## Three-dimensional color object visualization and recognition using multi-wavelength computational holography

Optics Express, Vol. 15, Issue 15, pp. 9394-9402 (2007)

http://dx.doi.org/10.1364/OE.15.009394

Acrobat PDF (1285 KB)

### Abstract

In this paper, we address 3D object visualization and recognition with multi-wavelength digital holography. Color features of 3D objects are obtained by the multiple-wavelengths. Perfect superimposition technique generates reconstructed images of the same size. Statistical pattern recognition techniques: principal component analysis and mixture discriminant analysis analyze multi-spectral information in the reconstructed images. Class-conditional probability density functions are estimated during the training process. Maximum likelihood decision rule categorizes unlabeled images into one of trained-classes. It is shown that a small number of training images is sufficient for the color object classification.

© 2007 Optical Society of America

## 1. Introduction

1. A. Mahalanobis and F. Goudail, “Methods for automatic target recognition by use of electro-optic sensors: introduction to the feature Issue,” Appl. Opt. **43**, 207–209 (2004). [CrossRef]

10. S. Yeom, I Moon, and B. Javidi, “Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms,” Proc. IEEE **94**, 550–566 (2006). [CrossRef]

5. B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. **25**, 610–612 (2000). [CrossRef]

17. P. Almoro, W. Garcia, and C. Saloma, “Colored object recognition by digital holography and a hydrogen Raman shifter,” Opt. Express **15**, 7176–7181 (2007). [CrossRef] [PubMed]

17. P. Almoro, W. Garcia, and C. Saloma, “Colored object recognition by digital holography and a hydrogen Raman shifter,” Opt. Express **15**, 7176–7181 (2007). [CrossRef] [PubMed]

12. I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. **27**, 1108–1110 (2002). [CrossRef]

15. D. Alfieri, G. Coppola, S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattini, and B. Javidi, “Method for superposing reconstructed images from digital holograms of the same object recorded at different distance and wavelength,” Opt. Commun. **260**, 113–116 (2006). [CrossRef]

14. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**, 854–855 (2004). [CrossRef] [PubMed]

27. B. J. Frey and N. Jojic, “Transformation-invariant clustering using the EM algorithm,” IEEE Trans. on Pattern Anal. Mach. Intell. **25**, 1–17 (2003). [CrossRef]

22. C. Fraley and A. E. Raftery, “Model-based clustering, discriminant analysis, and density estimation,” J. of Am. Stat. Assoc. **97**, 611–631 (2002). [CrossRef]

27. B. J. Frey and N. Jojic, “Transformation-invariant clustering using the EM algorithm,” IEEE Trans. on Pattern Anal. Mach. Intell. **25**, 1–17 (2003). [CrossRef]

## 2. Multi-wavelength digital holography

12. I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. **27**, 1108–1110 (2002). [CrossRef]

15. D. Alfieri, G. Coppola, S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattini, and B. Javidi, “Method for superposing reconstructed images from digital holograms of the same object recorded at different distance and wavelength,” Opt. Commun. **260**, 113–116 (2006). [CrossRef]

*l*

_{1}and

*l*

_{2}) with different wavelengths are used. One is in the red region (λ

_{1}=632.8 nm), and the other is in the green region (λ

_{2}=532.0 nm). The optical configuration is arranged to allow the two lasers to propagate along the same paths either for the reference or the object beams. The reflecting prism which is in the path of the red laser beam permits the matching of the optical paths of the two interfering beams inside the optical coherent length of the laser. The object (toy warrior) as shown in Fig. 1 is placed at a distance 500 mm from the CCD (charge-coupled detector) array. Two holograms are recorded with two wavelengths. The holograms are reconstructed separately by the Fresnel transformation method [16

16. U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**, 179–181 (1994). [CrossRef] [PubMed]

14. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**, 854–855 (2004). [CrossRef] [PubMed]

*N*

_{x1}is an increased size of the hologram plane by zero-padding in the

*x*direction. If one hologram has been recorded with wavelength λ

_{1}, the other one with λ

_{2}, and λ

_{1}>λ

_{2}, then, the number of pixels of the hologram of the wavelength λ

_{1}is changed to

*N*

_{x2}is the size of the hologram plane in the

*x*direction with the wavelength

*λ*

_{2}. Consequently, we obtain the same resolution for reconstructed images for the holograms of different wavelengths:

*Δx*

_{1}′ and

*Δx*

_{2}′ are the resolutions of the image plane in the

*x*direction with wavelength

*λ*

_{1}and

*λ*

_{2}, respectively. The reconstructed image size in the

*y*direction is controlled in the same way. In the experiments, the size of the hologram plane (

*N*

_{x2}) is 1024 pixels, therefore,

*N*

_{x1}becomes 1218 pixels.

## 3. Statistical pattern recognition techniques

### 3.1 Principal component analysis

**x**∈

**R**

^{d×1}, where

**R**

^{d×1}is

*d*-dimensional Euclidean space, and

*d*is the same with the number of pixels in the reconstructed image. The PCA projects the

*d*-dimensional vectors onto the

*l*dimension subspace (

*l*≤

*d*) [18–21]. For a real

*d*-dimensional random vector

**x**, let the mean vector be µ

*=E(*

_{x}**x**), and the covariance matrix be Σ

*=E(*

_{xx}**x**-µ

*)(*

_{x}**x**-µ

_{x})

*, where the superscript*

^{t}*t*denotes transpose. The space for the PCA is spanned by the orthonormal eigenvectors of the covariance matrix, that is, Σ

*E=EV where the column vectors of E are normalized eigenvectors*

_{xx}**e**

*’s, i.e. E=[*

_{i}**e**

_{1},…,

**e**

*], and the diagonal matrix V is composed of eigenvalues*

_{d}*v*’s, i.e. V=diag(

_{i}*v*

_{1},…,

*v*). For the PCA, the projection matrix

_{d}*W*is the same as the eigenvector matrix E. Therefore, a projected vector

_{p}**y**by

*W*is

_{p}**y**, i.e. Σ

*=E(*

_{yy}**y**-µ

*)(*

_{y}**y**-µ

*)*

_{y}*=V where µ*

^{t}*=E(*

_{y}**y**). If we choose the projection matrix

*W*=[

_{p}**e**

_{1},…,

**e**

*], the subspace is spanned by the corresponding*

_{l}*l*eigenvectors. It is a well known property of the PCA that by choosing

*l*eigenvectors of the largest eigenvalues, the mean-squared error between a vector

**x**and a restored vector

**x**̂ is minimized. The mean-squared error is defined as

**x**̂ is defined as

*v*’s are eigenvalues of

_{i}*v*≤

_{d}*v*

_{d-1},…,

*v*

_{2}≤

*v*

_{1}. The PCA can reduce the dimension of the vectors while retaining dominant features of the object structure and reducing redundant and noisy data.

### 3.2 Mixture discriminant analysis

22. C. Fraley and A. E. Raftery, “Model-based clustering, discriminant analysis, and density estimation,” J. of Am. Stat. Assoc. **97**, 611–631 (2002). [CrossRef]

27. B. J. Frey and N. Jojic, “Transformation-invariant clustering using the EM algorithm,” IEEE Trans. on Pattern Anal. Mach. Intell. **25**, 1–17 (2003). [CrossRef]

*j*is composed of

*G*components of the probability density functions, the class-conditional probability density function is

_{j}**y**is a projected vector onto the subspace by the PCA,

*w*denotes an event that the vector

_{jk}**y**belongs to the component

*k*of the class

*j*,

*N*is the number of classes under investigation,

_{c}*P*(

*w*) is the probability that the event

_{jk}*w*occurs.

_{jk}*N*(·) denotes the multivariate Gaussian distribution, and

**µ**

*and Σ*

_{jk}*are the mean vector and the covariance matrix of the component*

_{jk}*k*in the class

*j*, respectively. Therefore, solving the MDA with the Gaussian mixture model is equivalent to estimating three unknown parameters (

*P*(

*w*),

_{jk}**µ**

*, Σ*

_{jk}*) for each Gaussian component of the classes.*

_{jk}*n*training images of the class

_{j}*j*be

*n*observations from the class

_{j}*j*. The maximum likelihood solution of Eq. (8) is obtained as [19–21]

_{jk}**µ**̂, Σ̂

*, and*

_{jk}*P*̂(

*w*) are the estimators for the mean, covariance and mixing weight, respectively.

_{jk}*j*;

*i*represents the number of the iteration, and

*i*

_{max}and

*ε*are the termination criteria for the iteration. The EM procedure is as follows: for initialization, we set

*k*=1,…,

*G*; during E step, we compute

_{j}*P*;̂(

*w*|

_{jk}**y**

*) for*

_{jt}*k*=1,…,

*G*, and

_{j}*t*=1,…,

*n*as in Eq. (9); and during M step, we compute

_{j}**µ**;̂

*, Σ̂*

_{jk}*,*

_{jk}*P*;̂(

*w*) for

_{jk}*k*=1,…,

*G*as in Eqs. (10)–(12). The E and M steps are iterated until

_{j}*i*>

*i*

_{max}. In the experiment, we set

*ε*at 10

^{-50}and

*i*

_{max}at 100. These E-step and M-step are repeated for

*j*=1,…,

*N*.

_{c}**y**on the PCA subspace should be pre-determined for the PCA. In the experiments, several dimensions for

**y**are tested. The number of clusters

*G*in Eq. (6) is another factor to be considered carefully. We chose the cluster number to be the same as the number of wavelengths which is two in the experiments. The initialization of the parameters is the other important factor. We use the Linde-Buzo-Gray initialization satisfying the Lloyd’s optimality conditions [28

_{j}28. Y. Linde, A. Buzo, and R. M. Gray, “An algorithm for vector quantizer design,” IEEE Trans. Commun. **COM-28**84–95 (1980). [CrossRef]

29. J. Jang, S. Yeom, and B. Javidi, “Compression of ray information in three-dimensional integral imaging,” Opt. Eng. **44**, 12700-1~10 (2005). [CrossRef]

*k*-means clustering [18].

### 3.3 Maximum likelihood decision rule

**y**

*be the unlabeled vector on the PCA subspace as in Eq. (3). In the experiments, a test vector corresponds to a reconstructed image from one hologram. Therefore, a test vector*

_{test}**y**

*contains a single spectral feature of the object.*

_{test}*C*̂ is the set of the class

_{j}*j*̂, and

*p*̂

*(*

_{j}**y**) is the class-conditional probability density function with the estimated parameters.

## 4. Experiments and simulation results

_{1}and λ

_{2}are 35 mW and 50 mW, respectively, and the sizes of the beam diameter for λ

_{1}and λ

_{2}are 1.2 mm and 1.8 mm, respectively. We control the reflecting prism (RP) to equalize the paths of the object beam and the reference beam, in such a way that the optical path difference is inside the coherence length of the laser. It is noted that the coherence length of the red laser is about 20 cm, while the green is in meters. The CCD array is JAI Mod. CV M4. The spatial resolution of the CCD is 1280×1024 pixels and the size of each pixel is 6.7 µm.

14. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**, 854–855 (2004). [CrossRef] [PubMed]

_{1}) and the green component is the green laser (λ

_{2}).

## 5. Conclusions

## References and links

1. | A. Mahalanobis and F. Goudail, “Methods for automatic target recognition by use of electro-optic sensors: introduction to the feature Issue,” Appl. Opt. |

2. | F. A. Sadjadi, “IR target detection using probability density functions of wavelet transform subbands,” Appl. Opt. |

3. | B. Javidi, ed., |

4. | F. Goudail and P. Refregier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. |

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

6. | Y. Frauel, E. Tajahuerce, M. Castro, and B. Javidi, “Distortion-tolerant three-dimensional object recognition with digital holography,” Appl. Opt. |

7. | Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. |

8. | S. Yeom and B. Javidi, “Three-dimensional object feature extraction and classification with computational holographic imaging,” Appl. Opt. |

9. | 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 |

10. | S. Yeom, I Moon, and B. Javidi, “Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms,” Proc. IEEE |

11. | J. Maycock, T. Naughton, B. Hennely, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. |

12. | I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. |

13. | J. Kato, I. Yamaguchi, and T. Matsumura, “Multicolor digital holography with an achromatic phase shifter,” Opt. Lett. |

14. | P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. |

15. | D. Alfieri, G. Coppola, S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattini, and B. Javidi, “Method for superposing reconstructed images from digital holograms of the same object recorded at different distance and wavelength,” Opt. Commun. |

16. | U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

17. | P. Almoro, W. Garcia, and C. Saloma, “Colored object recognition by digital holography and a hydrogen Raman shifter,” Opt. Express |

18. | A. K. Jain, |

19. | R. O. Duda, P. E. Hart, and D. G. Stork, |

20. | K. Fukunaga, |

21. | C. M. Bishop, |

22. | C. Fraley and A. E. Raftery, “Model-based clustering, discriminant analysis, and density estimation,” J. of Am. Stat. Assoc. |

23. | T. Hastie and R. Tibshirani, “Discriminant analysis by Gaussian mixtures,” J. Royal Statistical Society B |

24. | G. J. McLachlan, |

25. | M. M. Dundar and D. Landgrebe, “A model-based mixture-supervised classification approach in hyperspectral data analysis,” IEEE. Trans. on Geoscience and remote sensing |

26. | M. H. C. Law, M. A. T. Figueiredo, and A. K. Jain, “Simultaneous feature selection and clustering using mixture models,” IEEE. Trans. on Pattern Anal. Mach. Intell. |

27. | B. J. Frey and N. Jojic, “Transformation-invariant clustering using the EM algorithm,” IEEE Trans. on Pattern Anal. Mach. Intell. |

28. | Y. Linde, A. Buzo, and R. M. Gray, “An algorithm for vector quantizer design,” IEEE Trans. Commun. |

29. | J. Jang, S. Yeom, and B. Javidi, “Compression of ray information in three-dimensional integral imaging,” Opt. Eng. |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.5010) Image processing : Pattern recognition

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Holography

**History**

Original Manuscript: April 17, 2007

Revised Manuscript: June 20, 2007

Manuscript Accepted: June 21, 2007

Published: July 16, 2007

**Citation**

Seokwon Yeom, Bahram Javidi, Pietro Ferraro, Domenico Alfieri, Sergio DeNicola, and Andrea Finizio, "Three-dimensional color object visualization and recognition using multi-wavelength computational holography," Opt. Express **15**, 9394-9402 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9394

Sort: Year | Journal | Reset

### References

- A. Mahalanobis and F. Goudail, "Methods for automatic target recognition by use of electro-optic sensors: introduction to the feature Issue," Appl. Opt. 43, 207-209 (2004). [CrossRef]
- F. A. Sadjadi, "IR target detection using probability density functions of wavelet transform subbands," Appl. Opt. 43, 315-323 (2004). [CrossRef] [PubMed]
- B. Javidi, ed., Optical Imaging Sensors and Systems for Homeland Security Applications (Springer, NewYork, 2005).
- F. Goudail and P. Refregier, "Statistical algorithms for target detection in coherent active polarimetric images," J. Opt. Soc. Am. 18, 3049-3060 (2001). [CrossRef]
- B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000). [CrossRef]
- Y. Frauel, E. Tajahuerce, M. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001). [CrossRef]
- Y. Frauel and B. Javidi, "Neural network for three-dimensional object recognition based on digital holography," Opt. Lett. 26, 1478-1480 (2001). [CrossRef]
- S. Yeom and B. Javidi, "Three-dimensional object feature extraction and classification with computational holographic imaging," Appl. Opt. 43, 442-451 (2004). [CrossRef] [PubMed]
- 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]
- S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proc. IEEE 94, 550-566 (2006). [CrossRef]
- J. Maycock, T. Naughton, B. Hennely, J. McDonald, and B. Javidi, "Three-dimensional scene reconstruction of partially occluded objects using digital holograms," Appl. Opt. 45, 2975-2985 (2006). [CrossRef] [PubMed]
- I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002). [CrossRef]
- J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2003). [CrossRef]
- P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-855 (2004). [CrossRef] [PubMed]
- D. Alfieri, G. Coppola, S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattini, and B. Javidi, "Method for superposing reconstructed images from digital holograms of the same object recorded at different distance and wavelength," Opt. Commun. 260, 113-116 (2006). [CrossRef]
- U. Schnars, and W. Juptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
- P. Almoro, W. Garcia, and C. Saloma, "Colored object recognition by digital holography and a hydrogen Raman shifter," Opt. Express 15, 7176-7181 (2007). [CrossRef] [PubMed]
- A. K. Jain, Fundamentals of digital image processing (Prentice-Hall Inc., 1989).
- R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification 2nd ed. (Wiley Interscience, New York, 2001).
- K. Fukunaga, Introduction to Statistical Pattern Recognition 2nd ed. (Academic Press, Boston, 1990).
- C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University Press, New York, 1995).
- C. Fraley and A. E. Raftery, "Model-based clustering, discriminant analysis, and density estimation," J. of Am. Stat. Assoc. 97, 611-631 (2002). [CrossRef]
- T. Hastie and R. Tibshirani, "Discriminant analysis by Gaussian mixtures," J. Royal Statistical Society B 58, 155-176 (1996).
- G. J. McLachlan, Discriminant analysis and statistical pattern recognition (Wiley, New York, 1992). [CrossRef]
- M. M. Dundar and D. Landgrebe, "A model-based mixture-supervised classification approach in hyperspectral data analysis," IEEE. Trans. Geoscience and remote sensing 40, 2692-2699 (2002). [CrossRef]
- M. H. C. Law, M. A. T. Figueiredo, and A. K. Jain, "Simultaneous feature selection and clustering using mixture models," IEEE. Trans. on Pattern Anal. Mach. Intell. 26, 1154-1166 (2004) [CrossRef]
- B. J. Frey and N. Jojic, "Transformation-invariant clustering using the EM algorithm," IEEE Trans. on Pattern Anal. Mach. Intell. 25, 1-17 (2003). [CrossRef]
- Y. Linde, A. Buzo, and R. M. Gray, "An algorithm for vector quantizer design," IEEE Trans. Commun. COM- 2884-95 (1980). [CrossRef]
- J. Jang, S. Yeom, and B. Javidi, "Compression of ray information in three-dimensional integral imaging," Opt. Eng. 44, 12700-1~10 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: AVI (2157 KB)

» Media 2: AVI (2157 KB)

» Media 3: AVI (2157 KB)

» Media 4: AVI (2157 KB)

» Media 5: AVI (2157 KB)

» Media 6: AVI (2157 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.