Journal of Scientific Computing

, Volume 50, Issue 3, pp 519–535 | Cite as

A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography

Article

Abstract

In this paper, we consider 3D Bioluminescence tomography (BLT) source reconstruction from Poisson data in three dimensional space. With a priori information of sources sparsity and MAP estimation of Poisson distribution, we study the minimization of Kullback-Leihbler divergence with 1 and 0 regularization. We show numerically that although several 1 minimization algorithms are efficient for compressive sensing, they fail for BLT reconstruction due to the high coherence of the measurement matrix columns and high nonlinearity of Poisson fitting term. Instead, we propose a novel greedy algorithm for 0 regularization to reconstruct sparse solutions for BLT problem. Numerical experiments on synthetic data obtained by the finite element methods and Monte-Carlo methods show the accuracy and efficiency of the proposed method.

Keywords

Source reconstruction Bioluminescence tomography 0 regularization Orthogonal matching pursuit Poisson noise 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexandrakis, G., Rannou, F.-R., Chatziioannou, A.-F.: Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study. Phys. Med. Biol. 50, 4225–4241 (2005) CrossRefGoogle Scholar
  2. 2.
    Brune, C., Sawatzky, A., Burger, M.: Bregman-EM-TV methods with application to optical nanoscopy. In: Proc. SSVM 2009. LNCS, vol. 5567. Springer, Berlin (2009) Google Scholar
  3. 3.
    Candès, E.J.: Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, vol. 3(26), pp. 1433–1452 (2006) Google Scholar
  4. 4.
    Candès, E.J.: Decoding by linear programing. IEEE Trans. Inf. Theory 15(12), 4203–4215 (2004) Google Scholar
  5. 5.
    Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(26), 1207–1223 (2005) Google Scholar
  6. 6.
    Cong, W., Wang, G., Kumar, D., Liu, Y., Jiang, M., Wang, L.V., Hoffman, E.A., McLennan, G., McCray, P.B., Zabner, J., Cong, A.: Practical reconstruction method for bioluminescence tomography. Opt. Express 13(18), 6756–6771 (2005) CrossRefGoogle Scholar
  7. 7.
    Chan, R.H., Chen, K.: Multilevel algorithm for a Poisson noise removal model with total-variation regularization. Int. J. Comput. Math. 84(8), 1183–1198 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Csiszár, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19(4), 2032–2066 (1991) MATHCrossRefGoogle Scholar
  9. 9.
    Davis, G., Mallat, S., Avellaneda, M.: Greedy adaptive approximation. Constr. Approx. 13, 57–98 (1997) MathSciNetMATHGoogle Scholar
  10. 10.
    Dupont, F., Fadili, J.M., Starck, J.: A proximal iteration for deconvolving Poisson noisy images using sparse representations. IEEE Trans. Image Process. 18(2), 310–321 (2009) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Figueiredo, M., Bioucas-Dias, J.: Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization. In: IEEE Workshop on Statistical Signal Processing, Cardiff, UK (2009) Google Scholar
  12. 12.
    Figueiredo, M., Bioucas-Dias, J.: Restoration of Poissonian images using alternating direction optimization. IEEE Trans. Image Process., to appear Google Scholar
  13. 13.
    Han, W., Cong, W., Wang, G.: Mathematical theory and numerical analysis of bioluminescence tomography. Inverse Probl. 22, 1659–1675 (2006) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Han, W., Cong, W., Wang, G.: Mathematical study and numerical simulation of multispectral bioluminescence tomography. Int. J. Biomed. Imaging 2006, 1–10 (2006) Google Scholar
  15. 15.
    Han, W., Wang, G.: Bioluminescence tomography: biomedical background, mathematical theory, and numerical approximation. J. Comput. Math. 26, 324–335 (2008) MathSciNetMATHGoogle Scholar
  16. 16.
    Gao, H., Zhao, H.: Multilevel bioluminescence tomography based on radiative transfer equation. Part 1: l1 regularization. Opt. Express 18(3), 1854–1871 (2010) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gao, H., Zhao, H.: Multilevel bioluminescence tomography based on radiative transfer equation. Part 2: total variation and l1 data fidelity. Opt. Express 18(3), 2894–2912 (2010) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gibson, A.P., Hebden, J.C., Arridge, S.R.: Recent advances in diffuse optical imaging. Phys. Med. Biol. 50, R1–R43 (2005) CrossRefGoogle Scholar
  19. 19.
    Gu, X., Zhang, Q., Larcom, L., Jiang, H.: Three-dimensional bioluminescence tomography with model-based reconstruction. Opt. Express 12(17), 3996–4000 (2004) CrossRefGoogle Scholar
  20. 20.
    Harmany, Z.T., Marcia, R.F., Willett, R.M.: This is SPIRAL-TAP: sparse Poisson intensity reconstruction algorithms—theory and practice. arXiv:1005.4274v1
  21. 21.
    Jonsson, E., Huang, S.-C., Chan, T.-F.: Total-variation regularization in positron emission tomography. UCLA CAM Report (98-48) (1998) Google Scholar
  22. 22.
    Kuo, C., Coquoz, O., Troy, T.L., Xu, H., Rice, B.W.: Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging. J. Biomed. Opt. 12, 024007 (2007) CrossRefGoogle Scholar
  23. 23.
    Kim, S., Lim, Y.T., Soltesz, E.G., De Grand, A.M., Lee, J., Nakayama, A., Parker, J.A., Mihaljevic, T., Laurence, R.G., Dor, D.M., Cohn, L.H., Bawendi, M.G., Frangioni, J.V.: Near-infrared fluorescent type II quantum dots for sentinel lymph node mapping. Nat. Biotechnol. 22(1), 93–97 (2004) CrossRefGoogle Scholar
  24. 24.
    Lange, K., Carson, R.: EM reconstruction algorithms for emission and transmission tomography. J. Comput. Assist. Tomogr. 8(2), 306–316 (1984) Google Scholar
  25. 25.
    Lingenfelter, D., Fessler, J., He, Z.: Sparsity regularization for image reconstruction with Poisson data. In: Proc. SPIE Computational Imaging VII, vol. 7246 (2009) Google Scholar
  26. 26.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lu, Y., Zhang, X., Douraghy, A., Stout, D., Tian, J., Chan, T.-F., Chatziioannou, A.-F.: Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information. Opt. Express 17(10), 8062–8080 (2009) CrossRefGoogle Scholar
  28. 28.
    Lu, Y., Machado, H.B., Bao, Q., Stout, D., Herschman, H., Chatziioannou, A.F.: In Vivo mouse bioluminescence tomography with radionuclide-based imaging validation. Mol. Imaging Biol. 13(1), 53–58 (2010) CrossRefGoogle Scholar
  29. 29.
    Lv, Y., Tian, J., Cong, W., Wang, G., Luo, J., Yang, W., Li, H.: A multilevel adaptive finite element algorithm for bioluminescence tomography. Opt. Express 14(18), 8211–8223 (2006) CrossRefGoogle Scholar
  30. 30.
    Needella, D., Tropp, J.A.: CoSaMP: Iterative signal recovery from incomplete and inaccurate samplesstar. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nowak, R.D., Kolaczyk, E.D.: A Bayesian multiscale framework for Poisson inverse problems. IEEE Trans. Inf. Theory 46(5), 1811–1825 (2000) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Raginsky, M., Willett, R.M., Harmany, Z.T., Marcia, R.F.: Compressed sensing performance bounds under Poisson noise. IEEE Trans. Signal Process. 58(8), 3990–4002 (2010) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rao, S.S.: The Finite Element Method in Engineering. Butterworth-Heinemann, Boston (1999) Google Scholar
  35. 35.
    Resmerita, E., Engl, H.W., Iusem, A.N.: The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis. Inverse Probl. 24(5), 059801 (2008) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schweiger, M., Arridge, S.R., Hiraoka, M., Delpy, D.T.: The finite element method for the propagation of light in scattering media: Boundary and source conditions. Med. Phys. 22, 1779–1792 (1995) CrossRefGoogle Scholar
  37. 37.
    Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Represent. 21, 193–199 (2010) CrossRefGoogle Scholar
  38. 38.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction in positron emission tomography. IEEE Trans. Med. Imaging 1, 113–122 (1982) CrossRefGoogle Scholar
  39. 39.
    Tropp, J.A., Gilbert, A.C.: Signal recovery from random measurements via Orthogonal Matching Pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007) MathSciNetCrossRefGoogle Scholar
  40. 40.
    Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for Positron emission tomography. J. Am. Stat. Assoc. 80(389), 8–20 (1995) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wang, G., Hoffman, E.A., McLennan, G., Wang, L.V., Suter, M., Meinel, J.F.: Development of the first bioluminescence CT scanner. Radiology 566, 229 (2003) Google Scholar
  42. 42.
    Wang, G., Li, Y., Jiang, M.: Uniqueness theorems in bioluminescence tomography. Med. Phys. 31(8), 2289–2299 (2004) CrossRefGoogle Scholar
  43. 43.
    Wang, G., Shen, H., Durairaj, K., Qian, X., Cong, W.: The first bioluminescence tomography system for simultaneous acquisition of multiview and multispectral data. Int. J. Biomed. Imaging 2006, 1–8 (2006) Google Scholar
  44. 44.
    Willett, R., Raginsky, M.: Performance bounds for compressed sensing with Poisson noise. In: Proc. of IEEE Int. Symp. on Inf. Theory (2009) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Center for Molecular Imaging, Institute of Molecular MedicineUniversity of Texas Health Science Center at HoustonHoustonUSA
  3. 3.HKUSTKowloonHong Kong

Personalised recommendations