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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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)

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]

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]

Esteve-Taboada JJ, Garcia J, Ferreira C, �??Rotation-invariant optical recognition of three-dimensional objects,�?? App. Opt. 39, 5998-6005 (2000
[CrossRef]

Hsu Y., Arsenault HH, �??Optical-pattern recognition using circular harmonic expansion,�?? App. Opt. 21, 4016-4019 (1982)
[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)

Huber DF, Hebert M, �??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)

P. Parrein, J. Taboury, P. Chavel, �??Evaluation of the shape conformity using correlation of range images,�?? Opt. Commun. 195 (5-6), 393-397 (2001).
[CrossRef]

Chang S, Rioux M, Grover CP, �??Range face recognition based on the phase Fourier transform,�?? Opt. Commun. 222, 143-153 (2003)
[CrossRef]

Opt. Eng. (2)

Hassebrook LG, Lhamon ME, Wang M, Chatterjee JP, �??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)

<a href="http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html">http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html</a>

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.

Fig. 2.
Fig. 2.

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

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)

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.

Equations (9)

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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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