## Phase Fourier vector model for scale invariant three-dimensional image detection

Optics Express, Vol. 15, Issue 12, pp. 7818-7825 (2007)

http://dx.doi.org/10.1364/OE.15.007818

Acrobat PDF (311 KB)

### Abstract

A scale invariant 3D object detection method based on phase Fourier transform (PhFT) is addressed. Three-dimensionality is expressed in terms of range images. The PhFT of a range image gives information about the orientations of the surfaces in the 3D object. When the object is scaled, the PhFT becomes a distribution multiplied by a constant factor which is related to the scale factor. Then 3D scale invariant detection can be solved as illumination invariant detection process. Several correlation operations based on vector space representation are applied. Results show the tolerance of detection method to scale besides discrimination against false objects.

© 2007 Optical Society of America

## 1. Introduction

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

02. J. Rosen, “Three-dimensional electro-optical correlation,” J. Opt. Soc. Am. A **15**, 430–436 (1998). [CrossRef]

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

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

07. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. **22**, 3977–3882 (1983). [CrossRef] [PubMed]

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

09. J. J. Esteve-Taboada, J. García, and C. Ferreira, “Rotation invariant optical recognition of three-dimensional objectes,” Appl. Opt. **39**, 5998–5352 (2000). [CrossRef]

10. J. García, 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]

11. J. J. Esteve-Taboada, N. Palmer, J.- Ch. Giannesini, J. García, and C. Ferreira, “Recognition of polychromatic three-dimensional objects,” Appl. Opt. **43**, 433–441 (2004). [CrossRef] [PubMed]

12. J. J. Esteve-Taboada, J. García, and C. Ferreira, “Optical recognition of three-dimensional objects with scale invariance using classical convergent correlator,” Opt. Eng. **41**, 1324–1330 (2002). [CrossRef]

*z*axis [13

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

*PhFT*) [14

14. 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]

15. 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]

*PhFT*operation. The idea is to detect the planar surface of a 3D range image when the Fourier transform of a phase-coded image (the phase being proportional to the elevation) is calculated using feed-forward neural networks [14

14. 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]

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

## 2. Scale changes for range image phase Fourier transforms

*z*(

*x, y*) , contains the depth information of an object from a given view line, that defines the

*z*axis. One of the main advantages of a range image is that three-dimensional information is stored in a 2D image containing only geometrical information. In fact, the range image is considered as a set of facets which may be described by their normals to the surface. The encoding of the depth information has been used in the literature to extend the possibilities of range images for pattern recognition [10

10. J. García, 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]

14. 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]

15. 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]

*m*is a constant that permits the adjustment of the phase slope of the object. From now, without losing generality, we will assume

*m*=1.

*PhFT*) is

*F*stands for two dimensional Fourier transform.

_{2D}*PhFT*is that it contains information of all the orientations of the surfaces that defines a given 3D object [14

**34**, 1178–1183 (1995). [CrossRef]

15. 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]

*PhFT*maps a facet into a peak .The position and distribution of the peak represent the orientation and the boundary of the facet, respectively. The intensity of the

*PhFT*exhibits a crucial property: it is invariant to arbitrary translations of the object. This property is obvious for translation in the

*(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 [see 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 [15

**29**, 35–39 (1998). [CrossRef]

*PhFTs*. This connection between rotation in spatial domain and translation in angular

*PhFT*domain makes it advantageous for 3D object correlation based recognition methods [9

09. J. J. Esteve-Taboada, J. García, and C. Ferreira, “Rotation invariant optical recognition of three-dimensional objectes,” Appl. Opt. **39**, 5998–5352 (2000). [CrossRef]

10. J. García, 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]

*PhFT*amplitude of Fig. 1(a) is shown in Fig. 1(b). The four facets (four normals) correspond to four location peaks in the Fourier plane. On the other hand, if the curvature of the object’s surface is a continous function (smooth object surface, i. e. with large number of facets), the

*PhFT*will be continuous.

*PhFT*domain by a constant factor, with no change in the pattern distribution of the

*PhFTs*. Only a global change in the intensity is observed.

*PhFT*of Fig. 1(c). Both

*PhFTs*[Fig. 1(b) and Fig. 1(c)] are the same except for a global multiplicative constant factor.

*PhFTs*shown in Figs. 2(b) and 2(d), respectively, is not a global constant factor. In section 4 we will deal with this factor in more details. But, except for those digitalization errors, changes in scale involve changes in intensity or amplitude. There are several methods for intensity invariant pattern recognition based on correlations [18–22

18. D. Lefebvre, H. H. Arsenault, P. Garcia-Martinez, and C. Ferreira, “Recognition of unsegmented targets invariant under transformations of intensity,” Appl. Opt. **41**, 6135–6142 (2002). [CrossRef] [PubMed]

24. F. M. Dickey and L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. **16**, 1186–1188 (1991). [CrossRef] [PubMed]

## 3. Intensity invariant correlation method

*PhFT*amplitude distributions.

18. D. Lefebvre, H. H. Arsenault, P. Garcia-Martinez, and C. Ferreira, “Recognition of unsegmented targets invariant under transformations of intensity,” Appl. Opt. **41**, 6135–6142 (2002). [CrossRef] [PubMed]

*PhFT*amplitude in the Fourier domain. So, a linear transformation of intensity over a target can be expressed as

*u,v*) is the binary support which is equal to unity over the support of the target

*PhFT*(

_{z}*u,v*), and equals to zero everywhere else, and

*α, β*, are unknown constants. An orthogonal basis for the subspace is selected. We define

*PhFT*(

_{z}^{o}*u,v*) as

*μ*is the mean of

_{f}*PhFT*(

_{z}*u, v*). Note that

*PhFT*(

_{z}^{o}*u,v*) is a zero-mean target in the region of support. Then, the target can be defined as a linear combination of two orthogonal images (a silhouette and a zero-mean target) as

*α’*and

*β’*are constants. The basis defined by {

*PhFT*(

^{o}_{z}*u,v*),◊(

*u,v*)} is not orthonormal, but we can normalize it to unit length as {

*ϕ*(

*u,v*) = ◊(

*u,v*)/∥◊(

*u,v*)∥;

*ϕ*

_{2}(

*u,v*) =

*PhFT*(

^{o}_{z}*u,v*) /∥

*PhFT*(

^{o}_{z}*u ,v*)∥}, where ∥

*PhFT*(

^{o}_{z}*u,v*)∥ =

*ϕ*

_{1}(

*u,v*),

*ϕ*

_{2}(

*u,v*)}, the target can now be defined as

18. D. Lefebvre, H. H. Arsenault, P. Garcia-Martinez, and C. Ferreira, “Recognition of unsegmented targets invariant under transformations of intensity,” Appl. Opt. **41**, 6135–6142 (2002). [CrossRef] [PubMed]

19. H. H. Arsenault and P. García-Martínez, “Intensity-invariant nonlinear filtering for detection in camouflage” Appl. Opt. **44**, 5483–5490 (2005). [CrossRef] [PubMed]

*N*is the number of pixels inside the region of support. Then for a given range target

*s*(

*x, y*), if

*PhFT*(

_{s}*u,v*) is a linear combination of the orthonormal basis, then the correlation peak will be equal to one, and it will be smaller than one if it is not. We have applied LACIF filtering for scale invariant 3D object detection codified in terms of

*PhFTs*. Note that for our 3D detection process only constant α is considered (see Eq. (3), where

*α*=

*k*

^{2}), whereas the constant

*β*equals zero. It means that there is no global constant added to the

*PhFT*, but multiplied. Moreover, the definition of Eq. (8) in terms of correlations makes the approach feasible to be implemented optoelectronically. The correlations can be performed by conventional optical correlators like Vander Lugt correlator or joint transform correlator architectures and the local calculations can be obtained using computer interface.

## 4. Results of detection

*PhFT*amplitude distribution as the region of support for all the experiments. We have chosen as reference target the average between different scaled reference targets in order to minimize the possible sampling errors due to scaling digitalization process. Results are shown in Fig. 3. As we see from Fig. 3(a), the LACIF is almost constant for all scale factors (60%-120%) and its value oscillates between 0.7 and almost 0.9 correlation peak value. A multimedia file shows the appearance of the range images and the result of the detection.

*PhFT*of the human face provides a new signature of the face. Ref. [16

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

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

*PhFT*of a range images is a multiplication of a constant factor in the

*PhFT*of the reference range target. LACIF filtering detects targets which have been multiplied by constant values. Then, applying LACIF to Fourier phase encoded range images will solve the scale changes in 3D range images detection. Various experiments were carried out to validate the scale invariance. For real applications, the method has applied to face human recognition. We successfully tested the method when other false targets are tested.

## Appendix A

*PhFT*distribution. The

*PhFT*contains information of all the orientations of the surfaces that defines a 3D object. It can be assumed that a 3D object is composed by multiples planar facets in different position and orientation. All of them define the object. Then, the

*PhFT*of the complete 3D object is the contribution of all the

*PhFTs*of the different facets.

*w*(

*x,y*) is a boundary function equals to one if the point

*(x,y)*belongs to the plane and zero otherwise. Its phase Fourier transform is

*u,v*) is the delta function, ⊗ is the convolution and

*W*(

*u,v*) =

*FT*{

*w*(

*x,y*)} . Note from Eq. (A.2) that the integral outside the boundary area is infinite and the exponential is a periodical function, so the integral of the second term of Eq. (A.2) is zero. Then, the

*PhFT*of an individual facet is centered in a peak given by the orientation of the normal to the plane and the shape of the peak is given by a certain factor of form which is the Fourier transform of the boundary. This result is similar to the interpretation of diffraction limited coherent optical imaging due to the pupil of the lenses.

*k*is the scale factor. Note that a scale change has no influence in the orientation of the individual facets. The

*PhFT*of the scaled object,

*f*′(

*x,y*), is

*PhFT*of the scaled object is given by a peak in the same position than the non-scaled object, but it is convolved by a scaled factor of form. Considering that a continuous 3D object is formed by numerous facets and the range of scale factors is not too large (i.e.

*k*varies from 0.6-1.2), it can be shown from the experiments (see Fig. 1) that a scale change in the factor of form will be negligible in comparison with the influence of the peaks position, so in a first approximation we can assume that

## Acknowledgments

## References and links

01. | B. Javidi, ed., |

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

03. | J. Rosen, “Three-dimensional joint transform correlator,” Appl. Opt. |

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

05. | B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. |

06. | B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express |

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

08. | J. J. Esteve-Taboada, D. Mas, and J. Garcia, “Three-dimensional object recognition by Fourier transform profilometry,” Appl. Opt. |

09. | J. J. Esteve-Taboada, J. García, and C. Ferreira, “Rotation invariant optical recognition of three-dimensional objectes,” Appl. Opt. |

10. | J. García, J. J. Vallés, and C. Ferreira, “Detection of three-dimensional objects under arbitrary rotations based on range images,” Opt. Express. |

11. | J. J. Esteve-Taboada, N. Palmer, J.- Ch. Giannesini, J. García, and C. Ferreira, “Recognition of polychromatic three-dimensional objects,” Appl. Opt. |

12. | J. J. Esteve-Taboada, J. García, and C. Ferreira, “Optical recognition of three-dimensional objects with scale invariance using classical convergent correlator,” Opt. Eng. |

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

14. | 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. |

15. | E. Paquet, P. Garcia-Martinez, and J. Garcia, “Tridimensional invariant correlation based on phase-coded and sine-coded range images,” J. Opt. |

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

17. | Y. Li and J. Rosen, “Scale invariant recognition of three-dimensional objects by use of a quasi-correlator,” Appl. Opt. |

18. | D. Lefebvre, H. H. Arsenault, P. Garcia-Martinez, and C. Ferreira, “Recognition of unsegmented targets invariant under transformations of intensity,” Appl. Opt. |

19. | H. H. Arsenault and P. García-Martínez, “Intensity-invariant nonlinear filtering for detection in camouflage” Appl. Opt. |

20. | H. H. Arsenault and D. Lefebvre, “Homomorphic cameo filter for pattern recognition that is invariant with change of illumination,” Opt. Lett. |

21. | D. Lefebvre, H. H. Arsenault, and S. Roy, “Nonlinear filter for pattern recognition invariant to illumination and to out-ot-plane rotations,” Appl. Opt. |

22. | S. Roy, D. Lefebvre, and H. H. Arsenault, “Recognition invariant under unknown affine transformations of intensity,” Opt. Commun. |

23. | J. J. Vallés, J. García, P. García-Martínez, and H. H. Arsenault, “Three-dimensional object detection under arbitrary lighting conditions,” Appl. Opt. |

24. | F. M. Dickey and L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. |

**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:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: April 2, 2007

Revised Manuscript: May 11, 2007

Manuscript Accepted: May 15, 2007

Published: June 8, 2007

**Citation**

José J. Vallés, Pascuala Garcia-Martinez, Javier García, and Carlos Ferreira, "Phase Fourier vector model for scale invariant three-dimensional image detection," Opt. Express **15**, 7818-7825 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7818

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### References

- B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications, (Marcel Dekker, New York, 2002). [CrossRef]
- .J. Rosen, ‘Three-dimensional electro-optical correlation,’’J. Opt. Soc. Am. A 15, 430-436 (1998). [CrossRef]
- J. Rosen, ‘‘Three-dimensional joint transform correlator,’’Appl. Opt. 37, 7538-7544 (1998). [CrossRef]
- T. Poon and T. Kim, 'Optical image recognition of three-dimensional objects,' Appl. Opt. 38, 370-381 (1999). [CrossRef]
- B. Javidi and E. Tajahuerce, ‘‘Three-dimensional object recognition by use of digital holography,’’Opt. Lett. 25, 610-612 (2000). [CrossRef]
- B. Javidi, I. Moon, S. Yeom, and E. Carapezza, "Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography," Opt. Express 13, 4492-4506 (2005). [CrossRef] [PubMed]
- M. Takeda and K. Mutoh, ‘‘Fourier transform profilometry for the automatic measurement of 3-D object shapes,’’Appl. Opt. 22, 3977-3882 (1983).Q1 [CrossRef] [PubMed]
- J. J. Esteve-Taboada, D. Mas, and J. Garcia, ‘‘Three-dimensional object recognition by Fourier transform profilometry,’’Appl. Opt. 38, 4760-4765 (1999). [CrossRef]
- J. J. Esteve-Taboada, J. García and C. Ferreira, "Rotation invariant optical recognition of three-dimensional objectes," Appl. Opt. 39, 5998-5352 (2000).Q2 [CrossRef]
- J. García, 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]
- J. J. Esteve-Taboada, N. Palmer, J.- Ch. Giannesini, J. García and C. Ferreira, "Recognition of polychromatic three-dimensional objects," Appl. Opt. 43, 433-441 (2004). [CrossRef] [PubMed]
- J. J. Esteve-Taboada, J. García and C. Ferreira, "Optical recognition of three-dimensional objects with scale invariance using classical convergent correlator," Opt. Eng. 41, 1324-1330 (2002). [CrossRef]
- M. Rioux, "Laser range finder based on synchronized scanners," Appl. Opt. 23, 3837-3844 (1984). [CrossRef] [PubMed]
- 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]
- 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]
- S. Chang, M. Rioux, and C. P. Grover, "Range face recognition based on the phase Fourier transform," Opt. Commun. 222, 143-153 (2003). [CrossRef]
- Y. Li and J. Rosen, "Scale invariant recognition of three-dimensional objects by use of a quasi-correlator," Appl. Opt. 42, 811-819 (2003). [CrossRef] [PubMed]
- D. Lefebvre, H. H. Arsenault, P. Garcia-Martinez, and C. Ferreira, "Recognition of unsegmented targets invariant under transformations of intensity," Appl. Opt. 41, 6135-6142 (2002). [CrossRef] [PubMed]
- H. H. Arsenault and P. García-Martínez, "Intensity-invariant nonlinear filtering for detection in camouflage" Appl. Opt. 44, 5483-5490 (2005). [CrossRef] [PubMed]
- H. H. Arsenault and D. Lefebvre, "Homomorphic cameo filter for pattern recognition that is invariant with change of illumination," Opt. Lett. 25, 1567-1569 (2000). [CrossRef]
- D. Lefebvre, H. H. Arsenault, and S. Roy, "Nonlinear filter for pattern recognition invariant to illumination and to out-ot-plane rotations," Appl. Opt. 42, 4658-4662 (2003). [CrossRef] [PubMed]
- S. Roy, D. Lefebvre, and H. H. Arsenault, "Recognition invariant under unknown affine transformations of intensity," Opt. Commun. 238, 69-77 (2004). [CrossRef]
- J. J. Vallés, J. García, P. García-Martínez, and H. H. Arsenault, "Three-dimensional object detection under arbitrary lighting conditions," Appl. Opt. 45, 5237-5247 (2006). [CrossRef] [PubMed]
- F. M. Dickey and L. A. Romero, "Normalized correlation for pattern recognition," Opt. Lett. 16, 1186-1188 (1991). [CrossRef] [PubMed]

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