Abstract

Compressive holography applies sparsity priors to data acquired by digital holography to infer a small number of object features or basis vectors from a slightly larger number of discrete measurements. Compressive holography may be applied to reconstruct three-dimensional (3D) images from two-dimensional (2D) measurements or to reconstruct 2D images from sparse apertures. This paper is a tutorial covering practical compressive holography procedures, including field propagation, reference filtering, and inverse problems in compressive holography. We present as examples 3D tomography from a 2D hologram, 2D image reconstruction from a sparse aperture, and diffuse object estimation from diverse speckle realizations.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2011

2010

Q. Xu, K. Shi, H. Li, K. Choi, R. Horisaki, D. Brady, D. Psaltis, and Z. Liu, “Inline holographic coherent anti-Stokes Raman microscopy,” Opt. Express 18, 8213–8219 (2010).
[CrossRef] [PubMed]

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
[CrossRef]

A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express 18, 10510–10523(2010).
[CrossRef] [PubMed]

M. Suezen, A. Giannoula, and T. Durduran, “Compressed sensing in diffuse optical tomography,” Opt. Express 18, 23676–23690 (2010).
[CrossRef]

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

C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A 27, 1856–1862 (2010).
[CrossRef]

M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35, 871–873 (2010).
[CrossRef] [PubMed]

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010).
[CrossRef] [PubMed]

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, H1–H10 (2010).
[CrossRef] [PubMed]

X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A 27, 1630–1637 (2010).
[CrossRef]

2009

2008

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

X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
[CrossRef] [PubMed]

2007

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

2006

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

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

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

2003

2002

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1997

T. Kreis, M. Adams, and W. Juptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

1992

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

1962

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Adams, M.

T. Kreis, M. Adams, and W. Juptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Angelini, E.

Atlan, M.

Bioucas-Dias, J. M.

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

Brady, D.

Brady, D. J.

Candes, E. J.

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

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

Choi, K.

Coskun, A. F.

Cull, C. F.

Denis, L.

Donoho, D. L.

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

Durduran, T.

Fatemi, E.

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

Figueiredo, M. A. T.

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

Fournel, T.

Fournier, C.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Giannoula, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts and Company, 2005).

J. W. Goodman, Speckle Phenomena in Optics, Theory and Applications (Roberts and Company, 2007).

Hahn, J.

Horisaki, R.

Javidi, B.

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

Jueptner, W.

W. Jueptner and U. Schnars, Digital Holography (Springer-Verlag, 2005).

Juptner, W.

T. Kreis, M. Adams, and W. Juptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer, 2005).

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Kreis, T.

T. Kreis, M. Adams, and W. Juptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Lam, E. Y.

Leith, E. N.

Li, H.

Lim, S.

Liu, Z.

Lorenz, D.

Mait, J. N.

Marim, M.

Marim, M. M.

Marks, D. L.

Mattheiss, M.

Moon, T. K.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Moran, J. M.

A. R. Thompson, J. M. Moran, and J. G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Neifeld, M. A.

Olivo-Marin, J.-C.

Osher, S.

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

Ozcan, A.

Pitsianis, N.

D. J. Brady, N. Pitsianis, X. Sun, and P. Potuluri, “Compressive sampling and signal inference,” U.S. patent 7,283,231(16 October 2007).

Poon, T.-C.

Potuluri, P.

D. J. Brady, N. Pitsianis, X. Sun, and P. Potuluri, “Compressive sampling and signal inference,” U.S. patent 7,283,231(16 October 2007).

Psaltis, D.

Rivenson, Y.

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

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

Romberg, J. K.

E. J. Candes, 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.

Rudin, L.

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

Schnars, U.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer, 2005).

W. Jueptner and U. Schnars, Digital Holography (Springer-Verlag, 2005).

Schulz, T. J.

Sencan, I.

Shankar, P.

Shi, K.

Stern, A.

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

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

Stirling, W. C.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Su, T.-W.

Suezen, M.

Sun, X.

D. J. Brady, N. Pitsianis, X. Sun, and P. Potuluri, “Compressive sampling and signal inference,” U.S. patent 7,283,231(16 October 2007).

Swenson, J. G. W.

A. R. Thompson, J. M. Moran, and J. G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Tao, T.

E. J. Candes, 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.

Thompson, A. R.

A. R. Thompson, J. M. Moran, and J. G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Trede, D.

Tropp, J.

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

Upatnieks, J.

Wakin, M. B.

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

Wikner, D. A.

Xu, Q.

Xu, Z.

Zhang, X.

Appl. Opt.

Commun. Pure Appl. Math.

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

IEEE J. Display Technol.

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

IEEE Signal Process. Mag.

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

IEEE Trans. Image Process.

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

IEEE Trans. Inf. Theory

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Nature

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Physica D

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

Proc. SPIE

T. Kreis, M. Adams, and W. Juptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Other

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).
[CrossRef]

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer, 2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts and Company, 2005).

A. R. Thompson, J. M. Moran, and J. G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics, Theory and Applications (Roberts and Company, 2007).

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

W. Jueptner and U. Schnars, Digital Holography (Springer-Verlag, 2005).

D. J. Brady, N. Pitsianis, X. Sun, and P. Potuluri, “Compressive sampling and signal inference,” U.S. patent 7,283,231(16 October 2007).

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Figures (5)

Fig. 1
Fig. 1

Comparison of field propagations: (a) phase of object-scattered field, (b) backpropagation w/ ASM, and (c) backpropagation w/ FSASM.

Fig. 2
Fig. 2

Simulations for holographic tomography: (a) 3D object (View 1), (b) phase of transfer functions (View 2), (c) scattered field, (d) backpropagation (View 3), and (e) compressive reconstruction (View 4).

Fig. 3
Fig. 3

Simulations for diffuse object tomography: (a) 3D object (View 5), (b) phase of propagation kernels (View 6), (c) backpropagation of single speckled realization (View 7), (d) backpropagation averaged by 30 speckled realizations (View 8), (e) backpropagation using the Tikhonov regularization w/ 30 speckled realizations (View 9), and (f) compressive reconstruction w/ 30 speckled realizations (View 10).

Fig. 4
Fig. 4

Coherent and incoherent bandpasses in sparse aperture design: (a) coherent bandpass, (b) incoherent bandpass, and (c) cross section of incoherent bandpass. Note that Δ is the space of subapertures.

Fig. 5
Fig. 5

Simulations for sparse aperture holography: (a) 2D object, (b) sparse aperture, (c) averaged backpropagation of coherent estimation model, (d) compressive reconstruction of coherent estimation model, (e) averaged backpropagation of incoherent estimation model, and (f) compressive reconstruction of incoherent estimation model.

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Equations (40)

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

E o ( x , y ) = k e j k z j 2 π z e j k 2 z ( x 2 + y 2 ) d u d v e j k 2 z ( 2 x u + 2 y v ) E s ( u , v ) e j k 2 z ( u 2 + v 2 ) ,
δ x = λ z N δ u ,
Δ x = λ z δ u ,
E o ( x , y ) = 1 ( 2 π ) 2 d u d v e j ( k u u + k v v ) E s ( u , v ) d k u d k v e j z k 2 k u 2 k v 2 e j ( k u x + k v y ) ,
δ x = δ u ,
Δ x = N δ u ,
E o ( x , y ) = S ( x , y ) k e j k z j 2 π z e j k 2 z ( x 2 + y 2 ) ,
S ( x , y ) = j z e j k z 2 π k d u d v e j ( k u u + k v v ) E s ( u , v ) d k u d k v e j z k 2 k u 2 k v 2 e j ( k u x + k v y ) e j k 2 z ( x 2 + y 2 ) .
S ˜ ( k x , k y ) = j z e j k z 2 π k d u d v e j ( k u u + k v v ) E s ( u , v ) d k u d k v e j z k 2 k u 2 k v 2 d x d y e j ( k x x + k y y ) e j ( k u x + k v y ) e j k 2 z ( x 2 + y 2 ) .
S ˜ ( k x , k y ) = j z e j k z 2 π k d u d v e j ( k u u + k v v ) E s ( u , v ) d k u d k v e j z k 2 k u 2 k v 2 [ 2 π j z k e j z 2 k [ ( k u k x ) 2 + ( k v k y ) 2 ] ] .
e j z k 2 k u 2 k v 2 = e j z R ( k u , k v ) e j k z e j z 2 k ( k u 2 + k v 2 ) ,
S ˜ ( k x , k y ) = z 2 k 2 e j z 2 k ( k x 2 + k y 2 ) d u d v e j ( k u u + k v v ) E s ( u , v ) d k u d k v e j z R ( k u , k v ) e j z k ( k u k x + k v k y ) .
E o ( x , y ) = z e j k z j ( 2 π ) k e j k 2 z ( x 2 + y 2 ) d k x d k y e j ( k x x + k y y ) e j z 2 k ( k x 2 + k y 2 ) D ( k x , k y ) , with     D ( k x , k y ) = 1 ( 2 π ) 2 d k u d k v e j k u z k k x j k v z k k y e j z R ( k u , k v ) [ d u d v e j ( k u u + k v v ) E s ( u , v ) ] .
E o ( x , y ) = k e j k z j 2 π z e j k 2 z ( x 2 + y 2 ) d k x d k y e j k z ( u x + v y ) e j k 2 z ( u 2 + v 2 ) D ( u , v ) , with   D ( u , v ) = 1 ( 2 π ) 2 d k u d k v e j ( k u u + k v v ) e j z R ( k u , k v ) [ d u d v e j ( k u u + k v v ) E s ( u , v ) ] .
e j z k 2 k u 2 k v 2 .
I ( u , v ) = | R ( u , v ) + E s ( u , v ) | 2 ,
= | R ( u , v ) | 2 + | E s ( u , v ) | 2 + R * ( u , v ) E s ( u , v ) + R ( u , v ) E s * ( u , v ) ,
E s ( u , v ) = d x d y d z η ( x , y , z ) h ( u x , v y , z ) ,
E n 1 n 2 = F 2 D 1 { l η ^ m 1 m 2 l e i k l Δ z e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 } ,
g ¯ = G 2 D Q B f ,
[ P l ] m 1 m 2 = e i k l Δ z e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 .
g = 2 Re { g ¯ } = 2 Re { G 2 D Q B f } = 2 Re { H f } + q + n ,
f ^ = argmin f f T V such that g = H f ,
f k T V = k n 1 n 2 | ( f k ) n 1 , n 2 | ,
g k = H f k + w k ,
I s = Σ k = 1 K I k ,
σ s = Σ k = 1 K I ¯ k 2
C = σ s I s ¯ = Σ k = 1 K I ¯ k 2 Σ k = 1 K I ¯ k .
p ( f ) = 1 π N det ( R f ) exp ( f H R f 1 f ) ,
f ^ k = H H ( H H H ) 1 g k = H H g k = H H H f k + H H w k .
s ^ n = 1 K Σ k = 1 K | f ^ n k | 2 .
d = E [ s ^ ] = 1 K Σ k = 1 K Diag ( E [ f k ^ f k ^ H ] ) = Diag ( E [ f ^ f ^ H ] ) = Diag ( H H H R f H H H + σ 2 H H H ) ,
E [ s ^ n ] = m = 1 N | h n , h m | 2 α m + σ 2 h n , h n = m = 1 N | h n H h m | 2 α m + σ 2 h n 2 ,
d = E [ s ^ ] = B α + σ 2 W ,
α * = argmin α 1 2 d B α 2 2 + β Φ ( α ) .
α * = argmin α 1 2 d ˜ B ˜ α 2 2 + β Φ ( α ) ,
Δ ϕ R ( u ; z r ) = 2 π λ { z r 2 + ( u + δ u ) 2 z r 2 + u 2 } .
Δ ϕ R ( u ; z r ) = 2 π λ { 2 u δ u + δ u 2 2 z r } = π λ z r 2 u δ u < π ,
Δ ϕ H ( f u ; z d ) = 2 π λ z d { 1 λ 2 ( f u + δ f u ) 2 1 λ 2 f u 2 } .
Δ ϕ H ( f u ; z d ) = 2 π λ z d λ 2 { 2 f u δ f u + δ f u 2 2 } = π z d λ 1 δ u 1 u < π ,

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