Abstract

Compressive holography is a combination of compressive sensing and holography. In this paper, an approach to expand the amplification ratio and enhance the axial resolution in in-line compressive holography is proposed. Firstly the basic principle of 4f amplified in-line compressive holography is described. Next the feasibility of reconstructing object and analysis of reconstruction quality is verified. Finally, both simulated and real experiments on multilayer objects with non-overlapping and overlapping patterns are demonstrated to validate the approach.

© 2014 Optical Society of America

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References

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2013 (6)

2012 (1)

2011 (2)

2010 (3)

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49(19), E67–E82 (2010).
[Crossref] [PubMed]

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref] [PubMed]

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

2009 (1)

2008 (2)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346(9), 589–592 (2008).
[Crossref]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

2006 (1)

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

2004 (1)

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

2002 (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

2000 (1)

Andrés, P.

Angelini, E.

Arce, G. R.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref] [PubMed]

Atlan, M.

Banerjee, P. P.

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Brady, D. J.

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346(9), 589–592 (2008).
[Crossref]

Choi, K.

Clemente, P.

Climent, V.

Cull, C. F.

Di, H.

Donoho, D. L.

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

Durán, V.

Glennon, B.

Hahn, J.

Horisaki, R.

Javidi, B.

Jin, H. Z.

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

Kim, T.

Kim, Y. S.

Lam, E. Y.

Lancis, J.

Li, Y.

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

Lim, S.

Ma, L. H.

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

Mait, J. N.

Marim, M.

Marks, D. L.

Mattheiss, M.

McDonnell, S.

Nehmetallah, G.

Olivo-Marin, J. C.

Poon, T.-C.

Rivenson, Y.

Ryle, J. P.

Sheridan, J. T.

Stern, A.

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]

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]

C. Wen, C. Xudong, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics Journal 5(2), 6900113 (2013).

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

Tajahuerce, E.

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Wang, H.

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

Wang, Z.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref] [PubMed]

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Wen, C.

C. Wen, C. Xudong, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics Journal 5(2), 6900113 (2013).

Wikner, D. A.

Williams, L.

Xudong, C.

C. Wen, C. Xudong, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics Journal 5(2), 6900113 (2013).

Zhang, X.

Zheng, K.

Zhou, C.

Appl. Opt. (6)

C. R. Math. (1)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346(9), 589–592 (2008).
[Crossref]

IEEE Photonics Journal (1)

C. Wen, C. Xudong, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics Journal 5(2), 6900113 (2013).

IEEE Signal Process. Lett. (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

IEEE Signal Process. Mag. (1)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19(1), 264–270 (2010).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (1)

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

J. Disp. Technol. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

L. H. Ma, H. Wang, Y. Li, and H. Z. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A, Pure Appl. Opt. 6(4), 396–400 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Other (1)

L. Tian, Y. Liu, and G. Barbastathis, “Improved Axial Resolution of Digital Holography Via Compressive Reconstruction [A],” in Digital Holography and Three-Dimensional Imaging, Optical Society of American (OSA), DW4C.3–DW4C.3 (2012).

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of a 4f amplified in-line compressive holography (4FICH) recording system Mo - Microscopic Objective, P - Pinhole, L - collimating Lens, M - Mirror, L1 and L2 - Lenses with focal length f1 and f2 respectively.
Fig. 2
Fig. 2 The sampling pattern in the frequency domain with sampling ratio of 25%.
Fig. 3
Fig. 3 A flow chart for illustrating the whole processes of 4FICH.
Fig. 4
Fig. 4 Simulated experimental setup.
Fig. 5
Fig. 5 Simulated object and its image: (a) two slices with line width δ l d ' = 13.95 μ m placed at z 1 ' = 402 m m , z 2 ' = 402.02 μ m , (b) the images of two slices with line width δ l d = 41.85 μ m imaged at z 1 = 2 m m , z 2 = 2.18 m m from the detector respectively.
Fig. 6
Fig. 6 The reconstruction results of different holograms by different reconstruction techniques (a) BP-H. (b) CH-H. (c) BP-4FH and (d) CH-4FH.
Fig. 7
Fig. 7 Experimental setup of recording 4f amplifying in-line hologram: ① laser light source, ② microscopic objective, ③ pin hole, ④ collimating lens, ⑤ mirror, ⑦ tested object, ⑧⑨ two lenses with focal length f1 and f2, ⑩ CCD.
Fig. 8
Fig. 8 The two tested objects: (a) Object with two no-overlapping patterns, (b) Object with two overlapping patterns.
Fig. 9
Fig. 9 Recorded in-line holograms of the objects: (a) Hologram of two no-overlapping patterns without 4f amplification. (b) Hologram of two no-overlapping patterns with 4f amplification. (c) Hologram of two overlapping patterns without 4f amplification. (d) Hologram of two overlapping patterns with 4f amplification.
Fig. 10
Fig. 10 Reconstruction results of the holograms in Figs. 9 (a) and 9(b): (a) Reconstruction by CH of the in-line hologram with normal system at two positions. (b) Reconstruction by BP of the in-line hologram with 4f amplification at two positions. (c) Reconstruction by CH of the in-line hologram with 4f amplification at two positions. (d) Highlighted cross line of Figs. 10(a)-10(c) at z 1 = 428.62 m m . (e) Highlighted cross line of Figs. 10(a)-10(c) at z 2 = 431.80 m m .
Fig. 11
Fig. 11 Reconstruction results of the holograms shown in Figs. 9(c) and 9(d): (a) Reconstruction results by CH of the in-line hologram with normal system at two positions. (b) Reconstruction results by BP of the in-line hologram with 4f amplifying system at two positions. (c) Reconstruction results by CH of the in-line hologram with 4f amplifying system at two positions. (d) Corresponding highlighted cross line of Figs. 11(a)-11(c) at z 1 = 428.62 m m . (e) Corresponding highlighted cross line of Figs. 11(a)-11(c) at z 2 = 431.80 m m .

Tables (1)

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Table1 ImQs of the Reconstruction Results

Equations (16)

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( 1 δ s ) x 2 2 Hψx 2 2 ( 1+ δ s ) x 2 2
g=Hf
x ^ =argmin x 1 , s.t.  g=Hf=Hψx
β x,y = f 2 f 1 , β z = ( β x,y ) 2 = ( f 2 f 1 ) 2
Δ x = f 2 f 1 Δ x ' , Δ y = f 2 f 1 Δ y ' , Δ z = β z Δ z ' = ( f 2 f 1 ) 2 Δ z '
I H ( x,y )= | R( x,y )+E( x,y ) | 2 =1+ | E( x,y ) | 2 +2Re( E( x,y ) )
I H ( x,y ) ¯ =2Re[ E( x,y ) ]+n( x,y )
E( x,y )=Img( x ' , y ' , z ' )h( x ' , y ' , z ' )= F x,y -1 { z ' F k x , k y [ Img( x ' , y ' ; z ' )h( x ' , y ' ; z ' ) ] } = F x,y -1 { z ' { F k x , k y [ Img( x ' , y ' ; z ' ) ] H( k x , k y ; z ' ) } }
E( k Δ x ,l Δ y )= F k,l 1 { q N Z F k x , k y [ Img( m Δ x ,n Δ y ;q Δ z ) ]H( k x , k y ;q Δ z ) } = F k,l 1 { q N Z F k x , k y [ Img( m Δ x ,n Δ y ;q Δ z ) ]exp(jq Δ z k 2 k x 2 k y 2 ) }
Y= T 2D QBX=HX
p( x,y )=exp( ( x 2 + y 2 ) α F 2 / β F 2 )
f ^ =argmin f TV , s.t. g=Hf
z 1 = z 1 ' 2 ( f 1 + f 2 ) , z 2 = z 1 + β z ( z 2 ' z 1 ' )
δ l d = β x , y δ l d '
ImQ= 4 σ ϕω ϕ ¯ ω ¯ ( σ ϕ 2 + σ ω 2 )[ ( ϕ ¯ ) 2 + ( ω ¯ ) 2 ]
M= Δ z 4f / Δ z inline = ( z 1 / β x,y z 1 ' ) 2

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