Abstract

In this paper, we show fast signal reconstruction for compressive holography using a graphics processing unit (GPU). We implemented a fast iterative shrinkage-thresholding algorithm on a GPU to solve the 1 and total variation (TV) regularized problems that are typically used in compressive holography. Since the algorithm is highly parallel, GPUs can compute it efficiently by data-parallel computing. For better performance, our implementation exploits the structure of the measurement matrix to compute the matrix multiplications. The results show that GPU-based implementation is about 20 times faster than CPU-based implementation.

© 2016 Optical Society of America

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References

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  1. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
    [Crossref]
  2. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
    [Crossref]
  3. E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
    [Crossref]
  4. E. J. Candes and M. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Proc. Mag. 25(2), 21–30 (2008).
    [Crossref]
  5. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive Holography,” Opt. Express 17(15), 13040–13049 (2009).
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    [Crossref] [PubMed]
  10. J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
    [Crossref]
  11. J. Nickolls and W. J. Dally, “The GPU Computing Era,” IEEE Micro 30(2), 56–69 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  14. Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
    [Crossref]
  15. D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12), 14871–14884 (2014).
    [Crossref] [PubMed]
  16. S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.
  17. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
    [Crossref]
  18. A. Chambolle, “An Algorithm for Total Variation Minimization and Applications,” J. Math. Imaging Vis. 20(1), 89–97 (2004).
    [Crossref]
  19. A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imag. Sci. 2(1), 183–202 (2009).
    [Crossref]
  20. A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
    [Crossref] [PubMed]
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  22. N. Parikh and S. Boyd, “Proximal Algorithms,” Foundations and Trends in Optimization 1(3), 127–239 (2014).
    [Crossref]
  23. D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130(1), 11–16 (1983).
    [Crossref]
  24. Y. Rivenson, A. Stern, and B. Javidi, “Overview of compressive sensing techniques applied in holography [Invited],” Appl. Opt. 52(1), A423–A432 (2013).
    [Crossref] [PubMed]
  25. Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52(1), A223–A231 (2013).
    [Crossref] [PubMed]
  26. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” J. Disp. Technol. 6(10), 506–509 (2010).
    [Crossref]
  27. R. Horisaki, J. Tanida, A. Stern, and B. Javidi, “Multidimensional imaging using compressive Fresnel holography,” Opt. Lett. 37(11), 2013–2015 (2012).
    [Crossref] [PubMed]

2014 (2)

2013 (3)

2012 (3)

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
[Crossref]

R. Horisaki, J. Tanida, A. Stern, and B. Javidi, “Multidimensional imaging using compressive Fresnel holography,” Opt. Lett. 37(11), 2013–2015 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (3)

K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49(34), H1–H10 (2010).
[Crossref] [PubMed]

J. Nickolls and W. J. Dally, “The GPU Computing Era,” IEEE Micro 30(2), 56–69 (2010).
[Crossref]

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

2009 (3)

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imag. Sci. 2(1), 183–202 (2009).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive Holography,” Opt. Express 17(15), 13040–13049 (2009).
[Crossref] [PubMed]

2008 (3)

E. J. Candes and M. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Proc. Mag. 25(2), 21–30 (2008).
[Crossref]

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

2006 (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

2004 (1)

A. Chambolle, “An Algorithm for Total Variation Minimization and Applications,” J. Math. Imaging Vis. 20(1), 89–97 (2004).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
[Crossref]

1983 (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130(1), 11–16 (1983).
[Crossref]

Beck, A.

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imag. Sci. 2(1), 183–202 (2009).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

Bernabe, S.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Bilen, Ç.

Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
[Crossref]

Bioucas-Dias, J. M.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Boyd, S.

N. Parikh and S. Boyd, “Proximal Algorithms,” Foundations and Trends in Optimization 1(3), 127–239 (2014).
[Crossref]

Brady, D. J.

Brandwood, D. H.

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130(1), 11–16 (1983).
[Crossref]

Buck, I.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Candes, E. J.

E. J. Candes and M. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Proc. Mag. 25(2), 21–30 (2008).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

Chambolle, A.

A. Chambolle, “An Algorithm for Total Variation Minimization and Applications,” J. Math. Imaging Vis. 20(1), 89–97 (2004).
[Crossref]

Choi, K.

Dally, W. J.

J. Nickolls and W. J. Dally, “The GPU Computing Era,” IEEE Micro 30(2), 56–69 (2010).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
[Crossref]

Garland, M.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3 Edition, (Roberts & Company, 2005).

Gore, J. C.

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

Green, S.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Hahn, J.

Horisaki, R.

Houston, M.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Huang, Y.

Javidi, B.

Kang, J. U.

Lim, S.

Luebke, D.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Marks, D. L.

Martin, G.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Nascimento, J. M. P.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Nickolls, J.

J. Nickolls and W. J. Dally, “The GPU Computing Era,” IEEE Micro 30(2), 56–69 (2010).
[Crossref]

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
[Crossref]

Owens, J. D.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Parikh, N.

N. Parikh and S. Boyd, “Proximal Algorithms,” Foundations and Trends in Optimization 1(3), 127–239 (2014).
[Crossref]

Phillips, J. C.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Plaza, A.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Rivenson, Y.

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Rosen, J.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
[Crossref]

Schulz, T. J.

Selesnick, I. W.

Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
[Crossref]

Silva, V.

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

Skadron, K.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Smith, D. S.

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

Stern, A.

Stone, J. E.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Tanida, J.

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

Teboulle, M.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imag. Sci. 2(1), 183–202 (2009).
[Crossref]

Wakin, M.

E. J. Candes and M. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Proc. Mag. 25(2), 21–30 (2008).
[Crossref]

Wang, Y.

Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
[Crossref]

Welch, E. B.

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

Xu, D.

Yankeelov, T. E.

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

Appl. Opt. (4)

Foundations and Trends in Optimization (1)

N. Parikh and S. Boyd, “Proximal Algorithms,” Foundations and Trends in Optimization 1(3), 127–239 (2014).
[Crossref]

IEE Proc. H Microwaves Opt. Antennas (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130(1), 11–16 (1983).
[Crossref]

IEEE J. Emerg. Sel. Topics Circuits Syst. (1)

Ç. Bilen, Y. Wang, and I. W. Selesnick, “High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing,” IEEE J. Emerg. Sel. Topics Circuits Syst. 2(3), 370–379 (2012).
[Crossref]

IEEE Micro (1)

J. Nickolls and W. J. Dally, “The GPU Computing Era,” IEEE Micro 30(2), 56–69 (2010).
[Crossref]

IEEE Signal Proc. Mag. (1)

E. J. Candes and M. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Proc. Mag. 25(2), 21–30 (2008).
[Crossref]

IEEE Trans. Image Process. (1)

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

Int. J. Biomed. Imaging (1)

D. S. Smith, J. C. Gore, T. E. Yankeelov, and E. B. Welch, “Real-Time Compressive Sensing MRI Reconstruction Using GPU Computing and Split Bregman Methods,” Int. J. Biomed. Imaging 2012, 864827 (2012).
[Crossref] [PubMed]

J. Disp. Technol. (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

J. Math. Imaging Vis. (1)

A. Chambolle, “An Algorithm for Total Variation Minimization and Applications,” J. Math. Imaging Vis. 20(1), 89–97 (2004).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1–4), 259–268 (1992).
[Crossref]

Proc. IEEE (1)

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Queue (1)

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

SIAM J. Imag. Sci. (1)

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imag. Sci. 2(1), 183–202 (2009).
[Crossref]

Other (2)

S. Bernabe, G. Martin, J. M. P. Nascimento, J. M. Bioucas-Dias, A. Plaza, and V. Silva, “GPU implementation of a hyperspectral coded aperture algorithm for compressive sensing,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2015), pp. 521–524.

J. W. Goodman, Introduction to Fourier Optics, 3 Edition, (Roberts & Company, 2005).

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Figures (6)

Fig. 1
Fig. 1 Setup for Gabor holography.
Fig. 2
Fig. 2 Schematic diagram of the implementation of GPU-accelerated compressive holography.
Fig. 3
Fig. 3 Multiplication of the measurement matrix H using its structure.
Fig. 4
Fig. 4 Computation time of compressive holography using FISTA with 600 iterations on (a) the GPU (GTX 980) and (b) the CPU (Core i7-4790K).
Fig. 5
Fig. 5 Computation time of each step in the GPU-accelerated compressive holography where Nx × Ny × Nz = 1024 × 1024 × 10.
Fig. 6
Fig. 6 (a) 3D object and reconstructed images by (b) backpropagation and (c) compressive holography.

Tables (1)

Tables Icon

Table 1 Computation time of compressive holography using FISTA with 600 iterations.

Equations (16)

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

u = Hf = F 1 QBf ,
Q = [ P 1 P 2 P N z ] ,
B = bdiag ( F , F , , F ) ,
g i = 2 Re ( r i u i ) + | r i | 2 + | u i | 2 , i = 1 , , N x y ,
g i = 2 Re ( u i ) + | u i | 2 = 2 Re ( u i ) + e i , i = 1 , , N x y ,
g = 2 Re ( Hf ) + e .
min f { ( f ) + τ ( f ) } : = min f { 2 Re ( Hf ) g 2 2 + τ ( f ) } ,
f 1 : = i = 1 N x y z | f i | ,
TV ( f ) : = j = 1 N z i = 1 N x y D i f j 2 ,
f k = prox τ / L ( h k 1 L ( h k ) ) ,
t k + 1 = 1 + 1 + 4 t k 2 2 ,
h k + 1 = f k + ( t k 1 t k + 1 ) ( f k f k 1 ) ,
prox τ / L ( h ) : = argmin f { τ L ( f ) + 1 2 f h 2 2 } .
( f ) = 2 H H ( 2 Re ( Hf ) g ) ,
prox τ / L ( h ) = 𝒯 τ / L ( h ) ,
𝒯 κ ( a ) : = ( 1 κ | a | ) + a ,

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