## 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

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### References

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- Publication

- 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. Express 17, 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. Express 18, 24825–24841 (2010).

[Crossref] [PubMed] -
W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A 27, 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. Express 18, 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. Express 18, 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. Imaging 2011, 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]

#### 2011 (4)

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]

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]

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]

#### 2010 (5)

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. Express 18, 1854–1871 (2010).

[Crossref]
[PubMed]

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

[Crossref]

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]

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]

#### 2009 (2)

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]

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]

#### 2007 (5)

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]

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]

#### 1997 (1)

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

[Crossref]
[PubMed]

#### Arridge, S.

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

[Crossref]
[PubMed]

#### Baraniuk, R. G.

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

[Crossref]

#### Candes, E.

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

[Crossref]

#### Carpenter, C. M.

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]

#### Chan, T. F.

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]

#### Chatziioannou, A. F.

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]

#### Chen, D.

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]

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]

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]

#### Chen, X.

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]

#### Cong, W.

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

[Crossref]

#### Coquoz, O.

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]

#### Davis, S. C.

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]

#### Dehghani, H.

[Crossref]

#### Donoho, D.

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]

#### Douraghy, A.

[Crossref]
[PubMed]

#### Eames, M. E.

[Crossref]

#### El-Ghussein, F.

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]

#### Figueiredo, M. A. T.

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]

#### Gao, H.

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]

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]

#### Han, D.

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]

#### Hanson, R.

C. Lawson and R. Hanson, Solving Least Squares Problems (SIAM, 1995).

[Crossref]

#### He, X.

[Crossref]

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]

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]

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]

#### Hebden, J.

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

[Crossref]
[PubMed]

#### Holt, R. W.

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]

#### Hou, Y.

[Crossref]

[Crossref]
[PubMed]

#### Hu, Z.

[Crossref]

#### Jiang, Y.

[Crossref]

#### Jiao, L.

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]

#### Kuo, C.

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]

#### Lawson, C.

C. Lawson and R. Hanson, Solving Least Squares Problems (SIAM, 1995).

[Crossref]

#### Leblond, F.

[Crossref]
[PubMed]

#### Li, W.

[Crossref]

#### Liang, J.

[Crossref]

[Crossref]
[PubMed]

#### Liu, F.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Liu, J.

[Crossref]

[Crossref]

#### Liu, K.

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]

#### Lu, Y.

[Crossref]
[PubMed]

#### Lustig, M.

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]

#### Nowak, R. D.

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]

#### Paulsen, K. D.

[Crossref]

#### Pauly, J. M.

[Crossref]
[PubMed]

#### Pogue, B. W.

[Crossref]
[PubMed]

[Crossref]

#### Qin, C.

[Crossref]
[PubMed]

#### Qu, X.

[Crossref]

[Crossref]
[PubMed]

#### Rauhut, H.

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]

#### Rice, B.

[Crossref]
[PubMed]

#### Romberg, J.

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

[Crossref]

#### Shen, M.

[Crossref]

#### Srinivasan, S.

[Crossref]

#### Stout, D.

[Crossref]
[PubMed]

#### Tian, J.

[Crossref]
[PubMed]

[Crossref]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Tichauer, K. M.

[Crossref]
[PubMed]

#### Troy, T.

[Crossref]
[PubMed]

#### Wang, G.

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

[Crossref]

#### Wang, X.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Wright, S. J.

[Crossref]

#### Wu, J.

[Crossref]
[PubMed]

#### Wu, P.

[Crossref]
[PubMed]

#### Xu, H.

[Crossref]
[PubMed]

#### Yalavarthy, P. K.

[Crossref]

#### Yang, X.

[Crossref]
[PubMed]

#### Yu, J.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Zhang, Q.

[Crossref]

[Crossref]

#### Zhang, X.

[Crossref]
[PubMed]

#### Zhao, H.

[Crossref]

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]

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]

#### Zhu, S.

[Crossref]
[PubMed]

#### Commun. Numer. Meth. En. (1)

[Crossref]

#### IEEE J. Sel. Top. Signal Process. (1)

[Crossref]

#### IEEE Signal. Process. Mag. (1)

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

[Crossref]

#### IEEE Trans. Biomed. Eng. (1)

[Crossref]
[PubMed]

#### Int. J. Biomed. Imaging (1)

[Crossref]

#### Inverse Probl. (1)

[Crossref]

#### J. Biomed. Opt. (2)

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### J. Opt. Soc. Am. A (1)

[Crossref]

#### Magn. Reson. Med. (1)

[Crossref]
[PubMed]

#### Opt. Commun. (1)

[Crossref]

#### Opt. Express (4)

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Opt. Lett. (1)

[Crossref]
[PubMed]

#### Phys. Med. Biol. (1)

[Crossref]
[PubMed]

#### Other (2)

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]

C. Lawson and R. Hanson, Solving Least Squares Problems (SIAM, 1995).

[Crossref]

### Cited By

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### Figures (6)

**Fig. 1**

(a) CSCG algorithm pseudocode. ^{5}. ^{−20} of the initial value of ^{−1/2}. (b) Flowchart of CSCG algorithm.

**Fig. 2**

Caliper XPM-2 Phantom Mouse and the torso region used in reconstructions. The XPM-2 is shown in light blue, and the torso region used is overlaid in dark blue. The positions of the measurements acquired are shown as red spheres.

**Fig. 3**

Reconstructions of two bioluminescent sources using simulated measurements in the absence of measurement noise. (a)Target (b)GN (c)NNLS (d)CSCG

**Fig. 4**

Reconstructions of two bioluminescent sources using simulated measurements, as Fig. 3, in the presence of 1% normally distributed noise.

**Fig. 5**

Reconstructions of two bioluminescent sources using simulated measurements, as Fig. 3, in the presence of 5% normally distributed noise.

**Fig. 6**

Reconstructions of experimental measurements of an XPM-2 Phantom Mouse (Caliper Life Sciences, Hopkinton, MA, USA). (a)GN (b)NNLS (c)CSCG

### Tables (2)

**Table 1** Simulation localisation error (mm) of mean reconstructed source centre. Source locations were calculated by taking the centres of fitted Gaussian distributions. Source A is the left-most source, and source B is the right-most source, as displayed in Fig. 3, Fig. 4, and Fig. 5. At each noise level, 30 samples of noisy measurements were generated. The location of each source was taken to be the location of the maximum value in a region (a sphere of radius 8mm) around the true source centre.

**Table 2** Simulation mean volume (mm^{3}) (see Table 1) as a percentage of the true volume. The volume of each source was taken to be the connected volume enclosed by half the maximum value in the region, containing the location of the maximum value.

### Equations (8)

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