Abstract

In recent years compressive sensing (CS) has been successfully introduced in digital holography (DH). Depending on the ability to sparsely represent an object, the CS paradigm provides an accurate object reconstruction framework from a relatively small number of encoded signal samples. DH has proven to be an efficient and physically realizable sensing modality that can exploit the benefits of CS. In this paper, we provide an overview of the theoretical guidelines for application of CS in DH and demonstrate the benefits of compressive digital holographic sensing.

© 2012 Optical Society of America

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2013 (1)

Y. Rivenson, A. Stern, and B. Javidi, “Improved three-dimensional resolution by single exposure in-line compressive holography,” Appl. Opt. 52, xxx–xxx (2013).

2012 (4)

2011 (9)

2010 (7)

2009 (9)

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2, 1098–1128 (2009).
[CrossRef]

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. 31, 890–912 (2009).
[CrossRef]

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[CrossRef]

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

S. M. Khan and M. Shah, “Tracking multiple occluding people by localizing on multiple scene planes,” IEEE Trans. Pattern. Anal. Mach. Intell. 31, 505–519 (2009).
[CrossRef]

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48, H120–H136(2009).
[CrossRef]

I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
[CrossRef]

2008 (4)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008).
[CrossRef]

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

2007 (5)

A. Stern and B. Javidi, “Random projections image with extended space-bandwidth product,” J. Display Technology 3, 315–320 (2007).
[CrossRef]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

A. Stern and B. Javidi, “Theoretical analysis of three-dimensional imaging and recognition of micro-organisms with a single-exposure on-line holographic microscope,” J. Opt. Soc. Am. A 24, 163–168 (2007).
[CrossRef]

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Y. S. Hwang, S.-H. Hong, and B. Javidi, “Free view 3-d visualization of occluded objects by using computational synthetic aperture integral imaging,” J. Display Technology 3, 64–70 (2007).
[CrossRef]

2006 (6)

2005 (2)

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492–4506 (2005).
[CrossRef]

2004 (2)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
[CrossRef]

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

1990 (1)

Abolbashari, M.

Angelini, E.

Araújo, F.

Ashok, A.

Atlan, M.

Avidan, S.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, “Synthetic aperture tracking: tracking through occlusions,” in IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Balber, S.

Baraniuk, R.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Barbastathis, G.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Bolas, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

Brady, D.

Brady, D. J.

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Burger, M.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

Candès, E.

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

Candès, E. J.

E. J. Candès and Y. Plan, “A probabilistic and RIPless theory of compressed sensing,” IEEE Trans. Inf. Theory 57, 7235–7254 (2011).
[CrossRef]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math 59, 1207–1223 (2006).
[CrossRef]

Carapezza, E.

Chan, W.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charan, K.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charrière, F.

Chen, B.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

Choi, K.

Clemente, P.

Colomb, T.

Correia, M.

Coskun, A. F.

Cuche, E.

Cull, C. F.

Daneshpanah, M.

I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Denis, L.

Depeursinge, C.

Donoho, D.

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

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008).
[CrossRef]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Durán, V.

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Farahi, F.

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Fernández-Alonso, M.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Fessler, J. A.

Fienup, J.

A. Tippie and J. Fienup, “Weak-object image reconstructions with single-shot digital holography,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest(Optical Society of America, 2012), paper DM4C.5.

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Figueiredo, M. A. T.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

Fournier, C.

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. 31, 890–912 (2009).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Goldfarb, D.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

Hahn, J.

Hennelly, B. M.

Hong, S.-H.

Y. S. Hwang, S.-H. Hong, and B. Javidi, “Free view 3-d visualization of occluded objects by using computational synthetic aperture integral imaging,” J. Display Technology 3, 64–70 (2007).
[CrossRef]

B. Javidi, R. Ponce-Díaz, and S.-H. Hong, “Three-dimensional recognition of occluded objects by using computational integral imaging,” Opt. Lett. 31, 1106–1108 (2006).
[CrossRef]

Horisaki, R.

Horowitz, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

Huang, H.

Hwang, Y. S.

Y. S. Hwang, S.-H. Hong, and B. Javidi, “Free view 3-d visualization of occluded objects by using computational synthetic aperture integral imaging,” J. Display Technology 3, 64–70 (2007).
[CrossRef]

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Improved three-dimensional resolution by single exposure in-line compressive holography,” Appl. Opt. 52, xxx–xxx (2013).

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

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

Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18, 15094–15103 (2010).
[CrossRef]

I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
[CrossRef]

A. Stern and B. Javidi, “Theoretical analysis of three-dimensional imaging and recognition of micro-organisms with a single-exposure on-line holographic microscope,” J. Opt. Soc. Am. A 24, 163–168 (2007).
[CrossRef]

Y. S. Hwang, S.-H. Hong, and B. Javidi, “Free view 3-d visualization of occluded objects by using computational synthetic aperture integral imaging,” J. Display Technology 3, 64–70 (2007).
[CrossRef]

A. Stern and B. Javidi, “Random projections image with extended space-bandwidth product,” J. Display Technology 3, 315–320 (2007).
[CrossRef]

J. Maycock, C. P. McElhinney, B. M. Hennelly, T. J. Naughton, J. B. McDonald, and B. Javidi, “Reconstruction of partially occluded objects encoded in three-dimensional scenes by using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
[CrossRef]

B. Javidi, R. Ponce-Díaz, and S.-H. Hong, “Three-dimensional recognition of occluded objects by using computational integral imaging,” Opt. Lett. 31, 1106–1108 (2006).
[CrossRef]

S. Yeom and B. Javidi, “Automatic identification of biological microorganisms using three-dimensional complex morphology,” J. Biomed. Opt. 11, 024017 (2006).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492–4506 (2005).
[CrossRef]

A. Stern, Y. Rivenson, and B. Javidi, “Single exposure optically compressed imaging and visualization using random aperture,” in Seventh International Workshop on Information Optics (IOP, 2008), paper 12018.

Joshi, N.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, “Synthetic aperture tracking: tracking through occlusions,” in IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Katz, B.

Kelly, K.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Khan, S. M.

S. M. Khan and M. Shah, “Tracking multiple occluding people by localizing on multiple scene planes,” IEEE Trans. Pattern. Anal. Mach. Intell. 31, 505–519 (2009).
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry, 1st ed.(Wiley-VCH, 2004), Chap. 3.

Kriegman, D. J.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, “Synthetic aperture tracking: tracking through occlusions,” in IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Lam, E. Y.

Lancis, J.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Lee, J.

Levoy, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

Lim, S.

Liu, Y.

Lorenz, D.

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008).
[CrossRef]

Magalhães, F.

Mait, J. N.

Marcia, R.

R. Willett, R. Marcia, and J. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50, 072601 (2011).
[CrossRef]

Marim, M.

Marim, M. M.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Marks, D.

Marks, D. L.

Marquet, P.

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mattheiss, M.

Matusik, W.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, “Synthetic aperture tracking: tracking through occlusions,” in IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Maycock, J.

McDonald, J. B.

McDowall, I.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[CrossRef]

McElhinney, C. P.

Mittleman, D.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Montfort, F.

Moon, I.

I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492–4506 (2005).
[CrossRef]

Naughton, T. J.

Neifeld, M.

Nichols, J.

R. Willett, R. Marcia, and J. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50, 072601 (2011).
[CrossRef]

Nowak, R. D.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

Olivo-Marin, J.-C.

Osher, S.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Ozcan, A.

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008).
[CrossRef]

Plan, Y.

E. J. Candès and Y. Plan, “A probabilistic and RIPless theory of compressed sensing,” IEEE Trans. Inf. Theory 57, 7235–7254 (2011).
[CrossRef]

Ponce-Díaz, R.

Rivenson, Y.

Romberg, J.

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2, 1098–1128 (2009).
[CrossRef]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

Romberg, J. K.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math 59, 1207–1223 (2006).
[CrossRef]

Rosen, J.

Rot, A.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008).
[CrossRef]

Schulz, T. J.

Sencan, I.

Shah, M.

S. M. Khan and M. Shah, “Tracking multiple occluding people by localizing on multiple scene planes,” IEEE Trans. Pattern. Anal. Mach. Intell. 31, 505–519 (2009).
[CrossRef]

Shaked, N. T.

Sotthivirat, S.

Stern, A.

Y. Rivenson, A. Stern, and B. Javidi, “Improved three-dimensional resolution by single exposure in-line compressive holography,” Appl. Opt. 52, xxx–xxx (2013).

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

Y. Rivenson, A. Rot, S. Balber, A. Stern, and J. Rosen, “Recovery of partially occluded objects by applying compressive Fresnel holography,” Opt. Lett. 37, 1757–1759 (2012).
[CrossRef]

Y. Rivenson, A. Stern, and J. Rosen, “Compressive multiple view projection incoherent holography,” Opt. Express 19, 6109–6118 (2011).
[CrossRef]

Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011).
[CrossRef]

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

Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18, 15094–15103 (2010).
[CrossRef]

I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
[CrossRef]

A. Stern and B. Javidi, “Theoretical analysis of three-dimensional imaging and recognition of micro-organisms with a single-exposure on-line holographic microscope,” J. Opt. Soc. Am. A 24, 163–168 (2007).
[CrossRef]

A. Stern and B. Javidi, “Random projections image with extended space-bandwidth product,” J. Display Technology 3, 315–320 (2007).
[CrossRef]

A. Stern, Y. Rivenson, and B. Javidi, “Single exposure optically compressed imaging and visualization using random aperture,” in Seventh International Workshop on Information Optics (IOP, 2008), paper 12018.

Su, T.-W.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Tajahuerce, E.

Takhar, D.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Tanida, J.

Tao, T.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math 59, 1207–1223 (2006).
[CrossRef]

Thiébaut, E.

Tian, L.

Tippie, A.

A. Tippie and J. Fienup, “Weak-object image reconstructions with single-shot digital holography,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest(Optical Society of America, 2012), paper DM4C.5.

Trede, D.

Triantafyllou, M.

Vaish, V.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. 31, 890–912 (2009).
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Wakin, M.

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
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R. Willett, R. Marcia, and J. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50, 072601 (2011).
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S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
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S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
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Yeom, S.

S. Yeom and B. Javidi, “Automatic identification of biological microorganisms using three-dimensional complex morphology,” J. Biomed. Opt. 11, 024017 (2006).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492–4506 (2005).
[CrossRef]

Yin, W.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

Zhang, X.

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W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Commun. Pure Appl. Math (1)

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math 59, 1207–1223 (2006).
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E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
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M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
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IEEE Trans. Inf. Theory (2)

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
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IEEE Trans. Pattern. Anal. Mach. Intell. (1)

S. M. Khan and M. Shah, “Tracking multiple occluding people by localizing on multiple scene planes,” IEEE Trans. Pattern. Anal. Mach. Intell. 31, 505–519 (2009).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

Inverse Probl. (1)

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

J. Biomed. Opt. (1)

S. Yeom and B. Javidi, “Automatic identification of biological microorganisms using three-dimensional complex morphology,” J. Biomed. Opt. 11, 024017 (2006).
[CrossRef]

J. Display Technol. (1)

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A. Stern and B. Javidi, “Random projections image with extended space-bandwidth product,” J. Display Technology 3, 315–320 (2007).
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Y. S. Hwang, S.-H. Hong, and B. Javidi, “Free view 3-d visualization of occluded objects by using computational synthetic aperture integral imaging,” J. Display Technology 3, 64–70 (2007).
[CrossRef]

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S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4, 460–489 (2005).
[CrossRef]

Opt. Commun. (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Opt. Eng. (1)

R. Willett, R. Marcia, and J. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50, 072601 (2011).
[CrossRef]

Opt. Express (6)

Opt. Lett. (9)

Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011).
[CrossRef]

B. Javidi, R. Ponce-Díaz, and S.-H. Hong, “Three-dimensional recognition of occluded objects by using computational integral imaging,” Opt. Lett. 31, 1106–1108 (2006).
[CrossRef]

Y. Rivenson, A. Rot, S. Balber, A. Stern, and J. Rosen, “Recovery of partially occluded objects by applying compressive Fresnel holography,” Opt. Lett. 37, 1757–1759 (2012).
[CrossRef]

R. Horisaki, J. Tanida, A. Stern, and B. Javidi, “Multidimensional imaging using compressive Fresnel holography,” Opt. Lett. 37, 2013–2015 (2012).
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Y. Liu, L. Tian, J. Lee, H. Huang, M. Triantafyllou, and G. Barbastathis, “Scanning-free compressive holography for object localization with subpixel accuracy,” Opt. Lett. 37, 3357–3359 (2012).
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L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
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I. Moon, M. Daneshpanah, B. Javidi, and A. Stern, “Automated three-dimensional imaging, identification and tracking of micro/nano biological organisms by holographic microscopy,” Proc. IEEE 97, 990–1010 (2009).
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Other (7)

T. Kreis, Handbook of Holographic Interferometry, 1st ed.(Wiley-VCH, 2004), Chap. 3.

A. Stern, Y. Rivenson, and B. Javidi, “Single exposure optically compressed imaging and visualization using random aperture,” in Seventh International Workshop on Information Optics (IOP, 2008), paper 12018.

“Compressive sensing resources,” http://www.compressedsensing.com/ .

“Compressive sensing: The big picture,” https://sites.google.com/site/igorcarron2/cs#reconstruction .

“SPARCO: A toolbox for testing sparse reconstruction algorithms,” http://www.cs.ubc.ca/labs/scl/sparco .

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, “Synthetic aperture tracking: tracking through occlusions,” in IEEE 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

A. Tippie and J. Fienup, “Weak-object image reconstructions with single-shot digital holography,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest(Optical Society of America, 2012), paper DM4C.5.

Supplementary Material (4)

» Media 1: AVI (3448 KB)     
» Media 2: AVI (3448 KB)     
» Media 3: AVI (4137 KB)     
» Media 4: AVI (4137 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of numerical near and far field diffraction, and its relation with the coherence parameter, μ 1 .

Fig. 2.
Fig. 2.

Simulation results showing the normalized compressive sampling ratio for different sparsifying bases required for reconstruction of the USAF 1951 resolution target as a function of working distance.

Fig. 3.
Fig. 3.

Compressive digital holographic sensing (CDHS) for a reflection, single shot, off-axis hologram. (a) Fresnel hologram of a 5 NIS coin. (b) Backpropagation reconstruction from the fully sampled hologram in (a). (c) 8% uniformly random subsampling of (a). (d) 8% variable density random sampling of (a). (e) Backpropagation reconstruction from (c). (f) CS reconstruction from (c) yielding 32.2 dB PSNR.. (g) Backpropagation reconstruction from (d). CS reconstruction in (h) yielding 34 dB PSNR.

Fig. 4.
Fig. 4.

Schematic setup for the acquisition of a partially occluded object field. In the object and occluded planes, black represents totally opaque regions, white represents transparent regions, and gray level represent turbid regions.

Fig. 5.
Fig. 5.

Reconstruction of an occluded resolution chart from its 2 × 2 subsampled, noisy hologram. (a) Original object and (b) simulated occluding plane, which is composed from random phases and totally opaque, X-shaped region. (c) Reconstruction from the acquired hologram using backpropagation. Backpropagation with different opacity percentage is available in Media 1. (d) Reconstruction using the proposed compressive digital holographic sensing approach. Compressive digital holographic sensing reconstruction with different opacity percentage is available in Media 2.

Fig. 6.
Fig. 6.

Illustration of compressive MVP incoherent holography [26,51]. Using a CCD camera located at distance z 0 from a scene, K log N projections (denoted by p i ) are captured. Each acquired projection is digitally multiplied and summarized by a corresponding complex function to generate a subsampled hologram.

Fig. 7.
Fig. 7.

CDHS applied to MVP incoherent holography. (a) One of the captured views of the scene. (b) Acquired, subsampled hologram where only 5% of the nominal number views are acquired. (c) Standard numerical backpropagation of two of the object planes from (b). Backpropagation from different subsampling percentage is available in Media 3. (d) The two corresponding planes from (c), only this time reconstructed using the CS approach. The depth sectioning is evident. CS based reconstruction from different subsampling percentage is available in Media 4.

Fig. 8.
Fig. 8.

SEOL holography setup.

Fig. 9.
Fig. 9.

SEOL holography sectioning dependence on the reference beam intensity. (a) Original 3D object, distributed over 3 planes. (b) Noisy SEOL hologram recorded using (a) with an SNR of 10 dB. (c) Results of numerical backpropagation reconstruction from (b). (d) Compressive digital holographic sensing 3D object reconstruction approach from the noisy SEOL hologram in (b).

Equations (15)

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

g = Φ f = Φ Ψ α ,
μ 1 = max i , j | ϕ i , ψ j | 2 ,
M N C μ 1 S log N .
μ 2 = max i j | ω i * , ω j | ω i 2 ω j 2 ,
S 1 2 { 1 + 1 μ 2 } .
min α α 1 such that g = Φ Ψ α .
min f TV ( f ) such that g = Φ f with TV ( f ) = ( f i + 1 , j f i , j ) 2 + ( f i , j + 1 f i , j ) 2 .
g ( x , y ) = f ( x , y ) × exp { j π λ z ( x 2 + y 2 ) } = exp { j π λ z ( x 2 + y 2 ) } f ( ξ , η ) exp { j π λ z ( ξ 2 + η 2 ) } exp { j 2 π λ z ( x ξ + y η ) } d ξ d η .
g ( p Δ x o , q Δ x o ) = F 2 D 1 exp { j π λ z ( m 2 N Δ x 0 2 + n 2 N Δ y 0 2 ) } F 2 D { f ( l Δ x 0 , k Δ y 0 ) } ,
g ( p Δ x z , q Δ y z ) = exp { j π λ z ( p 2 Δ x z 2 + q 2 Δ y z 2 ) } F 2 D [ f ( k Δ x 0 , l Δ y 0 ) exp { j π λ z ( k 2 Δ x 0 2 + l 2 Δ y 0 2 ) } ] ,
M C N F 2 S N log N ,
M C S log N ,
g ( x , y ) = [ f ( x , y ) × exp { j π λ z 1 ( x 2 + y 2 ) } ] × o ( x , y ) × exp { j π λ z 2 ( x 2 + y 2 ) } .
g ˜ ( p Δ x z 1 , q Δ y z 1 ) = o ( p Δ x z 1 , q Δ y z 1 ) × exp { j π λ z ( p 2 Δ x z 1 2 + q 2 Δ y z 1 2 ) } × F 2 D [ f ( k Δ x 0 , l Δ y 0 ) exp { j π λ z 1 ( k 2 Δ x 0 2 + l 2 Δ y 0 2 ) } ] .
μ 2 F F = max m 1 | O ^ ( m l ) O ^ ( m l ) | o 2 2 ,

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