Abstract

Optical Diffraction Tomography has been recently introduced in fluid velocimetry to provide three dimensional information of seeding particle locations. In general, image reconstruction methods at visible wavelengths have to account for diffraction. Linear approximation has been used for three-dimensional image reconstruction, but a non-linear and iterative reconstruction method is required when multiple scattering is not negligible. Non-linear methods require the solution of the Helmholtz equation, computationally highly demanding due to the size of the problem. The present work shows the results of a non-linear method customized for spherical particle location using GPU computing and a made-to-measure storing format.

© 2015 Optical Society of America

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  1. M. P. Arroyo and K. D. Hinsch, “Recent developments of PIV towards 3D measurements,” in Particle Image Velocimetry, volume 112 of Topics in Applied Physics, A. Schroder and C.E. Willert, eds. (Springer, 2008), pp. 127–154.
  2. J. M. Coupland and N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocity measurements using holographic recording and optical correlation,” Appl. Opt. 31(8), 1005–1007 (1992).
    [Crossref] [PubMed]
  3. J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
    [Crossref]
  4. W. D. Koek, N. Bhattacharya, J. J. M. Braat, T. A. Ooms, and J. Westerweel, “Influence of virtual images on the signal-to-noise ratio in digital in-line particle holography,” Opt. Express 13(7), 2578–2589 (2005).
    [Crossref] [PubMed]
  5. J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19(7), 074002 (2008).
    [Crossref]
  6. J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42(2), 235–250 (2003).
    [Crossref] [PubMed]
  7. J. Lobera and J. M. Coupland, “Optical diffraction tomography in fluid velocimetry: the use of a priori information,” Meas. Sci. Technol. 19(7), 074013 (2008).
    [Crossref]
  8. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
    [Crossref]
  9. K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20(7), 1223–1229 (2003).
    [Crossref] [PubMed]
  10. P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
    [Crossref]
  11. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A 24(4), 1164–1171 (2007).
    [Crossref] [PubMed]
  12. China’s TianhE-2 Supercomputer Maintains Top Spot on 42nd. TOP500 List. June 2014.
  13. G. Ortega, J. Lobera, M. P. Arroyo, I. García, and E. M. Garzón, “High performance computing for optical diffraction tomography”. In Proceedings of High Performance Computing Simulation, (IEEE, 2012), pp. 195–201.
  14. G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
    [Crossref]
  15. Mathworks. MATLAB with MEX Files. http://www.mathworks.es/es/help/distcomp/run-mex-functions-containing-cuda-code.html .
  16. M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, 2001).
  17. J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering,” Meas. Sci. Technol. 19(7), 074012 (2008).
    [Crossref]
  18. F. Ihlenburg and I. Babuska, “Finite element solution of the Helmholtz equation with high wave number Part I: The H-version of the FEM,” Comput. Math. Appl. 30(9), 9–37 (1995).
    [Crossref]
  19. I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM Avoidable for the Helmholtz Equation considering high wave numbers?” SIAM Rev. 42(3), 451–484 (2000).
    [Crossref]
  20. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).
  21. G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
    [Crossref]
  22. G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “Exploiting the regularity of differential operators to accelerate solutions of PDEs on GPUs,” InProceedings of the 11th International Computational and Mathematical Methods in Science and Engineering, J. Vigo-Aguiar, ed. (CMMSE, 2011), pp. 908–917.
  23. NVIDIA. CUDA Toolkit. https://developer.nvidia.com/cuda-toolkit
  24. CUBLAS user guide (du-06702–001 v5.5). http://docs.nvidia.com/cuda/pdf/CUBLAS_Library.pdf .
  25. G. Ortega, I. García, and E. M. Garzón, “A Hybrid Approach for Solving the 3D Helmholtz Equation on Heterogeneous Platforms,” In Euro-Par 2013: Parallel Processing Workshops, volume 8374 of LNCS, D. an Mey et al., eds. (Springer, 2014), pp. 198–207.

2013 (1)

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

2010 (2)

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

2008 (3)

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19(7), 074002 (2008).
[Crossref]

J. Lobera and J. M. Coupland, “Optical diffraction tomography in fluid velocimetry: the use of a priori information,” Meas. Sci. Technol. 19(7), 074013 (2008).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

2007 (1)

2005 (1)

2003 (2)

2000 (1)

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM Avoidable for the Helmholtz Equation considering high wave numbers?” SIAM Rev. 42(3), 451–484 (2000).
[Crossref]

1995 (1)

F. Ihlenburg and I. Babuska, “Finite element solution of the Helmholtz equation with high wave number Part I: The H-version of the FEM,” Comput. Math. Appl. 30(9), 9–37 (1995).
[Crossref]

1992 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Arroyo, M. P.

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Atkinson, C.

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19(7), 074002 (2008).
[Crossref]

Babuska, I.

F. Ihlenburg and I. Babuska, “Finite element solution of the Helmholtz equation with high wave number Part I: The H-version of the FEM,” Comput. Math. Appl. 30(9), 9–37 (1995).
[Crossref]

Babuska, I. M.

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM Avoidable for the Helmholtz Equation considering high wave numbers?” SIAM Rev. 42(3), 451–484 (2000).
[Crossref]

Belkebir, K.

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20(7), 1223–1229 (2003).
[Crossref] [PubMed]

Bhattacharya, N.

Braat, J. J. M.

Chaumet, P. C.

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

Coupland, J. M.

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

J. Lobera and J. M. Coupland, “Optical diffraction tomography in fluid velocimetry: the use of a priori information,” Meas. Sci. Technol. 19(7), 074013 (2008).
[Crossref]

J. M. Coupland and N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocity measurements using holographic recording and optical correlation,” Appl. Opt. 31(8), 1005–1007 (1992).
[Crossref] [PubMed]

Denis, L.

Fournier, C.

García, I.

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Garzón, E. M.

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Giovannini, H.

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

Goepfert, C.

Halliwell, N. A.

Ihlenburg, F.

F. Ihlenburg and I. Babuska, “Finite element solution of the Helmholtz equation with high wave number Part I: The H-version of the FEM,” Comput. Math. Appl. 30(9), 9–37 (1995).
[Crossref]

Katz, J.

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42(2), 235–250 (2003).
[Crossref] [PubMed]

Koek, W. D.

Lobera, J.

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

J. Lobera and J. M. Coupland, “Optical diffraction tomography in fluid velocimetry: the use of a priori information,” Meas. Sci. Technol. 19(7), 074013 (2008).
[Crossref]

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Maire, G.

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

Malkiel, E.

Ooms, T. A.

Ortega, G.

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Sauter, S. A.

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM Avoidable for the Helmholtz Equation considering high wave numbers?” SIAM Rev. 42(3), 451–484 (2000).
[Crossref]

Sentenac, A.

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20(7), 1223–1229 (2003).
[Crossref] [PubMed]

Sheng, J.

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42(2), 235–250 (2003).
[Crossref] [PubMed]

Soria, J.

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19(7), 074002 (2008).
[Crossref]

Soulez, F.

Thiébaut, É.

Vázquez, F.

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

Westerweel, J.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

Appl. Opt. (2)

Comput. Math. Appl. (1)

F. Ihlenburg and I. Babuska, “Finite element solution of the Helmholtz equation with high wave number Part I: The H-version of the FEM,” Comput. Math. Appl. 30(9), 9–37 (1995).
[Crossref]

J. Mod. Opt. (1)

P. C. Chaumet, A. Sentenac, K. Belkebir, G. Maire, and H. Giovannini, “Improving the resolution of grating-assisted optical diffraction tomography using a priori information in the reconstruction procedure,” J. Mod. Opt. 57(9), 798–808 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Supercomput. (1)

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “The BiConjugate gradient method on GPUs,” J. Supercomput. 64(1), 49–58 (2013).
[Crossref]

Meas. Sci. Technol. (3)

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19(7), 074002 (2008).
[Crossref]

J. Lobera and J. M. Coupland, “Optical diffraction tomography in fluid velocimetry: the use of a priori information,” Meas. Sci. Technol. 19(7), 074013 (2008).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Opt. Express (1)

SIAM Rev. (1)

I. M. Babuska and S. A. Sauter, “Is the pollution effect of the FEM Avoidable for the Helmholtz Equation considering high wave numbers?” SIAM Rev. 42(3), 451–484 (2000).
[Crossref]

Other (11)

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

M. P. Arroyo and K. D. Hinsch, “Recent developments of PIV towards 3D measurements,” in Particle Image Velocimetry, volume 112 of Topics in Applied Physics, A. Schroder and C.E. Willert, eds. (Springer, 2008), pp. 127–154.

G. Ortega, E. M. Garzón, F. Vázquez, and I. García, “Exploiting the regularity of differential operators to accelerate solutions of PDEs on GPUs,” InProceedings of the 11th International Computational and Mathematical Methods in Science and Engineering, J. Vigo-Aguiar, ed. (CMMSE, 2011), pp. 908–917.

NVIDIA. CUDA Toolkit. https://developer.nvidia.com/cuda-toolkit

CUBLAS user guide (du-06702–001 v5.5). http://docs.nvidia.com/cuda/pdf/CUBLAS_Library.pdf .

G. Ortega, I. García, and E. M. Garzón, “A Hybrid Approach for Solving the 3D Helmholtz Equation on Heterogeneous Platforms,” In Euro-Par 2013: Parallel Processing Workshops, volume 8374 of LNCS, D. an Mey et al., eds. (Springer, 2014), pp. 198–207.

China’s TianhE-2 Supercomputer Maintains Top Spot on 42nd. TOP500 List. June 2014.

G. Ortega, J. Lobera, M. P. Arroyo, I. García, and E. M. Garzón, “High performance computing for optical diffraction tomography”. In Proceedings of High Performance Computing Simulation, (IEEE, 2012), pp. 195–201.

G. Ortega, J. Lobera, I. García, M. P. Arroyo, and E. M. Garzón, “Parallel resolution of the 3D Helmholtz equation based on multi-GPU clusters,” Concurr. Comput. (2014), doi:.
[Crossref]

Mathworks. MATLAB with MEX Files. http://www.mathworks.es/es/help/distcomp/run-mex-functions-containing-cuda-code.html .

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, 2001).

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Figures (9)

Fig. 1
Fig. 1

Algorithm of the NLODT-P model.

Fig. 2
Fig. 2

Schematic process of NLODT-P model.

Fig. 3
Fig. 3

Algorithm of the Biconjugate Gradient Method.

Fig. 4
Fig. 4

Comparison of the NLODT-P runtime for MATLAB + GPU (GPU computing and CRF format) and the approach only with MATLAB.

Fig. 5
Fig. 5

(a) 3D view of the particle distribution problem with the full optical access recording configuration. LODT image: (b) 3D view of the brightest voxels and (c) 2D view of the plane at z = 64 pixels.

Fig. 6
Fig. 6

LODT image of one isolated particle (sample): (a) 3D view and (b) 2D view of the central plane (at z = 81 pixels). Filtered LODT image of the particle distribution problem: (c) 3D view and (d) 2D view of the plane at z = 64 pixels.

Fig. 7
Fig. 7

Filtered NLODT gradients (images of the particles that remain to be located) for the full optical access configuration.

Fig. 8
Fig. 8

(a) 3D view of the configuration with one observation direction. Filtered LODT image: (b) 3D view of the brightest voxels and (c) 2D view at z = 64 pixels.

Fig. 9
Fig. 9

Filtered NLODT gradient for the configuration with only one observation direction: (top) 3D view and (bottom) 2D view at z = 64 pixels.

Tables (1)

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Table 1 Memory Requirements (in GB) to Store the Sparse Matrix M with Several Sizes, Using the COO Format (MATLAB) and CRF

Equations (3)

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( 2 + k 0 2 n 2 ( r ) ) E sth ( r )=f( r ) E r ( r ),
cost(f)= i | E m i (r) E c i (f,r) | 2 .
g * ( r )= i E m i * ( r ) E r i (r).

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