Abstract

Optical tomography has been widely investigated for biomedical imaging applications. In recent years optical tomography has been combined with digital holography and has been employed to produce high-quality images of phase objects such as cells. In this paper we describe a method for imaging 3D phase objects in a tomographic configuration implemented by training an artificial neural network to reproduce the complex amplitude of the experimentally measured scattered light. The network is designed such that the voxel values of the refractive index of the 3D object are the variables that are adapted during the training process. We demonstrate the method experimentally by forming images of the 3D refractive index distribution of Hela cells.

© 2015 Optical Society of America

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References

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    [Crossref]

2015 (1)

2014 (1)

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

2013 (1)

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

2012 (1)

W. Van den Broek, C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

2010 (2)

O. Haeberlé, K. Belkebir, H. Giovaninni, A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

E. Y. Sidky, M. A. Anastasio, X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced X-ray phase-contrast tomography,” Opt. Express 18, 10404–10422 (2010).
[Crossref]

2009 (2)

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

2008 (2)

W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Extended depth of focus in tomographic phase microscopy using a propagation algorithm,” Opt. Lett. 33, 171–173 (2008).
[Crossref]

E. J. Candes, M. B. Wakin, S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).

2007 (2)

M. Lustig, D. Donoho, J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref]

2006 (1)

2002 (2)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[Crossref]

A. Abubakar, P. M. Van den Berg, J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theor. Tech. 50, 1761–1771 (2002).
[Crossref]

1997 (1)

1994 (1)

1993 (2)

1991 (1)

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[Crossref]

1986 (1)

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986).
[Crossref]

1983 (1)

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[Crossref]

1982 (1)

1981 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1967 (1)

B. Widrow, P. E. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” J. Acoust. Soc. Am. 42, 1175–1176 (1967).
[Crossref]

Abubakar, A.

A. Abubakar, P. M. Van den Berg, J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theor. Tech. 50, 1761–1771 (2002).
[Crossref]

Anastasio, M. A.

Badizadegan, K.

Beck, A.

A. Beck, M. Teboulle, “Gradient-based algorithms with applications to signal recovery problems,” in Convex Optimization in Signal Processing and Communications, D. Palomar, Y. Eldar, eds. (Cambridge University, 2010), pp. 42–88.

Belkebir, K.

O. Haeberlé, K. Belkebir, H. Giovaninni, A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford, 1995).

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Bottou, L.

L. Bottou, “Stochastic gradient descent tricks,” in Neural Networks: Tricks of the Trade, 2nd ed. (Springer, 2012), pp. 421–437.

Boyd, S. P.

E. J. Candes, M. B. Wakin, S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).

Candes, E. J.

E. J. Candes, M. B. Wakin, S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).

Charrière, F.

Chaumet, P. C.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Choi, W.

Colomb, T.

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Cuche, E.

Dao, M.

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

Dasari, R. R.

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Extended depth of focus in tomographic phase microscopy using a propagation algorithm,” Opt. Lett. 33, 171–173 (2008).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref]

Depeursinge, C.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178–180 (2006).
[Crossref]

Devaney, A. J.

Diez-Silva, M.

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

Donk, J. V.

Donoho, D.

M. Lustig, D. Donoho, J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref]

Draine, B. T.

Drsek, F.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Fang-Yen, C.

Feld, M. S.

Fienup, J. R.

Flatau, P. J.

Giovaninni, H.

O. Haeberlé, K. Belkebir, H. Giovaninni, A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Giovannini, H.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Girard, J.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Goode, B.

B. Widrow, P. E. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” J. Acoust. Soc. Am. 42, 1175–1176 (1967).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Griffiths, L.

B. Widrow, P. E. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” J. Acoust. Soc. Am. 42, 1175–1176 (1967).
[Crossref]

Haeberlé, O.

O. Haeberlé, K. Belkebir, H. Giovaninni, A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986).
[Crossref]

Hugonin, J.-P.

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[Crossref]

Joachimowicz, N.

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[Crossref]

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Jueptner, W.

U. Schnars, W. Jueptner, Digital Holography (Springer, 2005).

Kim, K.

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

Koch, C. T.

W. Van den Broek, C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

Konan, D.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Kuehn, J.

Lagasse, P. E.

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[Crossref]

Lewitt, R. M.

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[Crossref]

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref]

Lustig, M.

M. Lustig, D. Donoho, J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref]

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Maire, G.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Maleki, M. H.

Mallorqui, J. J.

A. Abubakar, P. M. Van den Berg, J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theor. Tech. 50, 1761–1771 (2002).
[Crossref]

Mantey, P. E.

B. Widrow, P. E. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” J. Acoust. Soc. Am. 42, 1175–1176 (1967).
[Crossref]

Marian, A.

Marquet, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178–180 (2006).
[Crossref]

Montfort, F.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref]

Pan, X.

Park, Y.

K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19, 011005 (2014).
[Crossref]

Pauly, J. M.

M. Lustig, D. Donoho, J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref]

Pavillon, N.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Pichot, C.

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[Crossref]

Roey, J. V.

Rumelhart, D. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986).
[Crossref]

Schnars, U.

U. Schnars, W. Jueptner, Digital Holography (Springer, 2005).

Sentenac, A.

O. Haeberlé, K. Belkebir, H. Giovaninni, A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Sidky, E. Y.

Sung, Y.

Talneau, A.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[Crossref]

Teboulle, M.

A. Beck, M. Teboulle, “Gradient-based algorithms with applications to signal recovery problems,” in Convex Optimization in Signal Processing and Communications, D. Palomar, Y. Eldar, eds. (Cambridge University, 2010), pp. 42–88.

Tian, L.

Toy, F.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Van den Berg, P. M.

A. Abubakar, P. M. Van den Berg, J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theor. Tech. 50, 1761–1771 (2002).
[Crossref]

Van den Broek, W.

W. Van den Broek, C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

Wakin, M. B.

E. J. Candes, M. B. Wakin, S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).

Waller, L.

Widrow, B.

B. Widrow, P. E. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” J. Acoust. Soc. Am. 42, 1175–1176 (1967).
[Crossref]

Williams, R. J.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986).
[Crossref]

Wolf, E.

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Supplementary Material (2)

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Figures (7)

Fig. 1.
Fig. 1.

Experimental setup (BS, beam splitter; GM, galva mirror; L, lens; OB, objective; M, mirror).

Fig. 2.
Fig. 2.

Schematic diagram of object reconstruction by learning the 3D index distribution that minimizes the error ϵ, defined at the mean squared difference between the experimental measurement and the prediction of a computational model based on the BPM.

Fig. 3.
Fig. 3.

Experimental reconstruction of two 10 μm beads of refractive index 1.588 at λ=561nm in immersion oil with n0=1.516. (a)–(c) x–y, y–z, and x–z slices using the inverse Radon transform reconstruction. (d)–(f) Same slices for our learning-based reconstruction method. (The lines indicate the locations of the slices.)

Fig. 4.
Fig. 4.

Simulation geometry comprising two spherical beads with a refractive index difference of 0.04 compared to the background. (b)–(j) Cross-sectional views on x–y, x–z, and y–z planes of (b)–(d) original refractive index, (e)–(g) reconstruction with optical diffraction tomography, and (h)–(j) reconstruction with learning tomography. Because of the scattering, since the Born approximation (single scattering) is not valid, the diffraction tomography fails to reconstruct the refractive index inhomogeneity. However, the learning tomography is capable of correctly reconstructing the object.

Fig. 5.
Fig. 5.

Comparison of the proposed method initialized with the inverse Radon transform (left) versus initialization with a constant value (Δn^=0.007) (right). (a) and (e) plot the error fall-off for 80 illumination angles initialized with the inverse Radon and constant values, respectively. The horizontal dotted line shows the inverse Radon performance for comparison. (b)–(d), x–y, y–z, and x–z stacks for, respectively, the first, tenth, and hundredth iterations of the proposed method initialized by inverse Radon. (d)–(f) Same figures for the proposed method initialized by constant value. (See also Media 1.)

Fig. 6.
Fig. 6.

Error between the experimental measurements and the predictions of the computational mode plotted as a function of the number of iterations for four different initial conditions: constant index (black), Radon tomographic reconstruction (red), diffraction tomography [31] (green), and the iterative method described in [18] (blue).

Fig. 7.
Fig. 7.

Images of two hTERT-RPE1 cells. x–y slices corresponding to different depths of, respectively, +9, +6, +3, 0, and 3μm (positive being toward the detector) from the focal plane of the lens OB2 in Fig. 1 for (a)–(e) inverse Radon-transform-based reconstruction and (f)–(j) same slices for our learning-based reconstruction method.

Equations (2)

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minΔn^{12Kk=1KEk(Δn^)Mk(Δn)2+τS(Δn^)}subject to0Δn^.
Δn^Δn^(αKk=1KϵkϵkΔn^+τS(Δn^)Δn^),

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