Abstract

In this work compressive holography (CH) with multiple projection tomography is applied to solve the inverse ill-posed problem of reconstruction of three-dimensional (3D) objects with high axial accuracy. To visualize the 3D shape, we propose digital tomographic CH, where projections from more than one direction, as in tomographic imaging, can be employed, so that a 3D shape with improved axial resolution can be reconstructed. Also, we propose possible schemes for shadow elimination when the same object is illuminated at multiple angles using a single illuminating beam and using a single CCD. Finally, we adapt CH designed for a Gabor-type setup to a reflective geometry and apply the technique to reflective objects.

© 2013 Optical Society of America

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    [CrossRef]
  4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
    [CrossRef]
  5. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  6. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [CrossRef]
  7. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978).
    [CrossRef]
  8. 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]
  9. J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).
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    [CrossRef]
  11. T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).
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  17. G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010).
    [CrossRef]
  18. L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.
  19. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

2013 (1)

2011 (1)

2010 (3)

2009 (1)

2007 (2)

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

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]

2006 (4)

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]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

1978 (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978).
[CrossRef]

1949 (2)

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949).
[CrossRef]

Banerjee, P. P.

G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010).
[CrossRef]

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

Baraniuk, R.

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

Barbastathis, G.

L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.

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]

Brady, D. J.

Candes, E. J.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (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]

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.

Denis, L.

Devaney, A. J.

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978).
[CrossRef]

Donoho, D. L.

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

Fournel, T.

Fournier, C.

Golay, M.

Horisaki, R.

Javidi, B.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

Lee, J.

L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.

Lim, S.

Marks, D. L.

Nehmetallah, G.

G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010).
[CrossRef]

Poon, T.-C.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

Rivenson, Y.

Romberg, J.

J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

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]

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]

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

Shannon, C. E.

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Stern, A.

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]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (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]

Tian, L.

L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.

Wakin, M.

J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

Appl. Opt. (1)

Commun. Pure Appl. Math. (2)

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]

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 Signal Process. Mag. (1)

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

IEEE Trans. Image Process. (1)

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]

IEEE Trans. Inf. Theory (2)

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

J. Display Technol. (1)

J. Math. Phys. Sci. (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (1)

Proc. IRE (1)

C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Proc. SPIE (1)

G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010).
[CrossRef]

Other (4)

L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of the different matrices used for CS.

Fig. 2.
Fig. 2.

Typical Gabor-type setup using the transmissive geometry of an object surrounded by small objects. FPA, focal plane array.

Fig. 3.
Fig. 3.

(a) Experimental setup of typical SHOT-MT recording scheme and (b) schematic showing the principle of SHOT-MT reconstruction.

Fig. 4.
Fig. 4.

Typical DiTCH setup, transmissive geometry.

Fig. 5.
Fig. 5.

(a) Single-beam hologram of two bubbles. The top two are for the first bubble and the bottom two for the second bubble. The holograms to the left are from illumination of the bubbles along the x axis (90° with respect to the normal to the CCD), while the holograms to the right are from illumination of the bubbles along z axis (0° with respect to the normal to the CCD). The left holograms look larger because the objects are farther from the CCD, while the right holograms look smaller because the objects are nearer to the CCD. (b) 3D reconstruction, λ=543nm, 6.7 μm pixels, (c) reconstructed hologram at 61.8 cm, (d) y-z projection of the 3D view in (b), (e) reconstructed hologram at 20.6 cm, and (f) x-y projection of the 3D view in (b).

Fig. 6.
Fig. 6.

(a) Schematic of lab setup, (b) ball-point pen spring 450 μm thick, (c) three representative holograms at angles 0°, 90°, and 180°, out of a total of 13 angles recorded, 0°–180° in 15° increments, (d) TwIST reconstruction from 90° to 180° recordings at 33 cm (distance of object from CCD during recording); tomographic reconstruction using (e) 7 angles, 0°–180° in 30° increments, and (f) 13 angles, 0°–180° in 15° increments.

Fig. 7.
Fig. 7.

Geometry of multiangle illumination of an object with a circular cross-section, where αR is the illumination angle (angle of object rotation), α is the angle between adjacent illumination beams, α=180°αR, R is the cross-section radius; C is the chord length, C=2Rsin(αR/2), S is the sag height of the arc, S=RR2(C/2)2, and h is the excess height (tomographic measurement error).

Fig. 8.
Fig. 8.

Schematic of the shadow experimental setup.

Fig. 9.
Fig. 9.

(a) Shadow recording, (b) object recording from two angles, (c) plane wave reconstruction, (d) reconstruction using shadow illumination profile, and (e) reconstruction after direct subtraction.

Fig. 10.
Fig. 10.

(a) Experimental setup with diverging lens to provide demagnification and (b) CH reconstruction of a dime using the TWIST algorithm in the reflective mode. Feature size in the reconstructed hologram is 28.6 μm for a CCD camera with pixel size 6.7 μm, λ=633nm, d=31cm, and demagnification M=0.315.

Equations (12)

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

Akjcj=bk,Akj=ϕkiψij,
c^=argmincc1,such thatb=Φf=ΦΨc=Ac.
I(x,y)=|ER+EO(x,y)|2=|ER|2+|EO(x,y)|2+ER*EO(x,y)+EREO*(x,y)2Re{EO(x,y)}+e(x,y),
EO(x,y;z)=O(ξ,η,ς)gPSF(xξ,yη,zς)dξdηdς,
gPSF(x,y,z)jk02πexp(jk0x2+y2+z2)x2+y2+z2,
b=2Re{b¯}=2Re{I1PIf}+e2Re{Φf}+e,
f^=argminfO(f)=argminf[12b2Re(Φf)l22+λΓ(f)]=argminf[12b2Re(Φf)l22+λΨfl1],
f^=argminfO(f)=argminf[12b2Re(Φf)l22+λΓ(f)]=argminf[12b2Re(Φf)l22+λfTV]
fTV=ln1n2|(fl)n1,n2|=ln1n2(fl,n1+1,n2fl,n1,n2)2+(fl,n1,n2+1fl,n1,n2)2,
I=1MIj
Δη=λdNΔx,
h=C2tan(α/2)S,

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