Abstract

We propose the use of compressive holography for two–dimensional (2D) subpixel motion localization. Our approach is based on computational implementation of edge–extraction using a Fourier–plane spiral phase mask, followed by compressive reconstruction of the edge of the object. Using this technique and relatively low–cost computer and piezo motion stage to establish ground truth for the motion, we demonstrated localization within 1/30th of a camera pixel in each linear dimension.

© 2014 Optical Society of America

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  1. B. K. Horn, B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
    [CrossRef]
  2. T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
    [CrossRef]
  3. S. Zhu, K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Trans. Image Processing 9, 287–290 (2000).
    [CrossRef]
  4. Y. Liu, L. Tian, J. W. Lee, H. Y. H. Huang, M. S. Triantafyllou, G. Barbastathis, “Scanning-free compressive holography for object localization with subpixel accuracy,” Opt. Lett. 37, 3357–3359 (2012).
    [CrossRef]
  5. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [CrossRef] [PubMed]
  6. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  7. E. Candès, J. Romberg, T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
    [CrossRef]
  8. E. J. Candès, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67(6), 906–956 (2014).
    [CrossRef]
  9. G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.
  10. A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef] [PubMed]
  11. G. Situ, G. Pedrini, W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26, 1788–1797 (2009).
    [CrossRef]
  12. T. J. McIntyre, C. Maurer, S. Bernet, M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009).
    [CrossRef] [PubMed]
  13. S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
    [CrossRef]
  14. K. G. Larkin, D. J. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  15. Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.
  16. P. Bouchal, Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light,” Opt. Lett. 37, 2949–2951 (2012).
    [CrossRef] [PubMed]
  17. A. Sakdinawat, Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett. 32, 2635–2637 (2007).
    [CrossRef] [PubMed]
  18. W. M. Lee, X.-C. Yuan, W. C. Cheong, “Optical vortex beam shaping by use of highly efficientirregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004).
    [CrossRef] [PubMed]
  19. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
  20. J. M. Bioucas-Dias, 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]
  21. E. Jones, “Improved digital image correlation (DIC),” http://www.mathworks.com/matlabcentral/fileexchange/43073-improved-digital-image-correlation-dic (2013).
  22. H. Beem, J. Dahl, M. Triantafyllou, “Harbor seal vibrissa morphology reduces vortex-induced vibrations,” in Proceedings of the 64th Annual Meeting of the DFD(2011), Vol. 56.

2014 (1)

E. J. Candès, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67(6), 906–956 (2014).
[CrossRef]

2012 (2)

2009 (3)

2007 (3)

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

A. Sakdinawat, Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett. 32, 2635–2637 (2007).
[CrossRef] [PubMed]

J. M. Bioucas-Dias, 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 (2)

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

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

2005 (1)

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

2004 (1)

2001 (1)

2000 (1)

S. Zhu, K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Trans. Image Processing 9, 287–290 (2000).
[CrossRef]

1981 (1)

B. K. Horn, B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[CrossRef]

Barbastathis, G.

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

Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.

Beem, H.

H. Beem, J. Dahl, M. Triantafyllou, “Harbor seal vibrissa morphology reduces vortex-induced vibrations,” in Proceedings of the 64th Annual Meeting of the DFD(2011), Vol. 56.

Bernet, S.

T. J. McIntyre, C. Maurer, S. Bernet, M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009).
[CrossRef] [PubMed]

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Bhaskar, B.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias, 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]

Bone, D. J.

Bouchal, P.

Bouchal, Z.

Brady, D. J.

Brox, T.

T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
[CrossRef]

Bruhn, A.

T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
[CrossRef]

Candès, E.

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

Candès, E. J.

E. J. Candès, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67(6), 906–956 (2014).
[CrossRef]

Cheong, W. C.

Choi, K.

Dahl, J.

H. Beem, J. Dahl, M. Triantafyllou, “Harbor seal vibrissa morphology reduces vortex-induced vibrations,” in Proceedings of the 64th Annual Meeting of the DFD(2011), Vol. 56.

Donoho, D. L.

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

Fernandez-Granda, C.

E. J. Candès, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67(6), 906–956 (2014).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias, 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]

Furhapter, S.

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Fürhapter, S.

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

Goodman, J. W.

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

Horisaki, R.

Horn, B. K.

B. K. Horn, B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[CrossRef]

Huang, H. Y. H.

Huang, Y.

Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.

Jesacher, A.

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Jones, E.

E. Jones, “Improved digital image correlation (DIC),” http://www.mathworks.com/matlabcentral/fileexchange/43073-improved-digital-image-correlation-dic (2013).

Larkin, K. G.

Lee, J. W.

Lee, W. M.

Lim, S.

Liu, Y.

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

A. Sakdinawat, Y. Liu, “Soft-x-ray microscopy using spiral zone plates,” Opt. Lett. 32, 2635–2637 (2007).
[CrossRef] [PubMed]

Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.

Ma, K.-K.

S. Zhu, K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Trans. Image Processing 9, 287–290 (2000).
[CrossRef]

Marks, D. L.

Maurer, C.

T. J. McIntyre, C. Maurer, S. Bernet, M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009).
[CrossRef] [PubMed]

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

McIntyre, T. J.

Oldfield, M. A.

Osten, W.

Papenberg, N.

T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
[CrossRef]

Pedrini, G.

Recht, B.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.

Ritsch-Marte, M.

T. J. McIntyre, C. Maurer, S. Bernet, M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009).
[CrossRef] [PubMed]

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Romberg, J.

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

Sakdinawat, A.

Schunck, B. G.

B. K. Horn, B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[CrossRef]

Shah, P.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.

Situ, G.

Tang, G.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.

Tao, T.

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

Tian, L.

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

Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.

Triantafyllou, M.

H. Beem, J. Dahl, M. Triantafyllou, “Harbor seal vibrissa morphology reduces vortex-induced vibrations,” in Proceedings of the 64th Annual Meeting of the DFD(2011), Vol. 56.

Triantafyllou, M. S.

Weickert, J.

T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
[CrossRef]

Yuan, X.-C.

Zhu, S.

S. Zhu, K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Trans. Image Processing 9, 287–290 (2000).
[CrossRef]

Adv. Imaging Electron Phys. (1)

S. Fürhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte, “Spiral phase microscopy,” Adv. Imaging Electron Phys. 146, 1–56 (2007).
[CrossRef]

Artificial Intelligence (1)

B. K. Horn, B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[CrossRef]

Commun. Pure Appl. Math. (2)

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

E. J. Candès, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67(6), 906–956 (2014).
[CrossRef]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias, 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. Image Processing (1)

S. Zhu, K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Trans. Image Processing 9, 287–290 (2000).
[CrossRef]

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Other (6)

E. Jones, “Improved digital image correlation (DIC),” http://www.mathworks.com/matlabcentral/fileexchange/43073-improved-digital-image-correlation-dic (2013).

H. Beem, J. Dahl, M. Triantafyllou, “Harbor seal vibrissa morphology reduces vortex-induced vibrations,” in Proceedings of the 64th Annual Meeting of the DFD(2011), Vol. 56.

T. Brox, A. Bruhn, N. Papenberg, J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 25–36.
[CrossRef]

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in 50th Annual Allerton Conference on Communication, Control, and Computing (2012), pp. 778–785.

Y. Liu, L. Tian, Y. Huang, G. Barbastathis, “Two dimensional sub-pixel movement detection using spiral phase filtering and compressive holography,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2012), p. DW4C.4.

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

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Figures (7)

Fig. 1
Fig. 1

Phase distribution of spiral phase mask.

Fig. 2
Fig. 2

In–line DH geometry. The object was moved along x and y directions by a piezo-driven motion stage.

Fig. 3
Fig. 3

Subpixel notation. In this example, each camera pixel is divided into 4 × 4 pixels. (nx, ny) denotes the camera pixel index and (mx, my) represents the pixel index after subpixel division.

Fig. 4
Fig. 4

(a) Real part of effective transfer function for edge-extraction holography; (b) Real part of transfer function for traditional in–line DH.

Fig. 5
Fig. 5

Summary of the computational reconstruction procedure outlined in Section 2, together with representative intermediate reconstruction results from an actual experiment with a star object.

Fig. 6
Fig. 6

(a) Microscopic image of a star shape binary amplitude object; (b) Hologram of the star object; (c) Compressive reconstruction of star hologram; (d) Traditional back–propagation of star hologram; (e) Image of a ring shape amplitude object; (f) Hologram of the ring; (g) Compressive reconstruction of the ring hologram; (h) Traditional back–propagation of the ring hologram; (i) “M” shape phase object; (j) Hologram of the “M”; (k) Compressive reconstruction of “M” hologram; (l) Traditional back–propagation of “M” hologram.

Fig. 7
Fig. 7

(a) Star trajectory characterization from compressive localization algorithm; (b) The histogram of estimated step size distribution using compressive reconstruction; (c) Star trajectory characterization using standard DIC approach; (d) The histogram of the estimated step size using standard DIC approach. In both (a) and (c), red dots indicate the localization results from the corresponding hologram using either method, and the blue “Z” pattern indicates the ground truth of the motion established using the piezo stage.

Equations (5)

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S ( ϕ ) = exp ( i ϕ ) ,
I ( n x Δ , n y Δ ; z c ) = m x = ( n x 1 ) G + 1 n x G m y = ( n y 1 ) G + 1 n y G | A r + a ( m x d , m y d ; z c ) | 2 = ( G ) [ | A r + a ( m x d , m y d ; z c ) | 2 q G ] = ( G ) { [ | A r 2 + a ( m x d , m y d ; z c ) | 2 + A r a * ( m x d , m y d ; z c ) + A r * a ( m x d , m y d ; z c ) ] q G } ,
a ( m x d , m y d ; z c ) = a ( m x d , m y d ; 0 ) h ( m x d , m y d ; z c ) ,
I ( n x Δ , n y Δ ; z c ) = ( G ) [ a ( m x d , m y d ; 0 ) h ( m x d , m y d ; z c ) q G ] + e ,
( G ) S I = Q G H a edge + S E ,

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