## Detection of three-dimensional objects under arbitrary rotations based on range images

Optics Express, Vol. 11, Issue 25, pp. 3352-3358 (2003)

http://dx.doi.org/10.1364/OE.11.003352

Acrobat PDF (385 KB)

### 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

## 1. Introduction

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]

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]

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]

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. Fourier transforms of phase encoded range images

*z*=

*f*(

*x, y*), contains the depth information of an object from a given view line, that defines the z axis. Note that the range image is a single valued function, therefore only the part of the surface closer to the positive z axis is contained on it. The encoding of the depth information has been used in the literature to extend the possibilities of range images [7

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]

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]

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]

*w*is a constant that permits the adjustment of the phase slope of the object.

*PhFT*) is then:

*F*

_{2D}stands for two dimensional Fourier transform. Note that

*w*determines the scaling of the Fourier transform frequencies. From now, we will assume

*w*=1, that correspond to the PhFT mapping detailed in the following.

*z*axis defines the view line, the polar axis is the

*y*axis and the azimuth angle is measured with respect to the

*z*axis. A facet of the object is determined by the angles (

*α*

_{x},

*α*

_{y}), that are the angles of the

*z*axis with the projection of the normal to the planar surface. The tangents of these angles are proportional to the location of the peak in the Fourier domain:

*θ*,

*φ*) as:

*w*=1. A lower value will concentrate the information in low frequencies, while a higher one will expand the scale of the PhFT. From Eq. (4), the PhFT can be coordinate transformed to obtain a distribution

*PhFT*

_{sph}(

*θ, φ*), expressed in spherical coordinates, where the location of a peak gives directly the orientation of the corresponding facet. In general, for non-planar surfaces, the PhFT will contain the information of the orientations of the surfaces in the object. Figs. 1(b) and (c) show a sample object and the corresponding PhFT expressed in angular coordinates. It is worth noting that the angular variation for

*θ*and

*φ*has maximum span of ±45 degrees. For higher angles the energy contents of a surface will be small because the apparent size of the facet will be reduced.

*(x,y)*plane, as they will produce just a linear phase factor in the PhFT. On the other hand, from the definition of the phase encoded range image (Eq. 1), a shift along the view line (

*z*axis) will influence just as a constant phase factor. In both cases the PhFT is just altered by a phase that is irrelevant in intensity. This property makes the PhFT advantageous for 3D object matching, because the translation invariance is automatically obtained, when performing the correlation between the intensities of the PhFTs [8

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]

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]

## 3. Correlation under limited rotations

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]

*z*axis). This result is extended in [15

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

*(x,y)*plane].

*(x,y,z)*is attached to the object and rotates with it, whilst the second one

*(x’,y’,z’)*is fixed in space, the

*z’*axis defined by the fixed view line. Assuming a rotation of the object (and its attached coordinate system) around the

*y’*axis of angle

*ω*, from the definition of angular coordinates, it is obvious that a normal defined by

*(θ, φ)*in the rotated system will be represented by the angles

*(θ’, φ’)*=

*(θ, φ+ω)*in the fixed system [See Fig. 1(a)]. Therefore,

*PhFT(θ’, φ’)*, expressed in angular coordinates, will undergo just a displacement. Even in spatial frequencies,

*(u,v)*, at first order approximation of

*ω*the variations are linear with the rotation angle

*ω*, as can be checked using Eq. (4). A rotation around any axis contained in the

*(x,y)*plane can be reduced to this case simply by choosing the

*y*axis as the rotation axis. This property has been used to obtain detection of 3D objects by correlation of the PhFT intensity with limited tolerance to rotations with respect to an axis perpendicular to the view line [15

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]

*θ*smaller than 45 degrees), because larger nodding of the object was not permitted and thus no information of the facet pointing to the poles was available.

## 3. Object orientations map on the unit sphere

*α*around the y axis (b) a rotation of angle

*β*around the rotated

*x*axis (c) a rotation of angle

*γ*around the new

*z*axis.

*γ*) is not needed for scanning over all

*(θ, φ)*range. The procedure for building the 3DOOM starts by preparing an image for the full

*(θ, φ)*range. Then we scan the view line for all angles

*(θ, φ)*, by means of the two rotations around the

*y*and

*x’*axes. For every angle of the view line the following procedure is performed: (a) the range image is computed (b) the PhFT is obtained in the coordinates of the rotated system

*(θ’, φ’)*(c) the Intensity of the PhFT is coordinate transformed into the object axes

*(θ, φ)*(d) The resulting image is pasted (by averaging) on the full 3DOOM image. Figure 2 shows a sample 3D image and the corresponding map in spherical angles.

*(θ’, φ’)*, where the PhFT is obtained, into object spherical coordinates

*(θ, φ)*, that are the variables for the 3DOOM. This coordinate transformation is highly non linear [18], except for the case of rotations around

*y*axis, as previously discussed. The distortion is indeed an artifact due to the representation of the 3DOOM as a planar map. As in the case of cartography there is no singular point in the representation if the 3DOOM is represented on a unit radius sphere.

## 3. Correlations on the unit sphere

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

*D(α,β,γ)*is the rotation characterized by the proper Euler angles. Note that is not the only definition in the literature. In [20

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]

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

*Y*

_{lm}are the spherical harmonics functions. Following Wendelt

*et al.*[19

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

*f*

_{lm}and

*g*

_{lm}are the spherical Fourier transforms. In our case

*f*and

*g*are the PhFT from a given (unknown) point of view and the 3DOOM, respectively. Therefore, following this definition, the output of the correlation will be three dimensional, and the location of the correlation peak will give the

*(α, β,γ)*triplet with the object orientation. This correlation admits a rapid implementation using the fast Fourier transform algorithm [20

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]

## 4. Conclusions

## Acknowledgments

## References and links

1. | R. Campbell and P. Flynn, “A survey of free-form object representation and recognition techniques,” Computer Vision and Image Understanding |

2. | M. Rioux, “Laser range finder based on synchronized scanners,” Appl. Opt. |

3. | M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. |

4. | J. Rosen, “Three-dimensional electro-optical correlation,” J. Opt. Soc. Am. A |

5. | T. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. |

6. | E. Tajahuerce, O. Matoba, and B. Javidi, “Shift-Invariant Three-Dimensional Object Recognition by Means of Digital Holography,” App. Opt. |

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. |

8. | P. Parrein, J. Taboury, and P. Chavel, “Evaluation of the shape conformity using correlation of range images,” Opt. Commun. |

9. | DF Huber and M Hebert, “Fully automatic registration of multiple 3D data sets,” Image And Vision Computing |

10. | M. Rioux, P. Boulanger, and T. Kasvand, “Segmentation of range images using sine wave coding and Fourier transformation,” App. Opt. |

11. | E. Paquet, M. Rioux, and H. H. Arsenault, “Range image segmentation using the Fourier transform,” Opt. Eng. |

12. | J. J. Esteve-Taboada, D. Mas, and J. García, “Three-dimensional object recognition by Fourier transform profilometry,” App. Opt. |

13. | JJ Esteve-Taboada, J Garcia, and C Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. |

14. | Y. Hsu and HH Arsenault, “Optical-pattern recognition using circular harmonic expansion,” App. Opt. |

15. | S Chang, M Rioux, and CP Grover, “Range face recognition based on the phase Fourier transform,” Opt. Commun. |

16. | LG Hassebrook, ME Lhamon, M Wang, and JP Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. |

17. | J. J. Esteve-Taboada and J. García, “Detection and orientation evaluation for three-dimensional objects,” Opt. Com. |

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

19. | B. D. Wandelt and K. M. Górski, “Fast convolution on the sphere,” Phys. Rev. D |

20. | J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-Sphere,” Adv. In App. Math. |

21. | J.J. Sakurai, |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.5010) Fourier optics and signal processing : Pattern recognition

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 7, 2003

Revised Manuscript: November 24, 2003

Published: December 15, 2003

**Citation**

Javier García, Jose Valles, and Carlos Ferreira, "Detection of three-dimensional objects under arbitrary rotations based on range images," Opt. Express **11**, 3352-3358 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-25-3352

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### References

- 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]
- M. Rioux, �??Laser range finder based on synchronized scanners,�?? Appl. Opt. 23, 3837-3844 (1984). [CrossRef] [PubMed]
- 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]
- J. Rosen, �??Three-dimensional electro-optical correlation,�?? J. Opt. Soc. Am. A 15, 430-436 (1998). [CrossRef]
- T. Poon and T. Kim, �??Optical image recognition of three-dimensional objects,�?? Appl. Opt. 38, 370-381 (1999). [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]
- 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]
- 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]
- Huber DF, Hebert M, �??Fully automatic registration of multiple 3D data sets,�?? Image And Vision Computing 21 (7): 637-650 (2003) [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]
- E. Paquet, M. Rioux and H. H. Arsenault, �??Range image segmentation using the Fourier transform ,�?? Opt. Eng. 32, 2173-2180 (1994) [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]
- Chang S, Rioux M, Grover CP, �??Range face recognition based on the phase Fourier transform,�?? Opt. Commun. 222, 143-153 (2003) [CrossRef]
- Hassebrook LG, Lhamon ME, Wang M, Chatterjee JP, �??Postprocessing of correlation for orientation estimation,�?? Opt. Eng. 36, 2710-2718 (1997) [CrossRef]
- J. J. Esteve-Taboada and J. García, �??Detection and orientation evaluation for three-dimensional objects,�?? Opt. Com. 217, 123-131 (2002). [CrossRef]
- <a href="http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html">http://scienceworld.wolfram.com/astronomy/EquatorialCoordinates.html</a>
- B. D. Wandelt and K. M. Górski, �??Fast convolution on the sphere,�?? Phys. Rev. D 63, 123002 (2001). [CrossRef]
- 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]
- J.J.Sakurai,Modern Quantum Mechanics (Adisson-Wesley, New York, 1985), pp. 221-223

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