Abstract

In this paper a unique map or signature of three dimensional objects is defined. The map is obtained locally, for every possible rotation of the object, by the Fourier transform of the phase-encoded range-image at each specific rotation. From these local maps, a global map of orientations is built that contains the information about the surface normals of the object. The map is defined on a unit radius sphere and permits, by correlation techniques, the detection and orientation evaluation of three dimensional objects with three axis translation invariance from a single range image.

©2003 Optical Society of America

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References

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  1. R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding 81 2 (2001), pp. 166–210.
    [Crossref]
  2. M. Rioux, “Laser range finder based on synchronized scanners,” Appl. Opt. 23, 3837–3844 (1984).
    [Crossref] [PubMed]
  3. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [Crossref] [PubMed]
  4. J. Rosen, “Three-dimensional electro-optical correlation,” J. Opt. Soc. Am. A 15, 430–436 (1998).
    [Crossref]
  5. T. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
    [Crossref]
  6. E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
    [Crossref]
  7. E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
    [Crossref]
  8. P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
    [Crossref]
  9. DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing 21 (7): 637–650 (2003)
    [Crossref]
  10. M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
    [Crossref]
  11. E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
    [Crossref]
  12. J. J. Esteve-Taboada, D. Mas, and J. García, “Three-dimensional object recognition by Fourier transform profilometry,” App. Opt. 38, 4760–4765 (1999).
    [Crossref]
  13. JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
    [Crossref]
  14. Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
    [Crossref]
  15. S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
    [Crossref]
  16. LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
    [Crossref]
  17. J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. 217, 123–131 (2002).
    [Crossref]
  18. http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html
  19. B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D 63, 123002 (2001).
    [Crossref]
  20. J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. 15, 202–250 (1994).
    [Crossref]
  21. J.J. Sakurai,Modern Quantum Mechanics (Adisson-Wesley, New York, 1985), pp. 221–223

2003 (2)

DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing 21 (7): 637–650 (2003)
[Crossref]

S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
[Crossref]

2002 (1)

J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. 217, 123–131 (2002).
[Crossref]

2001 (4)

B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D 63, 123002 (2001).
[Crossref]

P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
[Crossref]

E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
[Crossref]

R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding 81 2 (2001), pp. 166–210.
[Crossref]

2000 (1)

JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[Crossref]

1999 (2)

T. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
[Crossref]

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

1998 (1)

1997 (1)

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

1995 (1)

E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
[Crossref]

1994 (2)

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. 15, 202–250 (1994).
[Crossref]

E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
[Crossref]

1987 (1)

M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
[Crossref]

1984 (1)

1983 (1)

1982 (1)

Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[Crossref]

Arsenault, H H

E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
[Crossref]

Arsenault, H. H.

E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
[Crossref]

Arsenault, HH

Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[Crossref]

Boulanger, P.

M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
[Crossref]

Campbell, R.

R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding 81 2 (2001), pp. 166–210.
[Crossref]

Chang, S

S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
[Crossref]

Chatterjee, JP

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

Chavel, P.

P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
[Crossref]

Driscoll, J. R.

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. 15, 202–250 (1994).
[Crossref]

Esteve-Taboada, J. J.

J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. 217, 123–131 (2002).
[Crossref]

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

Esteve-Taboada, JJ

JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[Crossref]

Ferreira, C

JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[Crossref]

Flynn, P.

R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding 81 2 (2001), pp. 166–210.
[Crossref]

Garcia, J

JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[Crossref]

García, J.

J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. 217, 123–131 (2002).
[Crossref]

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

Górski, K. M.

B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D 63, 123002 (2001).
[Crossref]

Grover, CP

S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
[Crossref]

Hassebrook, LG

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

Healy, D. M.

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. 15, 202–250 (1994).
[Crossref]

Hebert, M

DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing 21 (7): 637–650 (2003)
[Crossref]

Hsu, Y.

Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[Crossref]

Huber, DF

DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing 21 (7): 637–650 (2003)
[Crossref]

Javidi, B.

E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
[Crossref]

Kasvand, T.

M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
[Crossref]

Kim, T.

Lhamon, ME

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

Mas, D.

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

Matoba, O.

E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
[Crossref]

Mutoh, K.

Paquet, E

E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
[Crossref]

Paquet, E.

E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
[Crossref]

Parrein, P.

P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
[Crossref]

Poon, T.

Rioux, M

S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
[Crossref]

E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
[Crossref]

Rioux, M.

E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
[Crossref]

M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
[Crossref]

M. Rioux, “Laser range finder based on synchronized scanners,” Appl. Opt. 23, 3837–3844 (1984).
[Crossref] [PubMed]

Rosen, J.

Sakurai, J.J.

J.J. Sakurai,Modern Quantum Mechanics (Adisson-Wesley, New York, 1985), pp. 221–223

Taboury, J.

P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
[Crossref]

Tajahuerce, E.

E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
[Crossref]

Takeda, M.

Wandelt, B. D.

B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D 63, 123002 (2001).
[Crossref]

Wang, M

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

Adv. In App. Math. (1)

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. 15, 202–250 (1994).
[Crossref]

App. Opt. (5)

E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. 40, 3877–3886 (2001).
[Crossref]

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

JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[Crossref]

Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[Crossref]

M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. 26, 287–292 (1987).
[Crossref]

Appl. Opt. (3)

Computer Vision and Image Understanding (1)

R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding 81 2 (2001), pp. 166–210.
[Crossref]

Image And Vision Computing (1)

DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing 21 (7): 637–650 (2003)
[Crossref]

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

Opt. Com. (1)

J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. 217, 123–131 (2002).
[Crossref]

Opt. Commun. (2)

S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. 222, 143–153 (2003).
[Crossref]

P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. 195 (5–6), 393–397 (2001).
[Crossref]

Opt. Eng. (2)

LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[Crossref]

E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. 32, 2173–2180 (1994).
[Crossref]

Phys. Rev. D (1)

B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D 63, 123002 (2001).
[Crossref]

Pure Appl. Opt. (1)

E Paquet, H H Arsenault, and M Rioux, “Recognition of faces from range images by means of the phase Fourier transform,” Pure Appl. Opt. 4, 709–721 (1995).
[Crossref]

Other (2)

http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html

J.J. Sakurai,Modern Quantum Mechanics (Adisson-Wesley, New York, 1985), pp. 221–223

Supplementary Material (1)

» Media 1: AVI (851 KB)     

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Figures (4)

Fig. 1.
Fig. 1. (a) Definition of angular coordinates. (b) Range image of a pyramid shaped object. (c) PhFT intensity.

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Fig. 2.
Fig. 2. (a) Definition of Euler angles. (b) Range image of an object. (c) 3DOOM on θ-ϕ Coordinates.

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Fig. 3.
Fig. 3. (a) PhFT of the target shown in figure 2 (b). (b) The same PhFT depicted on the unit sphere. (c) 3DOOM on the unit sphere. The correlation consist on the matching of distributions on figures (b) and (c)

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Fig. 4.
Fig. 4. (851KB) (a) Range image of the 3-D object rotated with α=-120°, β=53.4°. (b) PhFT of the range image given in (a). (c) Output of the correlation. (d) Range image of another object. (e) PhFT of the range image given in (d). (e) Output of the correlation.

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Equations (9)

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

P ( x , y ) = exp ⁡ [ iwz ( x , y ) ] .
PhFT ( u , v ) = F 2 D { exp ⁡ [ iwz ( x , y ) ] } .
( u , v ) ≡ ( tan ⁡ ( α x ) 2 π , tan ⁡ ( α y ) 2 π ) .
( u , v ) = ( tan ⁡ ( φ ) 2 π tan ⁡ ( θ ) 2 π cos ⁡ ( φ ) ) .
R ( α , β , γ ) ≡ R Y ′ ( γ ) R X ′ ( β ) R Y ( α ) .
T ( α , β , γ ) ∈ SO ( 3 ) = f ( θ , φ ) * g ( θ , φ ) ≡ ∫ [ D ( α , β , γ ) f ] ( θ , φ ) g ( θ , φ ) sin ⁡ ( θ ) d θ d φ .
f lm = ∫ f ( θ , φ ) Y lm * ( θ , φ ) sin ⁡ ( θ ) d θ d φ .
T ( α , β , γ ) = ∑ m , m ′ , m ″ T mm ′ m ″ e i m α + i m ′ β + m ″ γ .
T mm ′ m ″ = ∑ l g lm d mm ′ l ( π 2 ) d m ′ m ″ l ( π 2 ) f lm ″ * .

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