Abstract

We present an overview of three-dimensional (3D) object recognition techniques that use active sensing by interferometric imaging (digital holography) and passive sensing by integral imaging. We describe how each technique can be used to retrieve the depth information of a 3D scene and how this information can then be used for 3D object recognition. We explore various algorithms for 3D recognition such as nonlinear correlation and target distortion tolerance. We also provide a comparison of the advantages and disadvantages of the two techniques.

© 2004 Optical Society of America

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References

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  1. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications (Marcel Dekker, New York, 2002).
    [CrossRef]
  4. Ph. Réfrégier, F. Goudail, “A decision theoretical approach to nonlinear joint transform correlation,” J. Opt. Soc. Am. A 15, 61–67 (1998).
    [CrossRef]
  5. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  6. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  7. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
    [CrossRef] [PubMed]
  8. B. Javidi, J. Wang, “Optimum distortion invariant filters for detecting a noisy distorted target in background noise,” J. Opt. Soc. Am. A 12, 2604–2614 (1995).
    [CrossRef]
  9. A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. 65 of SPIE Critical Reviews Series (SPIE, Bellingham, Wash., 1996), 240–260.
  10. B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Orlando, Fla., 1994).
  11. A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
    [CrossRef]
  12. J. Rosen, “Three-dimensional joint transform correlator,” Appl. Opt. 37, 7538–7544 (1998).
    [CrossRef]
  13. M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “System level evaluation of LADAR ATR using correlation filters,” in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE4050, 69–77 (2000).
    [CrossRef]
  14. J. J. Esteve-Taboada, D. Mas, J. García, “Three-dimensional object recognition by Fourier transform profilometry,” Appl. Opt. 38, 4760–4765 (1999).
    [CrossRef]
  15. M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “LADAR automatic target recognition using correlation filters,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 388–396 (1999).
    [CrossRef]
  16. J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
    [CrossRef]
  17. F. Goudail, Ph. Réfrégier, “Target segmentation in active polarimetric images using statistical active countour,” Appl. Opt. 41, 874–883 (2002).
    [CrossRef] [PubMed]
  18. B. Javidi, E. Tajahuerce, “Three-dimensional object recognition using digital holography,” Opt. Lett. 25, 610–612 (2000).
    [CrossRef]
  19. E. Tajahuerce, O. Matoba, B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40, 3877–3886 (2001).
    [CrossRef]
  20. Y. Frauel, E. Tajahuerce, M. A. Castro, B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. 40, 3887–3893 (2001).
    [CrossRef]
  21. Y. Frauel, B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478–1480 (2001).
    [CrossRef]
  22. O. Matoba, E. Tajahuerce, B. Javidi, “Real-time three-dimensional object recognition with multiple perspectives imaging,” Appl. Opt. 40, 3318–3325 (2001).
    [CrossRef]
  23. Y. Frauel, B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488–5496 (2002).
    [CrossRef] [PubMed]
  24. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [CrossRef]
  25. U. Schnars, W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  26. G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).
  27. H. J. Caulfield, ed., Handbook of Optical Holography (Academic, London, 1979).
  28. I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef] [PubMed]
  29. H. Ritter, T. Martinez, K. Schulten, Neural Computation and Self-Organizing Maps (Addison-Wesley, Boston, Mass., 1992).
  30. H. Arimoto, B. Javidi, “Integral three-dimensional imaging with digital reconstruction,” Opt. Lett. 26, 157–159 (2001).
    [CrossRef]
  31. T. Okoshi, Three Dimensional Imaging Techniques (Academic, New York, 1971).
  32. F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
    [CrossRef]
  33. J.-S. Jang, B. Javidi, “Three-dimensional synthetic aperture integral imaging,” Opt. Lett. 27, 1144–1146 (2002).
    [CrossRef]
  34. J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
    [CrossRef]

2002

2001

2000

B. Javidi, E. Tajahuerce, “Three-dimensional object recognition using digital holography,” Opt. Lett. 25, 610–612 (2000).
[CrossRef]

J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
[CrossRef]

1999

J. J. Esteve-Taboada, D. Mas, J. García, “Three-dimensional object recognition by Fourier transform profilometry,” Appl. Opt. 38, 4760–4765 (1999).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

1998

1997

A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
[CrossRef]

I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

1996

1995

1994

1989

1987

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

1984

1908

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Arai, J.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Arimoto, H.

Casasent, D.

Castro, M. A.

Denkewalter, R.

A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
[CrossRef]

Esteve-Taboada, J. J.

Frauel, Y.

García, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Goudail, F.

Gualdrón, O.

J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
[CrossRef]

Guerrero-Bermúdez, J.

J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
[CrossRef]

Horner, J. L.

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Orlando, Fla., 1994).

Hoshino, H.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Jang, J.-S.

Javidi, B.

J.-S. Jang, B. Javidi, “Three-dimensional synthetic aperture integral imaging,” Opt. Lett. 27, 1144–1146 (2002).
[CrossRef]

Y. Frauel, B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488–5496 (2002).
[CrossRef] [PubMed]

Y. Frauel, E. Tajahuerce, M. A. Castro, B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. 40, 3887–3893 (2001).
[CrossRef]

O. Matoba, E. Tajahuerce, B. Javidi, “Real-time three-dimensional object recognition with multiple perspectives imaging,” Appl. Opt. 40, 3318–3325 (2001).
[CrossRef]

Y. Frauel, B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478–1480 (2001).
[CrossRef]

E. Tajahuerce, O. Matoba, B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40, 3877–3886 (2001).
[CrossRef]

H. Arimoto, B. Javidi, “Integral three-dimensional imaging with digital reconstruction,” Opt. Lett. 26, 157–159 (2001).
[CrossRef]

B. Javidi, E. Tajahuerce, “Three-dimensional object recognition using digital holography,” Opt. Lett. 25, 610–612 (2000).
[CrossRef]

B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
[CrossRef] [PubMed]

B. Javidi, J. Wang, “Optimum distortion invariant filters for detecting a noisy distorted target in background noise,” J. Opt. Soc. Am. A 12, 2604–2614 (1995).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Orlando, Fla., 1994).

Jung, S.

J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
[CrossRef]

Jüptner, W. P. O.

Lee, B.

J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
[CrossRef]

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Mahalanobis, A.

A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. 65 of SPIE Critical Reviews Series (SPIE, Bellingham, Wash., 1996), 240–260.

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “LADAR automatic target recognition using correlation filters,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 388–396 (1999).
[CrossRef]

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “System level evaluation of LADAR ATR using correlation filters,” in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE4050, 69–77 (2000).
[CrossRef]

Martinez, T.

H. Ritter, T. Martinez, K. Schulten, Neural Computation and Self-Organizing Maps (Addison-Wesley, Boston, Mass., 1992).

Mas, D.

Matoba, O.

Meneses, J.

J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
[CrossRef]

Min, S.-W.

J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
[CrossRef]

Norris-Zachery, K.

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “System level evaluation of LADAR ATR using correlation filters,” in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE4050, 69–77 (2000).
[CrossRef]

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “LADAR automatic target recognition using correlation filters,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 388–396 (1999).
[CrossRef]

Okano, F.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Okoshi, T.

T. Okoshi, Three Dimensional Imaging Techniques (Academic, New York, 1971).

Onural, L.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Painchaud, D.

Park, J.-H.

J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
[CrossRef]

Perona, M. T.

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “System level evaluation of LADAR ATR using correlation filters,” in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE4050, 69–77 (2000).
[CrossRef]

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “LADAR automatic target recognition using correlation filters,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 388–396 (1999).
[CrossRef]

Psaltis, D.

A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
[CrossRef]

Pu, A.

A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
[CrossRef]

Réfrégier, Ph.

Ritter, H.

H. Ritter, T. Martinez, K. Schulten, Neural Computation and Self-Organizing Maps (Addison-Wesley, Boston, Mass., 1992).

Rosen, J.

Schnars, U.

Schulten, K.

H. Ritter, T. Martinez, K. Schulten, Neural Computation and Self-Organizing Maps (Addison-Wesley, Boston, Mass., 1992).

Scott, P. D.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Tajahuerce, E.

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

Wang, J.

Yamaguchi, I.

Yuyama, I.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Zhang, T.

Appl. Opt.

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
[CrossRef] [PubMed]

J. Rosen, “Three-dimensional joint transform correlator,” Appl. Opt. 37, 7538–7544 (1998).
[CrossRef]

J. J. Esteve-Taboada, D. Mas, J. García, “Three-dimensional object recognition by Fourier transform profilometry,” Appl. Opt. 38, 4760–4765 (1999).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Target segmentation in active polarimetric images using statistical active countour,” Appl. Opt. 41, 874–883 (2002).
[CrossRef] [PubMed]

E. Tajahuerce, O. Matoba, B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40, 3877–3886 (2001).
[CrossRef]

Y. Frauel, E. Tajahuerce, M. A. Castro, B. Javidi, “Distortion-tolerant 3D object recognition using digital holography,” Appl. Opt. 40, 3887–3893 (2001).
[CrossRef]

O. Matoba, E. Tajahuerce, B. Javidi, “Real-time three-dimensional object recognition with multiple perspectives imaging,” Appl. Opt. 40, 3318–3325 (2001).
[CrossRef]

Y. Frauel, B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488–5496 (2002).
[CrossRef] [PubMed]

U. Schnars, W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

C. R. Acad. Sci.

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

J. Opt. Soc. Am. A

Opt. Eng.

A. Pu, R. Denkewalter, D. Psaltis, “Real-time vehicle navigation using a holographic memory,” Opt. Eng. 36, 2737–2746 (1997).
[CrossRef]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

J. Guerrero-Bermúdez, J. Meneses, O. Gualdrón, “Object recognition using three-dimensional correlation of range images,” Opt. Eng. 39, 2828–2831 (2000).
[CrossRef]

Opt. Lett.

Other

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “LADAR automatic target recognition using correlation filters,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 388–396 (1999).
[CrossRef]

M. T. Perona, A. Mahalanobis, K. Norris-Zachery, “System level evaluation of LADAR ATR using correlation filters,” in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE4050, 69–77 (2000).
[CrossRef]

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications (Marcel Dekker, New York, 2002).
[CrossRef]

A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. 65 of SPIE Critical Reviews Series (SPIE, Bellingham, Wash., 1996), 240–260.

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Orlando, Fla., 1994).

T. Okoshi, Three Dimensional Imaging Techniques (Academic, New York, 1971).

H. Ritter, T. Martinez, K. Schulten, Neural Computation and Self-Organizing Maps (Addison-Wesley, Boston, Mass., 1992).

H. J. Caulfield, ed., Handbook of Optical Holography (Academic, London, 1979).

J.-H. Park, S.-W. Min, S. Jung, B. Lee, “New stereovision scheme using a camera and a lens array,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 73–80 (2001).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Setup for experimental digital holography: M’s, mirrors; BSs, beam splitters; SFs, spatial filters, L’s, lenses; RPs, retardation plates.

Fig. 2
Fig. 2

Reconstruction by use of windows in a digital hologram: (a) without a phase factor, (b) with a linear phase factor to recenter the reconstructed view.

Fig. 3
Fig. 3

Conventional 2D correlation between 2D intensity images of two model cars: (a) autocorrelation, (b) cross correlation.

Fig. 4
Fig. 4

(a) Autocorrelation and (b) cross correlation of the digital holograms of the two model cars used for Fig. 3.

Fig. 5
Fig. 5

Correlation peaks when images reconstructed from digital holograms of various objects are compared. Images 1–21 correspond to the reference object with different orientations. Images 22–28 are from other objects. (a) Filter made from one single view similar to Image 10. (b) Composite filter made from 9 views based on 3 different holograms.

Fig. 6
Fig. 6

Error rates for the recognition and orientation estimation of a die with 360° rotation obtained (a) by use of a bank of composite filters and (b) by use of a bank of filters and a neural network.

Fig. 7
Fig. 7

Two-layer neural network for recognition and orientation estimation of a die: b denotes bias.

Fig. 8
Fig. 8

Integral imaging experimental setup.

Fig. 9
Fig. 9

Principle of integral imaging: formation of the elemental images through the microlens array.

Fig. 10
Fig. 10

Example of the integral image of a 3D scene. The marked elemental images are those used to determine the depth information.

Fig. 11
Fig. 11

Views of the 3D objects used in the experiments: (a)–(c) planar reference objects; (d), (e) composite 3D input scenes with the reference objects at various distances; (f) map of the estimated depths for the 3D scene shown in (d).

Fig. 12
Fig. 12

Perspective views of the composite 3D scenes used in the experiments: (a) Scene 1, (b) Scene 2.

Fig. 13
Fig. 13

3D computer reconstructions of the scenes: (a) Scene 1, (b) Scene 2.

Fig. 14
Fig. 14

Parts of integral images of a die with different orientations: (a) reference object, (b) input object.

Fig. 15
Fig. 15

(a) Autocorrelation of the reference object. (b) Cross correlation between the reference and the input objects.

Fig. 16
Fig. 16

Maximum correlation values obtained in the planes at constant depths. The peaks are obtained in the planes containing the square and the circle in Scene 1.

Fig. 17
Fig. 17

Two correlation planes extracted from the 3D correlation between Scene 1 and the 3D square reference object: (a) correlation plane corresponding to Δz = -20 mm, (b) correlation plane corresponding to Δz = +10 mm.

Equations (8)

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

pφ-xz=Xp-pφd,
Xp=pφ1+dz-dz x.
Xq-Xp=q-pφ1+dz.
Cp, q, p, q=m=-44n=-44 IXp+m, Yq+nIXp+m, Yq+nm=-44n=-44 I2Xp+m, Yq+nm=-44n=-44 I2Xp+m, Yq+n1/2.
Mz=p=-33q=-23 Cp, q-1, p, q+q=-33p=-23 Cp-1, q, p, q,
x=-zd X0, y=-zd Y0.
SRI=|RI|2=|FT-1R˜Ĩ*|2,
SRIk=|R k I|2=|FT-1|R˜|k expiφR˜|Ĩ|k exp-iφĨ|2,

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