Abstract

We present “dynamic tomography” algorithms that allow for the high-resolution, time-resolved imaging of dynamic (i.e., continuously time evolving) complex systems at existing x-ray micro-CT facilities. The behavior of complex systems is constrained by the underlying physics. By exploiting a priori knowledge of the geometry of the physical process being studied to allow the use of sophisticated iterative reconstruction techniques that incorporate constraints, we improve on current frame rates by at least an order of magnitude. This allows time-resolved imaging of previously intractable processes, such as two-phase fluid flow. We present reconstructions from experimental data collected at the Australian National University x-ray micro-CT facility.

© 2011 Optical Society of America

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  1. F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  2. G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
    [CrossRef]
  3. S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
    [CrossRef]
  4. C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.
  5. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
    [CrossRef]
  6. G.T.Herman and A.Kuba, eds., Discrete Tomography: Foundations, Algorithms and Applications (Birkhäuser, 1999).
  7. K. J. Batenburg, “A network flow algorithm for binary image reconstruction from few projections,” Lect. Notes Comput. Sci. 4245, 86–97 (2006).
    [CrossRef]
  8. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  9. P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
    [CrossRef]
  10. S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
    [CrossRef]
  11. L. Hajdu and R. Tijdeman, “An algorithm for discrete tomography,” Linear Algebra Appl. 339, 147–169 (2001).
    [CrossRef]
  12. A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
    [CrossRef]
  13. G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
    [CrossRef] [PubMed]
  14. G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
    [CrossRef]
  15. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
    [CrossRef]
  16. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
    [CrossRef] [PubMed]

2010 (1)

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

2008 (3)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
[CrossRef] [PubMed]

G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
[CrossRef]

2006 (2)

K. J. Batenburg, “A network flow algorithm for binary image reconstruction from few projections,” Lect. Notes Comput. Sci. 4245, 86–97 (2006).
[CrossRef]

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

2005 (1)

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

2004 (1)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
[CrossRef]

2003 (1)

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

2001 (4)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

L. Hajdu and R. Tijdeman, “An algorithm for discrete tomography,” Linear Algebra Appl. 339, 147–169 (2001).
[CrossRef]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

1999 (1)

G.T.Herman and A.Kuba, eds., Discrete Tomography: Foundations, Algorithms and Applications (Birkhäuser, 1999).

1998 (1)

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

Alpers, A.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

Batenburg, K. J.

K. J. Batenburg, “A network flow algorithm for binary image reconstruction from few projections,” Lect. Notes Comput. Sci. 4245, 86–97 (2006).
[CrossRef]

Bonnet, S.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Caubit, C.

C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.

Chen, G.

G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
[CrossRef]

Clement, J. G.

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
[CrossRef]

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
[CrossRef]

de Vries, S.

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
[CrossRef]

Grangeat, P.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Gritzmann, P.

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

Guillemard, R.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Gureyev, T. E.

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
[CrossRef] [PubMed]

Hajdu, L.

L. Hajdu and R. Tijdeman, “An algorithm for discrete tomography,” Linear Algebra Appl. 339, 147–169 (2001).
[CrossRef]

Hamon, G.

C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.

Herman, G. T.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

Hornegger, J.

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

Hugonnard, P.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

Knudsen, E.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

Koenig, A.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Leng, S.

G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
[CrossRef]

Mayo, S. C.

Myers, G. R.

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
[CrossRef] [PubMed]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Øren, P. E.

C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.

Paganin, D. M.

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
[CrossRef] [PubMed]

Pan, X.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

Poulsen, H. F.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

Prangenberg, D.

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

Roux, S.

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Schnörr, C.

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

Schüle, T.

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

Sheppard, A. P.

C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.

Sidky, E. Y.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

Tang, J.

G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
[CrossRef]

Thomas, C. D. L.

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

Tijdeman, R.

L. Hajdu and R. Tijdeman, “An algorithm for discrete tomography,” Linear Algebra Appl. 339, 147–169 (2001).
[CrossRef]

Weber, S.

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

Wiegelmann, M.

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

Wübbeling, F.

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Appl. Phys. Lett. (1)

G. R. Myers, C. D. L. Thomas, D. M. Paganin, T. E. Gureyev, and J. G. Clement, “A general few-projection method for tomographic reconstruction of samples consisting of several distinct materials,” Appl. Phys. Lett. 96, 021105 (2010).
[CrossRef]

Commun. Pure Appl. Math. (1)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457(2004).
[CrossRef]

Int. J. Imag. Syst. Technol. (1)

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imag. Syst. Technol. 9, 101–109 (1998).
[CrossRef]

J. Appl. Cryst. (1)

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, “A discrete tomography algorithm for improving the quality of three-dimensional x-ray diffraction grain maps,” J. Appl. Cryst. 39, 582–588 (2006).
[CrossRef]

Lect. Notes Comput. Sci. (2)

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, “Binary tomography by iterating linear programs from noisy projections,” Lect. Notes Comput. Sci. 3322, 38–51 (2005).
[CrossRef]

K. J. Batenburg, “A network flow algorithm for binary image reconstruction from few projections,” Lect. Notes Comput. Sci. 4245, 86–97 (2006).
[CrossRef]

Linear Algebra Appl. (1)

L. Hajdu and R. Tijdeman, “An algorithm for discrete tomography,” Linear Algebra Appl. 339, 147–169 (2001).
[CrossRef]

Med. Phys. Lett. (1)

G. 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,” Med. Phys. Lett. 35, 660–663 (2008).
[CrossRef]

Opt. Express (1)

Phys. Med. Biol. (1)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

Proc. IEEE (1)

S. Bonnet, A. Koenig, S. Roux, P. Hugonnard, R. Guillemard, and P. Grangeat, “Dynamic x-ray computed tomography,” Proc. IEEE 91, 1574–1587 (2003).
[CrossRef]

Other (5)

C. Caubit, G. Hamon, A. P. Sheppard, and P. E. Øren, “Evaluation of the reliability of prediction of petrophysical data through imagery and pore network modelling,” presented at the 22nd International Symposium of the Society of Core Analysts, Abu Dhabi, United Arab Emirates, 29 October–2 November 2008, paper SCA2008-33.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

G.T.Herman and A.Kuba, eds., Discrete Tomography: Foundations, Algorithms and Applications (Birkhäuser, 1999).

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Typical cone-beam (i.e., lab-based) CT imaging geometry. The system has the geometric magnification of M = ( R 1 + R 2 ) / R 1 .

Fig. 2
Fig. 2

2D renderings of two 3D frames from a volumetric time series, showing a slice through the sample consisting of AlSiO 2 beads (white), and water (red). Frames are (a)  t = 15 m 21 s and (b)  t = 15 m 59 s after drainage (from the underside of the sample) commenced. The difference images are shown (c) with and (d) without the AlSiO 2 beads: red denotes voxels where the fluid has retreated, orange denotes voxels where the fluid has remained, and yellow denotes voxels where the fluid has advanced. As expected in a drainage experiment, the number of red voxels far surpasses the number of yellow voxels. Note the expanding void in the center of the field of view.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

μ ( r , t ) = { μ s ( r ) if     t < 0 μ s ( r ) + μ d ( r , t ) otherwise .
I ( M x , θ , t ) = I in exp [ g ( M x , θ , t ) ] ,
g ( M x , θ , t ) = ( P μ ) ( x , θ , t ) ,
( P μ ) ( x , θ , t ) 0 μ [ s ( θ ) + p | p | s , t ] d s , p = ( r 2 cos θ r 1 sin θ , R 1 r 1 cos θ r 2 sin θ , r 3 ) ,
μ s ( r ) = B ( F 1 { | ξ 1 | F [ R 1 g ( M x , θ , t ) R 1 2 + | x | 2 ] } ) , ( B g ) ( r ) 0 2 π R 1 2 p 2 2 g ( R 1 p 1 p 2 , R 1 p 3 p 2 , θ , t < 0 ) d θ ,
g d ( M x , θ , t ) = g ( M x , θ , t ) ( P μ s ) ( x , θ ) .
( S μ d ) ( r , t ) = μ d ( r , t ) + B [ g d ( M x , θ , t ) ( P μ d ) ( x , θ , t ) N ( x , θ , t ) ] ,
( P μ d ) ( M x , θ , T ) = P [ Δ t T + t Δ t μ d ( r , t ) + T t Δ t μ d ( r , t + Δ t ) ] .
( Z μ d ) ( r , t ) = { Ω μ d ( r t ) d r d t Ω d r d t ( r , t ) Ω 0 ( r , t ) Ω , Ω = { ( r , t ) : | μ d ( r , t ) | < | ϵ | } .
μ d ( n + 0.5 ) ( r , t ) = { t 1 T t ( S μ d ( n ) ) ( r , t ) inside support 0 outside support ,
μ d ( n + 1 ) ( r , t ) = { ( Z μ d ( n + 0.5 ) ) ( r , t ) if     μ d ( n + 0.5 ) ( r , t ) μ d ( n ) ( r , t ) L 1 < ϵ . μ d ( n + 0.5 ) ( r , t ) otherwise

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