Abstract

We define a nonlinear filtering based on correlations on unit spheres to obtain both rotation- and scale-invariant three-dimensional (3D) object detection. Tridimensionality is expressed in terms of range images. The phase Fourier transform (PhFT) of a range image provides information about the orientations of the 3D object surfaces. When the object is sequentially rotated, the amplitudes of the different PhFTs form a unit radius sphere. On the other hand, a scale change is equivalent to a multiplication of the amplitude of the PhFT by a constant factor. The effect of both rotation and scale changes for 3D objects means a change in the intensity of the unit radius sphere. We define a 3D filtering based on nonlinear operations between spherical correlations to achieve both scale- and rotation-invariant 3D object recognition.

© 2008 Optical Society of America

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References

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    [CrossRef]

2007

2006

2005

2003

J. J. Esteve-Taboada and J. García, "Detection and orientation evaluation for three-dimensional objects," Opt. Commun. 217, 123-131 (2003).
[CrossRef]

J. Garcia, J. J. Vallés, and C. Ferreira, "Detection of three-dimensional objects under arbitrary rotations based on range images," Opt. Express 11, 3352-3358 (2003).
[CrossRef] [PubMed]

2002

2001

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

1999

1998

E. Paquet, P. Garcia-Martinez, and J. Garcia, "Tridimensional invariant correlation based on phase-coded and sine-coded range images," J. Opt. 29, 35-39 (1998).
[CrossRef]

J. Rosen, "Three-dimensional electro-optical correlation," J. Opt. Soc. Am. A 15, 430-436 (1998).
[CrossRef]

1997

S. Chang, M. Rioux, and J. Domey, "Face recognition with range images and intensity images," Opt. Eng. 36, 1106-1112 (1997).
[CrossRef]

1995

E. Paquet, M. Rioux, and H. H. Arsenault, "Invariant pattern recognition for range images using the phase Fourier transform and a neural network," Opt. Eng. 34, 1178-1183 (1995).
[CrossRef]

1994

J. R. Driscoll and D. M. Healy, "Computing Fourier transforms and convolutions on the 2-sphere," Adv. Appl. Math 15, 202-250 (1994).
[CrossRef]

1984

1983

Adv. Appl. Math

J. R. Driscoll and D. M. Healy, "Computing Fourier transforms and convolutions on the 2-sphere," Adv. Appl. Math 15, 202-250 (1994).
[CrossRef]

Appl. Opt.

J. Opt.

E. Paquet, P. Garcia-Martinez, and J. Garcia, "Tridimensional invariant correlation based on phase-coded and sine-coded range images," J. Opt. 29, 35-39 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. J. Esteve-Taboada and J. García, "Detection and orientation evaluation for three-dimensional objects," Opt. Commun. 217, 123-131 (2003).
[CrossRef]

Opt. Eng.

S. Chang, M. Rioux, and J. Domey, "Face recognition with range images and intensity images," Opt. Eng. 36, 1106-1112 (1997).
[CrossRef]

E. Paquet, M. Rioux, and H. H. Arsenault, "Invariant pattern recognition for range images using the phase Fourier transform and a neural network," Opt. Eng. 34, 1178-1183 (1995).
[CrossRef]

Opt. Express

Phys. Rev. D

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

Other

J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1985), pp. 221-223.

B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications (Marcel Dekker, 2002).

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Figures (9)

Fig. 1
Fig. 1

(a) Definition of angular coordinates. (b) Example of a nonrotated and nonscaled range image. (c) Amplitude of the PhFT of the range image in ( θ , φ ) coordinates.

Fig. 2
Fig. 2

(a) Planar 3DOOM showing the position of Fig. 1(c) in a white square. (b) Full 3DOOM on the unit sphere for all possible orientations of Fig. 1(b).

Fig. 3
Fig. 3

(a) Range reference image at θ = 0 ° and φ = 0 ° . (b) Amplitude of the PhFT distribution of Fig. 3(a) drawn on the unit sphere. (c) Scaled range image (scale factor 0.6) at θ = 0 ° and φ = 0 ° . (d) Amplitude of the PhFT distribution of (c) drawn on the unit sphere.

Fig. 4
Fig. 4

Definition of Euler angles.

Fig. 5
Fig. 5

(a) Range image of a 3D object rotated 0° around the z axis. (b) Amplitude of the PhFT obtained for the range image shown in (a). (c) Range image of the 3D object rotated 17° around the z axis. (d) Amplitude of the PhFT obtained for the range image shown in (c).

Fig. 6
Fig. 6

Polar detection diagram showing LACIF output thresholded to a value of 0.7 for the reference range object. The radial coordinate is the scale factor [0.6–1.2] and the angular coordinate is φ varying between [ 0 2 π ] . The scale factor is changed every 0.05.

Fig. 7
Fig. 7

(a) Range image of another object (woman) rotated α = 0 ° and β = 0 ° . (b) Correlation output for different scale factors and a rotation of α = 0 ° and β = 0 ° for the two images.

Fig. 8
Fig. 8

(a) Range image of the 3D object rotated with α = 90 ° and β = 30 ° . (b) Amplitude of the PhFT of (a) drawn on the unit sphere. (c) Correlation output for that fixed orientation and different scale values.

Fig. 9
Fig. 9

(a) Range image of the 3D object rotated with α = 120 ° and β = 30 ° . (b) Amplitude of the PhFT of (a) on the unit sphere. (c) Correlation output for that fixed orientation and different scale values.

Equations (8)

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P ( y , z ) = exp [ i k x ( y , z ) ] ,
PhFT ( θ , φ ) = F 2 D { exp [ i k x ( y , z ) ] } ,
x ( y , z ) = x ( y m , z m ) ,
PhFT x ( θ , φ ) m 2 PhFT x ( θ , φ ) .
C f g ( α , β , γ ) S O ( 3 ) = f ( θ , φ ) * S O ( 3 ) g ( θ , φ ) j , m , m d m , m j ( β ) F j , m G j , m × exp ( i m γ ) exp ( i m α ) ,
F j , m = f ( θ , φ ) Y l m * ( θ , φ ) sin ( θ ) d θ d φ ,
g ( θ , φ ) = a f ( θ , φ ) + b w ( θ , φ ) ,
L g f ( α , β , γ ) = [ g ( θ , φ ) * S O ( 3 ) f o ( θ , φ ) ] 2 N [ g ( θ , φ ) * S O ( 3 ) w ( θ , φ ) ] [ g ( θ , φ ) * S O ( 3 ) w ( θ , φ ) ] 2 ,

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