## Abstract

Diffuse optical tomography (DOT) allows tomographic (3D), non-invasive reconstructions of tissue optical properties for biomedical applications. Severe under-sampling is a common problem in DOT which leads to image artifacts. A large number of measurements is needed in order to minimize these artifacts. In this work, we introduce a compressed sensing (CS) framework for DOT which enables improved reconstructions with under-sampled data. The CS framework uses a sparsifying basis, *ℓ*_{1}-regularization and random sampling to reduce the number of measurements that are needed to achieve a certain accuracy. We demonstrate the utility of the CS framework using numerical simulations. The CS results show improved DOT results in comparison to “traditional” linear reconstruction methods based on singular-value decomposition (SVD) with *ℓ*_{2}-regularization and with regular and random sampling. Furthermore, CS is shown to be more robust against the reduction of measurements in comparison to the other methods. Potential benefits and shortcomings of the CS approach in the context of DOT are discussed.

© 2010 OSA

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[Crossref] - A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol 50, 1–43 (2005).

[Crossref] - S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Problems 25, 123010 (2009).

[Crossref] - R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys Med Biol 45, 1051–70 (2000).

[Crossref] [PubMed] -
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[Crossref] - R. Baraniuk, “Compressive sensing,” Lecture notes in IEEE Signal Processing magazine 24, 118–120 (2007).

[Crossref] - J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on Information Theory 53, 4655 (2007).

[Crossref] - D. Needell and J. A. Tropp, “Cosamp: Iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis 26, 301–321 (2009).

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[Crossref] [PubMed] -
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[Crossref] -
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[Crossref] [PubMed] - J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in “Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium on,” (2008), pp. 1621–1624.

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[Crossref] [PubMed] -
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[Crossref] - P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM review 34, 561–580 (1992).

[Crossref] - E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems 23, 969–985 (2007).

[Crossref] - S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares. selected topics in signal processing,” IEEE Journal of 1, 606–617 (2007).

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[Crossref] - M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing mri,” IEEE Signal Processing Magazine 25, 72–82 (2008).

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[Crossref] [PubMed] - H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Applied optics 48, 137–143 (2009).

[Crossref] - H. Ponnekanti, J. Ophir, and Y. Huang, “Fundamental mechanical limitations on the visualization of elasticity contrast in elastography,” Ultrasound in Medicine and Biology 21, 553–543 (1995).

[Crossref] - X. M. Song, B. W. Pogue, S. D. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Applied optics 43, 1053–1062 (2004).

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[Crossref] - R. A. DeVore, “Deterministic constructions of compressed sensing matrices,” Journal of Complexity 23, 918–925 (2007).

[Crossref] - E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” Journal of Fourier Analysis and Applications 14, 877–905 (2008).

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[Crossref] - U. Gamper, P. Boesiger, and S. Kozerke, “Compressed sensing in dynamic mri,” Magnetic Resonance in Medicine 59, 365–373 (2008).

[Crossref] [PubMed]

#### 2010 (4)

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” Journal of Biomedical Optics 15, 021311 (2010).

[Crossref]
[PubMed]

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics 73, 076701 (2010).

[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]

Z. Xu and Y. L. Edmund, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).

[Crossref]

#### 2009 (7)

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).

[Crossref]
[PubMed]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Problems 25, 123010 (2009).

[Crossref]

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Applied optics 48, 137–143 (2009).

[Crossref]

D. Liang, H. F. Zhang, and L. Ying, “Compressed-sensing photoacoustic imaging based on random optical illumination,” International Journal of Functional Informatics and Personalised Medicine 2, 394–406 (2009).

[Crossref]

D. Needell and J. A. Tropp, “Cosamp: Iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis 26, 301–321 (2009).

[Crossref]

H. Yu and G. Wang, “Compressed sensing based interior tomography,” Physics in Medicine and Biology 54, 2791–2805 (2009).

[Crossref]
[PubMed]

D. Needell and R. Vershynin, “Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,” Foundations of computational mathematics 9, 317–334 (2009).

[Crossref]

#### 2008 (7)

J. Provost and F. Lesage, “The application of compressed sensing for photo-acoustic tomography.” IEEE transactions on medical imaging 28, 585–594 (2008).

[Crossref]

G. H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (piccs): a method to accurately reconstruct dynamic ct images from highly undersampled projection data sets,” Medical physics 35, 660 (2008).

[Crossref]
[PubMed]

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing mri,” IEEE Signal Processing Magazine 25, 72–82 (2008).

[Crossref]

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM Journal on Scientific Computing 31, 890–912 (2008).

[Crossref]

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” Journal of Fourier Analysis and Applications 14, 877–905 (2008).

[Crossref]

U. Gamper, P. Boesiger, and S. Kozerke, “Compressed sensing in dynamic mri,” Magnetic Resonance in Medicine 59, 365–373 (2008).

[Crossref]
[PubMed]

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review.” Breast Cancer Res Treat 108, 9–22 (2008).

[Crossref]

#### 2007 (8)

R. A. DeVore, “Deterministic constructions of compressed sensing matrices,” Journal of Complexity 23, 918–925 (2007).

[Crossref]

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

[Crossref]

S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares. selected topics in signal processing,” IEEE Journal of 1, 606–617 (2007).

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magnetic Resonance in Medicine 58, 1182–1195 (2007).

[Crossref]
[PubMed]

R. Baraniuk, “Compressive sensing,” Lecture notes in IEEE Signal Processing magazine 24, 118–120 (2007).

[Crossref]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on Information Theory 53, 4655 (2007).

[Crossref]

N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Optics Express 15, 13695–13707 (2007).

[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,” Applied Optics 46, 1679–1685 (2007).

[Crossref]
[PubMed]

#### 2006 (2)

D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution,” Communications on Pure and Applied Mathematics 59, 797–829 (2006).

[Crossref]

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207 (2006).

[Crossref]

#### 2005 (1)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol 50, 1–43 (2005).

[Crossref]

#### 2004 (1)

X. M. Song, B. W. Pogue, S. D. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Applied optics 43, 1053–1062 (2004).

[Crossref]
[PubMed]

#### 2003 (1)

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,” Medical Physics 30, 235 (2003).

[Crossref]
[PubMed]

#### 2001 (1)

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Optics Letters 26, 701–703 (2001).

[Crossref]

#### 2000 (1)

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys Med Biol 45, 1051–70 (2000).

[Crossref]
[PubMed]

#### 1999 (2)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse problems 15, R41–R93 (1999).

[Crossref]

B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Österberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Applied optics 38 (1999).

[Crossref]

#### 1998 (1)

S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998).

[Crossref]
[PubMed]

#### 1995 (1)

H. Ponnekanti, J. Ophir, and Y. Huang, “Fundamental mechanical limitations on the visualization of elasticity contrast in elastography,” Ultrasound in Medicine and Biology 21, 553–543 (1995).

[Crossref]

#### 1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proceedings of the National Academy of Sciences of the United States of America 91, 4887 (1994).

[Crossref]
[PubMed]

#### 1992 (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM review 34, 561–580 (1992).

[Crossref]

#### 1970 (1)

G. H. Golub and C. Reinsch, “Singular value decomposition and least squares solutions,” Numerische Mathematik 14, 403–420 (1970).

[Crossref]

#### 1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IRE 37, 10–21 (1949).

[Crossref]

#### Adibi, A.

P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Applied Optics 46, 1679–1685 (2007).

[Crossref]
[PubMed]

#### Arridge, S. R.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Problems 25, 123010 (2009).

[Crossref]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol 50, 1–43 (2005).

[Crossref]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse problems 15, R41–R93 (1999).

[Crossref]

S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998).

[Crossref]
[PubMed]

#### Athanasiou, T.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review.” Breast Cancer Res Treat 108, 9–22 (2008).

[Crossref]

#### Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics 73, 076701 (2010).

[Crossref]

#### Baraniuk, R.

R. Baraniuk, “Compressive sensing,” Lecture notes in IEEE Signal Processing magazine 24, 118–120 (2007).

[Crossref]

#### Boas, D. A.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys Med Biol 45, 1051–70 (2000).

[Crossref]
[PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proceedings of the National Academy of Sciences of the United States of America 91, 4887 (1994).

[Crossref]
[PubMed]

#### Boesiger, P.

U. Gamper, P. Boesiger, and S. Kozerke, “Compressed sensing in dynamic mri,” Magnetic Resonance in Medicine 59, 365–373 (2008).

[Crossref]
[PubMed]

#### Boyd, S.

S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares. selected topics in signal processing,” IEEE Journal of 1, 606–617 (2007).

#### Boyd, S. P.

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” Journal of Fourier Analysis and Applications 14, 877–905 (2008).

[Crossref]

#### Brady, D. J.

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).

[Crossref]
[PubMed]

#### Bresler, Y.

J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in “Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium on,” (2008), pp. 1621–1624.

#### Brooks, D. H.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys Med Biol 45, 1051–70 (2000).

[Crossref]
[PubMed]

#### Candes, E. J.

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” Journal of Fourier Analysis and Applications 14, 877–905 (2008).

[Crossref]

#### Candès, E.

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

[Crossref]

#### Candès, E. J.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207 (2006).

[Crossref]

#### Cao, N.

N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Optics Express 15, 13695–13707 (2007).

[Crossref]
[PubMed]

#### Chance, B.

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,” Medical Physics 30, 235 (2003).

[Crossref]
[PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proceedings of the National Academy of Sciences of the United States of America 91, 4887 (1994).

[Crossref]
[PubMed]

#### Chen, G. H.

G. H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (piccs): a method to accurately reconstruct dynamic ct images from highly undersampled projection data sets,” Medical physics 35, 660 (2008).

[Crossref]
[PubMed]

#### Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics 73, 076701 (2010).

[Crossref]

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,” Medical Physics 30, 235 (2003).

[Crossref]
[PubMed]

#### Choi, K.

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).

[Crossref]
[PubMed]

#### Culver, J. P.

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Applied optics 48, 137–143 (2009).

[Crossref]

[Crossref]
[PubMed]

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Optics Letters 26, 701–703 (2001).

[Crossref]

#### Darzi, A.

D. R. Leff, O. J. Warren, L. C. Enfield, A. Gibson, T. Athanasiou, D. K. Patten, J. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: a systematic review.” Breast Cancer Res Treat 108, 9–22 (2008).

[Crossref]

#### Dehghani, H.

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Applied optics 48, 137–143 (2009).

[Crossref]

X. M. Song, B. W. Pogue, S. D. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Applied optics 43, 1053–1062 (2004).

[Crossref]
[PubMed]

#### DeVore, R. A.

R. A. DeVore, “Deterministic constructions of compressed sensing matrices,” Journal of Complexity 23, 918–925 (2007).

[Crossref]

#### DiMarzio, C. A.

[Crossref]
[PubMed]

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### Figures (7)

**Fig. 1**

Illustrating the geometry of the simulated optical domain (infinite-medium). For the field-of-view, 3-D volume of size 4 × 4 × 3 cm^{3} and voxel dimensions 0.4 × 0.4 × 0.6 cm^{3} is considered. A spherical inhomogeneity is embedded in the medium at

**Fig. 2**

(a) Full set of S/D pairs (0% removal), (b) remaining S/D pairs after ∼ 90% “random” sampling (removal) for CS and SVD, (c) remaining S/D pairs after ∼ 90% “regular” sampling for SVD-reg.

**Fig. 3**

Cumulative coherence function (

**Fig. 4**

From top to bottom: “Forward” imaging data, SVD-reg, SVD, and CS reconstructed images for a simulated inhomogeneity of _{a,I}^{−1} and a case of ∼ 50% reduced S/D pairs. Five layers along the

**Fig. 5**

(a) Normalized observed contrast versus the investigated range of removed S/D pairs for CS, SVD and SVD-reg. The contrast was normalized according to that for 0% removal and the absorption coefficient of the inhomogeneity was _{a,I}^{−1}. (b) Observed contrast versus the true absorption coefficient of the inhomogeneity _{a,I}

**Fig. 6**

Normalized contrast-to-noise ratio (_{a,I}^{−1}.

**Fig. 7**

(a) Normalized root mean square error (_{a,I}^{−1}.

### Equations (12)

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