Abstract

We propose quantitative localization measurement of a known object with subpixel accuracy using compressive holography. We analyze the theoretical optimal solution in the compressive sampling framework and experimentally demonstrate localization accuracy of 1/45 pixel, in good agreement with the analysis.

© 2012 Optical Society of America

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References

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  1. M. J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).
    [CrossRef]
  2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
    [CrossRef]
  3. H. Beem, J. Dahl, and M. Triantafyllou, in APS Meeting Abstracts (American Physical Society, 2011), p. 27008.
  4. H. Ur and D. Gross, Comput. Vis. Graph. Image Process. 54, 181 (1992).
    [CrossRef]
  5. G. Vendroux and W. G. Knauss, Exp. Mech. 38, 86 (1998).
    [CrossRef]
  6. B. K. P. Horn and E. J. Weldon, Int. J. Comput. Vis. 2, 51 (1988).
    [CrossRef]
  7. E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
    [CrossRef]
  8. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  9. E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
    [CrossRef]
  10. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, Opt. Express 15, 18275 (2007).
    [CrossRef]
  11. L. Tian, N. Loomis, J. A. Domínguez-Caballero, and G. Barbastathis, Appl. Opt. 49, 1549 (2010).
    [CrossRef]
  12. E. Candès and Y. Plan, IEEE Trans. Inf. Theory 57, 7235 (2011).
    [CrossRef]
  13. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, Opt. Express 17, 13040 (2009).
    [CrossRef]
  14. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, Opt. Express 20, 8296 (2012).
    [CrossRef]
  15. A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, Opt. Express 18, 10510 (2010).
    [CrossRef]
  16. J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
    [CrossRef]
  17. Y. Rivenson and A. Stern, Opt. Lett. 36, 3365 (2011).
    [CrossRef]

2012 (1)

2011 (2)

E. Candès and Y. Plan, IEEE Trans. Inf. Theory 57, 7235 (2011).
[CrossRef]

Y. Rivenson and A. Stern, Opt. Lett. 36, 3365 (2011).
[CrossRef]

2010 (2)

2009 (1)

2007 (2)

2006 (3)

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
[CrossRef]

2003 (1)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

1998 (1)

G. Vendroux and W. G. Knauss, Exp. Mech. 38, 86 (1998).
[CrossRef]

1997 (1)

M. J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).
[CrossRef]

1992 (1)

H. Ur and D. Gross, Comput. Vis. Graph. Image Process. 54, 181 (1992).
[CrossRef]

1988 (1)

B. K. P. Horn and E. J. Weldon, Int. J. Comput. Vis. 2, 51 (1988).
[CrossRef]

Barbastathis, G.

Beem, H.

H. Beem, J. Dahl, and M. Triantafyllou, in APS Meeting Abstracts (American Physical Society, 2011), p. 27008.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Brady, D. J.

Candès, E.

E. Candès and Y. Plan, IEEE Trans. Inf. Theory 57, 7235 (2011).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Choi, K.

Coskun, A. F.

Dahl, J.

H. Beem, J. Dahl, and M. Triantafyllou, in APS Meeting Abstracts (American Physical Society, 2011), p. 27008.

Domínguez-Caballero, J. A.

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Grier, D. G.

Gross, D.

H. Ur and D. Gross, Comput. Vis. Graph. Image Process. 54, 181 (1992).
[CrossRef]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Horisaki, R.

Horn, B. K. P.

B. K. P. Horn and E. J. Weldon, Int. J. Comput. Vis. 2, 51 (1988).
[CrossRef]

Jacobson, K.

M. J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).
[CrossRef]

Kim, S. H.

Knauss, W. G.

G. Vendroux and W. G. Knauss, Exp. Mech. 38, 86 (1998).
[CrossRef]

Lee, J.

Lee, S. H.

Lim, S.

Loomis, N.

Marks, D. L.

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Oh, S. B.

Ozcan, A.

Plan, Y.

E. Candès and Y. Plan, IEEE Trans. Inf. Theory 57, 7235 (2011).
[CrossRef]

Rivenson, Y.

Roichman, Y.

Romberg, J.

E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Saxton, M. J.

M. J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).
[CrossRef]

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Sencan, I.

Stern, A.

Su, T.-W.

Tao, T.

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
[CrossRef]

Tian, L.

Triantafyllou, M.

H. Beem, J. Dahl, and M. Triantafyllou, in APS Meeting Abstracts (American Physical Society, 2011), p. 27008.

Ur, H.

H. Ur and D. Gross, Comput. Vis. Graph. Image Process. 54, 181 (1992).
[CrossRef]

van Blaaderen, A.

van Oostrum, P.

Vendroux, G.

G. Vendroux and W. G. Knauss, Exp. Mech. 38, 86 (1998).
[CrossRef]

Weldon, E. J.

B. K. P. Horn and E. J. Weldon, Int. J. Comput. Vis. 2, 51 (1988).
[CrossRef]

Yang, S. M.

Yi, G. R.

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Annu. Rev. Biophys. Biomol. Struct. (1)

M. J. Saxton and K. Jacobson, Annu. Rev. Biophys. Biomol. Struct. 26, 373 (1997).
[CrossRef]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

E. Candès, J. Romberg, and T. Tao, Commun. Pure Appl. Math. 59, 1207 (2006).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

H. Ur and D. Gross, Comput. Vis. Graph. Image Process. 54, 181 (1992).
[CrossRef]

Exp. Mech. (1)

G. Vendroux and W. G. Knauss, Exp. Mech. 38, 86 (1998).
[CrossRef]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

IEEE Trans. Inf. Theory (3)

E. Candès and Y. Plan, IEEE Trans. Inf. Theory 57, 7235 (2011).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Int. J. Comput. Vis. (1)

B. K. P. Horn and E. J. Weldon, Int. J. Comput. Vis. 2, 51 (1988).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Science (1)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef]

Other (1)

H. Beem, J. Dahl, and M. Triantafyllou, in APS Meeting Abstracts (American Physical Society, 2011), p. 27008.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Vector representation of the 1D object; (b) sparse derivative of the object.

Fig. 2.
Fig. 2.

In-line DH geometry. The object was moved along the y direction with a uniform step size of 267 nm (1/45 pixel) by a piezo-driven motion stage.

Fig. 3.
Fig. 3.

(a) Sample two-dimensional (2D) hologram; (b) sample row vector from the 2D hologram in (a); (c) real part of the compressive reconstruction of the 1D hologram in (b).

Fig. 4.
Fig. 4.

Compressive reconstructed positions of seven randomly chosen rows (in blue circles) and the “true” position (in red stair curve) at each step. The histogram on the bottom right combines the data of total 45 steps taken by the seven rows (45×7=315 data points total).

Equations (12)

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

x=[x0,x1,,xj,,xN1],
Iobj=[0,,0,1,,1,0,,0].
ΔIobj=[0,,0,1,0,,0,1,0,0],
I(mΔ,zD)=n|Ar+a(nd,zD)|2·p(mΔnd)=n[Ar2+|a(nd,zD)|2+Ara*(nd,zD)+Ar*a(nd,zD)]·p(mΔnd),
a(nd,zD)=g(nd)*h(nd,zD),
I(mΔ,zD)=g(nd)*h(nd,zD)*p(mΔ)+e,
iu·F·I=P·H·F·g+iu·e˜,
g^=argmingg1,such thatiu·F·I=P·H·F·g,
MCμslogN,
μ=MΔ2λzD=4NFM,
G=NM1Mexp(λzDCsΔ2).
dmin=ΔGmax=MΔexp(λzDCsΔ2).

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