Abstract

We use compressive in–line holography to image air bubbles in water and investigate the effect of bubble concentration on reconstruction performance by simulation. Our forward model treats bubbles as finite spheres and uses Mie scattering to compute the scattered field in a physically rigorous manner. Although no simple analytical bounds on maximum concentration can be derived within the classical compressed sensing framework due to the complexity of the forward model, the receiver operating characteristic (ROC) curves in our simulation provide an empirical concentration bound for accurate bubble detection by compressive holography at different noise levels, resulting in a maximum tolerable concentration much higher than the traditional back-propagation method.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (2)

2013 (2)

2012 (2)

2011 (2)

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

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

2010 (5)

2009 (5)

2008 (1)

2007 (1)

2006 (4)

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[Crossref] [PubMed]

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

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

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

2004 (2)

2002 (1)

1994 (1)

1992 (1)

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

Allano, D.

Amato-Grill, J.

Banerjee, P. P.

Barbastathis, G.

Y. Liu, L. Tian, C. H. Hsieh, and G. Barbastathis, “Compressive holographic two-dimensional localization with 1/302 subpixel accuracy,” Opt. Express 22, 9774–9782 (2014).
[Crossref] [PubMed]

Y. Liu, L. Tian, J. W. Lee, H. Y. H. Huang, M. S. Triantafyllou, and G. Barbastathis, “Scanning-free compressive holography for object localization with subpixel accuracy,” Opt. Lett. 37, 3357–3359 (2012).
[Crossref]

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express 20, 8296–8308 (2012).
[Crossref] [PubMed]

L. Tian, N. Loomis, J. A. Domínguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. 49, 1549–1554 (2010).
[Crossref] [PubMed]

L. Tian, Y. Liu, and G. Barbastathis, “Improved axial resolution of digital holography via compressive reconstruction,” in “Biomedical Optics and 3-D Imaging,” (Optical Society of America, 2012), p. DW4C.3.

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of inversion in digital holographic particle imaging: Theory and experimental validation,” in “Frontiers in Optics,” (Optical Society of America, 2008), p. FThV4.

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of the digital holographic inverse problem as a function of particle density,” in “Digital Holography and Three-Dimensional Imaging,” (Optical Society of America, 2008), p. PDPJMA6.

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[Crossref]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[Crossref]

Bohren, C. F.

C. F. Bohren and R. H. Donald, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Brady, D. J.

Callens, N.

Candès, E.

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

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

Cheong, F. C.

Choi, K.

Choo, Y. J.

Coëtmellec, S.

Cull, C. F.

Denis, L.

Dixon, L.

Dominguez-Caballero, J. A.

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of inversion in digital holographic particle imaging: Theory and experimental validation,” in “Frontiers in Optics,” (Optical Society of America, 2008), p. FThV4.

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of the digital holographic inverse problem as a function of particle density,” in “Digital Holography and Three-Dimensional Imaging,” (Optical Society of America, 2008), p. PDPJMA6.

Domínguez-Caballero, J. A.

Donald, R. H.

C. F. Bohren and R. H. Donald, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Donoho, D. L.

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

Dreyfus, R.

Dubois, F.

Erlinger, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Fatemi, E.

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

Ferraro, P.

Fessler, J. A.

Fournel, T.

Fournier, C.

Goepfert, C.

Grier, D. G.

Hennelly, B. M.

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Horisaki, R.

Hsieh, C. H.

Huang, H. Y. H.

Javidi, B.

Juptner, W. P.

Kang, B. S.

Katz, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid. Mech. 42, 531–555 (2010).
[Crossref]

Kelly, D. P.

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Lebrun, D.

Lee, J.

Lee, J. W.

Li, W.

Lim, S.

Liu, Y.

Loomis, N.

Lorenz, D.

Mait, J. N.

Malek, M.

Marks, D. L.

Mattheiss, M.

Memmolo, P.

Milgram, J. H.

Naughton, T. J.

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Nehmetallah, G.

Netti, P. A.

Oh, S. B.

Osher, S.

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

Ozcan, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Pandey, N.

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Paturzo, M.

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]

Rhodes, W. T.

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Rivenson, Y.

Romberg, J.

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

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

Rosen, J.

Rudin, L. I.

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

Schnars, U.

Schockaert, C.

Seo, S.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Sheng, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid. Mech. 42, 531–555 (2010).
[Crossref]

Sotthivirat, S.

Soulez, F.

Stern, A.

Su, T. W.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Sun, B.

Tao, T.

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

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

Thiébaut, E.

Thiébaut, Éric

Tian, L.

Trede, D.

Triantafyllou, M. S.

Tseng, D. K.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Verrier, N.

Wikner, D. A.

Williams, L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Xiao, K.

Yang, Y.

Yourassowsky, C.

Annu. Rev. Fluid. Mech. (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid. Mech. 42, 531–555 (2010).
[Crossref]

Appl. Opt. (6)

Comm. Pure Appl. Math. (1)

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

IEEE Trans. Inf. Theory (1)

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

IEEE Trans. Inform. Theory (2)

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

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

J. Display Technol. (1)

J. Opt. Soc. Am. A (3)

Lab Chip (1)

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009).
[Crossref] [PubMed]

Opt. Eng. (1)

D. P. Kelly, T. J. Naughton, W. T. Rhodes, B. M. Hennelly, and N. Pandey, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Physica D: Nonlinear Phenomena (1)

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

Other (6)

C. F. Bohren and R. H. Donald, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[Crossref]

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of inversion in digital holographic particle imaging: Theory and experimental validation,” in “Frontiers in Optics,” (Optical Society of America, 2008), p. FThV4.

J. A. Dominguez-Caballero and G. Barbastathis, “Stability of the digital holographic inverse problem as a function of particle density,” in “Digital Holography and Three-Dimensional Imaging,” (Optical Society of America, 2008), p. PDPJMA6.

L. Tian, Y. Liu, and G. Barbastathis, “Improved axial resolution of digital holography via compressive reconstruction,” in “Biomedical Optics and 3-D Imaging,” (Optical Society of America, 2012), p. DW4C.3.

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Figures (6)

Fig. 1
Fig. 1 Experimental geometry for in–line holography
Fig. 2
Fig. 2 The intensity at the focal plane reconstructed by BPM (a) and CHM (b), and the longitudinal cross section from BPM (c) and CHM (d), and the corresponding line profiles (e,f,g,h) across the center of the bubble.
Fig. 3
Fig. 3 The SBR in the CHM model decreases as the concentration Rg increases. Insets: sample holograms at the corresponding concentration.
Fig. 4
Fig. 4 Left: Ground truth of bubble distribution; middle: BPM reconstruction; right: CHM reconstruction. First row: Np = 16; second row: Np = 128; third row: Np = 512. Bubbles are represented as circles with colors representing their corresponding axial positions.
Fig. 5
Fig. 5 ROC curve of BPM (red curves)and CHM (blue curves) reconstruction of holograms without noise with number of bubbles Np = 8, 16, 32, 64, 128, 256, 512, 1024 as shown in (a)–(h).
Fig. 6
Fig. 6 Accuracy of discrimination of BPM and CHM reconstruction of holograms with noise level of SNR = 0, 10, and infinity at various concentration.

Equations (14)

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g ( x , y ) = 1 + | E ( x , y , 0 ) | 2 + E * ( x , y , 0 ) + E ( x , y , 0 ) .
E ( x , y , 0 ) = π λ 2 E r ( x , y , z ) f ( x , y , z ) h ( x x , y y , 0 z ) d x d y d z ,
h ( x x , y y , z z ) = exp [ i 2 π ( z z ) / λ ] ( z z ) exp { i π λ ( z z ) [ ( x x ) 2 + ( y y ) 2 ] } .
E ( x , y , 0 ) = i π λ [ f ( x , y z ) exp { i 2 π ( u x + v y ) } d x d y exp { i π λ z ( u 2 + v 2 ) } exp { i 2 π ( u x + v y ) } d u d v ] d z .
E ( l ) = H ( l ) f ( l ) ,
H ( l ) = 2 D 1 Q ( l ) 2 D
E = l E ( l ) = H f = [ H ( 1 ) H ( 2 ) H ( N z ) ] [ f ( 1 ) f ( 2 ) f ( N z ) ] ,
g = 1 + | H f | 2 + H * f * + H f ,
y = [ 2 H r 2 H i ] [ f r f i ] + n = A x + n ,
x ^ = arg min x y A x 2 2 + α x TV ,
x TV = l m 1 m 2 | x m 1 m 2 ( l ) | .
SBR BPM = power in the field from a single particle power in the field from all the rest particles , twin and halo terms ,
SBR CHM = power in total field from the real and twin image terms power in the halo term .
R g = total cross section areas of all bubbles area of the hologram N p π r ¯ 2 ( N Δ ) 2 ,

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