive sensing and structured random matrices,” in *Theoretical Foundations and Numerical Methods for Sparse Recovery*, M. Massimo, ed. (deGruyter, 2010), pp. 1–92. [CrossRef]

*norm. In the case discussed here where the solution is expected to be spatially sparse, CS uses the insight that the most spatially sparse solution satisfying Eq. (1) is the exact solution with high probability (if the problem satisfies the necessary conditions [4*

^{p}4. E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. **23**, 969–985 (2007). [CrossRef]

5. H. Rauhut, “Compressive sensing and structured random matrices,” in *Theoretical Foundations and Numerical Methods for Sparse Recovery*, M. Massimo, ed. (deGruyter, 2010), pp. 1–92. [CrossRef]

^{1}norm ||

**x**||

_{1}, which induces sparsity in the solution. The L

^{0}norm, which essentially counts the number of non-zero components of

**x**, induces sparsity even more strongly, but is a discontinuous function and so more difficult to minimise. In general, L

*norms for*

^{p}*p*≤ 1 can be used to maximise sparsity.

^{1}minimization and measurement error minimization as objectives to be optimized jointly, with specified weights on each objective, as in Eq. (2): where

*λ*is a term that weights the sparsity constraint against the measurement error. It can be shown [6

6. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. **1**, 586–597 (2007). [CrossRef]

7. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magn. Reson. Med. **58**, 1182–1195 (2007). [CrossRef] [PubMed]

*ε*, is some function of

*λ*.

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**, 8062–8080 (2009). [CrossRef] [PubMed]

^{1}regularisation. He et al. [9

9. X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express **18**, 24825–24841 (2010). [CrossRef] [PubMed]

^{1}regularisation. Cong et al. [10

10. W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A **27**, 174–179 (2010). [CrossRef]

^{1}sparsity minimisation algorithm, with the constraint being a measure of the measurement discrepancy. Yu et al. [11

11. J. Yu, F. Liu, J. Wu, L. Jiao, and X. He, “Fast source reconstruction for bioluminescence tomography based on sparse regularization,” IEEE Trans. Biomed. Eng. **57**, 2583–2586 (2010). [CrossRef] [PubMed]

^{1}regularisation. Gao et al. [12

12. H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,” Opt. Express **18**, 1854–1871 (2010). [CrossRef] [PubMed]

13. H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express **18**, 2894–2912 (2010). [CrossRef] [PubMed]

^{1}and Total Variation-based regularisation, to solve for sub-problems on meshes of different resolution and with an adaptive region of interest. The solution they proposed to choose the correct balance of L

^{1}and Total Variation weighting required additional prior knowledge of the bioluminescence distribution. He et al. [14

14. X. He, Y. Hou, D. Chen, Y. Jiang, M. Shen, J. Liu, Q. Zhang, and J. Tian, “Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method,” Int. J. Biomed. Imaging **2011**, 203537 (2011). [CrossRef]

^{1}regularisation. Liu et al. [15

15. K. Liu, J. Tian, C. Qin, X. Yang, S. Zhu, D. Han, and P. Wu, “Tomographic bioluminescence imaging reconstruction via a dynamically sparse regularized global method in mouse models,” J. Biomed. Opt. **16**, 046016 (2011). [CrossRef] [PubMed]

*norm, and explored how sensitive the reconstruction process is to the particular regularisation weight used. Zhang et al. [16*

^{p}16. Q. Zhang, H. Zhao, D. Chen, X. Qu, X. He, X. Chen, W. Li, Z. Hu, J. Liu, and J. Liang, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. **284**, 5871–5876 (2011). [CrossRef]

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**, 8062–8080 (2009). [CrossRef] [PubMed]

9. X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express **18**, 24825–24841 (2010). [CrossRef] [PubMed]

11. J. Yu, F. Liu, J. Wu, L. Jiao, and X. He, “Fast source reconstruction for bioluminescence tomography based on sparse regularization,” IEEE Trans. Biomed. Eng. **57**, 2583–2586 (2010). [CrossRef] [PubMed]

15. K. Liu, J. Tian, C. Qin, X. Yang, S. Zhu, D. Han, and P. Wu, “Tomographic bioluminescence imaging reconstruction via a dynamically sparse regularized global method in mouse models,” J. Biomed. Opt. **16**, 046016 (2011). [CrossRef] [PubMed]

6. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. **1**, 586–597 (2007). [CrossRef]

## 2. Methods

7. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magn. Reson. Med. **58**, 1182–1195 (2007). [CrossRef] [PubMed]

^{1}norm in the form of Eq. (4) was used to simplify reconstruction: with the magnitude of

*μ*insignificant relative to the magnitudes of non-zero

*x*. The algorithm solves a number of sub problems for decreasing values of

_{i}*λ*, using the solution to the previous sub problem as a prior for the next. As

*λ*becomes small, the problem tends to Eq. (1), and so the solution produced by CSCG is also locally optimal in terms of measurement error. The initial value of

*λ*is chosen as Eq. (5) so that the sparsity term is much larger than the measurement error term: The constant of 10

^{5}was selected empirically to ensure that the sparsity term dominates the measurement error term. If the value is higher than necessary then the reconstruction result is unaffected, but the reconstruction time may display a small increase, as

*λ*is decreased during the reconstruction. If the value is too low then the reconstruction process may be affected as the measurement error can reach the termination threshold before the solution is maximally sparsified. A set of sub-problems are then solved with decreasing

*λ*until the measurement discrepancy condition is satisfied or

*λ*satisfies inequality (6): The value of 10

^{20}was also selected empirically to ensure that in the case where the measurement error threshold cannot be met, the reconstruction process does not terminate before it is reduced to solving Eq. (1). CSCG is also constrained to be non-negative, due to the physical non-negativity of luminophore distributions. This constraint is implemented via projection within the algorithm, where the projected solution

**x**′ is calculated as Eq. (7).

17. C. Lawson and R. Hanson, *Solving Least Squares Problems* (SIAM, 1995). [CrossRef]

*α*.

18. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. En. **25**, 711–732 (2009). [CrossRef]

19. F. Leblond, K. M. Tichauer, R. W. Holt, F. El-Ghussein, and B. W. Pogue, “Toward whole-body optical imaging of rats using single-photon counting fluorescence tomography,” Opt. Lett. **36**, 3723–3725 (2011). [CrossRef] [PubMed]

### 2.1. Simulation

*ε*in Fig. 1) was set to zero to be consistent with the experimental reconstruction, and measurements were removed to leave the same subset of measurements that was used in the experimental reconstruction (see section 2.2). A single regularisation weight for GN and NNLS of 10

^{−5}was chosen empirically from simulations and used for all tests, as in practice one cannot know the best regularisation weight for data sets whose solution is unknown. The regularisation value was chosen to minimise regularisation whilst reducing artifacts at all noise levels.

### 2.2. Experiment

*ε*in Fig. 1) was set to zero. As the quantity being measured is the number of photons arriving at the detectors, all measurements should be positive. However, due to instrumental error some measurements were negative. The maximum negativity of measurements was used to estimate which measurements were noise dominated. All measurements with magnitudes less than five times the absolute value of the most negative measurement were considered noise-dominated and removed, leaving a total of 526 measurements from which to reconstruct 10171 unknowns.

## 3. Results and discussion

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. Arridge and J. Hebden, “Optical imaging in medicine: II. modelling and reconstruction,” Phys. Med. Biol. |

2. | C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. |

3. | R. G. Baraniuk, “Compressive sensing,” IEEE Signal. Process. Mag. |

4. | E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. |

5. | H. Rauhut, “Compressive sensing and structured random matrices,” in |

6. | M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. |

7. | M. Lustig, D. Donoho, and J. M. Pauly, “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magn. Reson. Med. |

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. | X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express |

10. | W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A |

11. | J. Yu, F. Liu, J. Wu, L. Jiao, and X. He, “Fast source reconstruction for bioluminescence tomography based on sparse regularization,” IEEE Trans. Biomed. Eng. |

12. | H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,” Opt. Express |

13. | H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express |

14. | X. He, Y. Hou, D. Chen, Y. Jiang, M. Shen, J. Liu, Q. Zhang, and J. Tian, “Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method,” Int. J. Biomed. Imaging |

15. | K. Liu, J. Tian, C. Qin, X. Yang, S. Zhu, D. Han, and P. Wu, “Tomographic bioluminescence imaging reconstruction via a dynamically sparse regularized global method in mouse models,” J. Biomed. Opt. |

16. | Q. Zhang, H. Zhao, D. Chen, X. Qu, X. He, X. Chen, W. Li, Z. Hu, J. Liu, and J. Liang, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. |

17. | C. Lawson and R. Hanson, |

18. | H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. En. |

19. | F. Leblond, K. M. Tichauer, R. W. Holt, F. El-Ghussein, and B. W. Pogue, “Toward whole-body optical imaging of rats using single-photon counting fluorescence tomography,” Opt. Lett. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: June 7, 2012

Revised Manuscript: June 27, 2012

Manuscript Accepted: July 4, 2012

Published: August 15, 2012

**Virtual Issues**

BIOMED 2012
(2012) *Biomedical Optics Express*

**Citation**

Hector R. A. Basevi, Kenneth M. Tichauer, Frederic Leblond, Hamid Dehghani, James A. Guggenheim, Robert W. Holt, and Iain B. Styles, "Compressive sensing based reconstruction in bioluminescence tomography improves image resolution and robustness to noise," Biomed. Opt. Express **3**, 2131-2141 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-9-2131

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### References

- S. Arridge and J. Hebden, “Optical imaging in medicine: II. modelling and reconstruction,” Phys. Med. Biol.42, 841–853 (1997). [CrossRef] [PubMed]
- C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007). [CrossRef] [PubMed]
- R. G. Baraniuk, “Compressive sensing,” IEEE Signal. Process. Mag.24, 118–124 (2007). [CrossRef]
- E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl.23, 969–985 (2007). [CrossRef]
- H. Rauhut, “Compressive sensing and structured random matrices,” in Theoretical Foundations and Numerical Methods for Sparse Recovery, M. Massimo, ed. (deGruyter, 2010), pp. 1–92. [CrossRef]
- M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process.1, 586–597 (2007). [CrossRef]
- M. Lustig, D. Donoho, and J. M. Pauly, “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magn. Reson. Med.58, 1182–1195 (2007). [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. Express17, 8062–8080 (2009). [CrossRef] [PubMed]
- X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express18, 24825–24841 (2010). [CrossRef] [PubMed]
- W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A27, 174–179 (2010). [CrossRef]
- J. Yu, F. Liu, J. Wu, L. Jiao, and X. He, “Fast source reconstruction for bioluminescence tomography based on sparse regularization,” IEEE Trans. Biomed. Eng.57, 2583–2586 (2010). [CrossRef] [PubMed]
- H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,” Opt. Express18, 1854–1871 (2010). [CrossRef] [PubMed]
- H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express18, 2894–2912 (2010). [CrossRef] [PubMed]
- X. He, Y. Hou, D. Chen, Y. Jiang, M. Shen, J. Liu, Q. Zhang, and J. Tian, “Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method,” Int. J. Biomed. Imaging2011, 203537 (2011). [CrossRef]
- K. Liu, J. Tian, C. Qin, X. Yang, S. Zhu, D. Han, and P. Wu, “Tomographic bioluminescence imaging reconstruction via a dynamically sparse regularized global method in mouse models,” J. Biomed. Opt.16, 046016 (2011). [CrossRef] [PubMed]
- Q. Zhang, H. Zhao, D. Chen, X. Qu, X. He, X. Chen, W. Li, Z. Hu, J. Liu, and J. Liang, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun.284, 5871–5876 (2011). [CrossRef]
- C. Lawson and R. Hanson, Solving Least Squares Problems (SIAM, 1995). [CrossRef]
- H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. En.25, 711–732 (2009). [CrossRef]
- F. Leblond, K. M. Tichauer, R. W. Holt, F. El-Ghussein, and B. W. Pogue, “Toward whole-body optical imaging of rats using single-photon counting fluorescence tomography,” Opt. Lett.36, 3723–3725 (2011). [CrossRef] [PubMed]

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