## Improved reflectance reconstruction for multispectral imaging by combining different techniques

Optics Express, Vol. 15, Issue 9, pp. 5531-5536 (2007)

http://dx.doi.org/10.1364/OE.15.005531

Acrobat PDF (106 KB)

### Abstract

In multispectral imaging system, one of the most important tasks is to accurately reconstruct the spectral reflectance from system responses. We propose such a new method by combing three most frequently used techniques, i.e., wiener estimation, pseudo-inverse, and finite-dimensional modeling. The weightings of these techniques are calculated by minimizing the combined standard deviation of both spectral errors and colorimetric errors. Experimental results show that, in terms of color difference error, the performance of the proposed method is better than those of the three techniques. It is found that the simple averaging of the reflectance estimates of these three techniques can also yield good color accuracy.

© 2007 Optical Society of America

## 1. Introduction

2. H. L. Shen and J. H. Xin, “Spectral characterization of a color scanner by adaptive estimation,” J. Opt. Soc. Am. A **21**, 1125–1130 (2004). [CrossRef]

3. H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the spectral reflectance of art paintings,” Appl. Opt. **39**, 6621–6632 (2000). [CrossRef]

4. V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A **22**, 1231–1240 (2005). [CrossRef]

*priori*knowledge about the imaging system.

9. H. Stokman and T. Gevers, “Selection and fusion of color models for image feature detection,” IEEE Trans. Pattern Anal. Mach. Intell. **29**, 371–381 (2007). [CrossRef] [PubMed]

## 2. Formulation of multispectral imaging process

**v**of

*L*channels can be formulated as [2

2. H. L. Shen and J. H. Xin, “Spectral characterization of a color scanner by adaptive estimation,” J. Opt. Soc. Am. A **21**, 1125–1130 (2004). [CrossRef]

*N*=31 denotes the number of samples in the visible wavelength range,

**r**denotes the spectral reflectance,

**M**represents the spectral responsivity incorporating the spectral power distribution of lighting source, the spectral transmittances of narrowband filters, and the spectral sensitivity of digital camera.

**b**denotes the bias response vector caused by camera dark current, and

**n**denotes zero-mean imaging noise. It is noted that if the system does not behave linearly, the optoelectronic conversion function needs to be further considered [2

2. H. L. Shen and J. H. Xin, “Spectral characterization of a color scanner by adaptive estimation,” J. Opt. Soc. Am. A **21**, 1125–1130 (2004). [CrossRef]

4. V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A **22**, 1231–1240 (2005). [CrossRef]

**M**and bias

**b**from training color samples, subject to the constraint of non-negativeness [10

10. K. Barnard and B. Funt, “Camera characterization for color research,” Color Res. Appl. **27**, 152–163 (2002). [CrossRef]

**21**, 1125–1130 (2004). [CrossRef]

## 3. Three techniques for spectral characterization of multispectral imaging

### 3.1 Wiener estimation

**r**̂ from the response vector

**u**=

**v**-

**b**through an

*N*×

*L*matrix

**W**, such that

**21**, 1125–1130 (2004). [CrossRef]

3. H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the spectral reflectance of art paintings,” Appl. Opt. **39**, 6621–6632 (2000). [CrossRef]

**W**is calculated by using the spectral responsivity

**M**:

*T*denotes transpose,

**K**

_{r}denotes the

*N*×

*N*covariance matrix of

**r**, and

**K**

_{n}denotes the covariance matrix of noise. In this study,

**K**

_{n}=

**0**is assumed.

### 3.2 Pseudo-inverse

**W**is directly solved as

**R**denotes the matrix of reflectance vector

**r**, and

**U**denotes the matrix of response vector

**u**.

### 3.3 Finite-dimensional modeling

**r**can always be represented by the linear combination of

*J*(<

*N*) basis functions

**b**

_{j}[12

12. L. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A **3**, 1673–1683 (1986). [CrossRef] [PubMed]

*a*is the coefficient of

_{j}**b**

_{j}.

**b**

_{j}can be calculated using principle component analysis of reflectance data. By combining Eq. (5) and (1), the response

**u**can then be represented as [4–7

4. V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A **22**, 1231–1240 (2005). [CrossRef]

*a*can be estimated by pseudo-inverse of the

_{j}*L*×

*J*matrix [

**Mb**

_{j}]. The reflectance

**r**ô can then be obtained by substituting the estimated

*a*into Eq. (5). In this study,

_{j}*J*=10 basis functions are used, as they are generally adequate for spectral reflectance construction [12

12. L. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A **3**, 1673–1683 (1986). [CrossRef] [PubMed]

5. M. Shi and G. Healey, “Using reflectance models for color scanner calibration,” J. Opt. Soc. Am. A **19**, 645–656 (2002). [CrossRef]

## 4. The proposed method

### 4.1 Statistical properties of different observations

*K*(

*K*=3 in this study) different techniques, and

*x*is the observation of the

_{k}*k*th technique. To obtain an improved estimation of that quantity, it is feasible to combine these observations as the following:

*w*is the weighting of the observation of the

_{k}*k*th technique. The estimate and standard deviation of

*x*can be expressed according to Eq. (8) and (9), respectively.

*E*(

*x*) is the averaged observation of the

_{k}*k*th technique, and σ(

*x*,

_{k}*x*) is the covariance between observations of the

_{l}*k*th and

*l*th techniques.

### 4.2 Combination of different spectral characterization techniques

**r**̂

_{k}be the estimated reflectance of the

*k*th technique, the spectral root mean square (rms) error can then be calculated as

13. R. McDonald and K. J. Smith, “CIE94 - a new colour difference formaula,” J. Soc. Dyers Colour. **111**, 376–379 (1995). [CrossRef]

*δr*is linear to reflectance

**r**, while the colorimetric error

*δe*is nonlinear to

**r**, due to the third-root-square transform between CIEXYZ space and CIELAB space. To account for both spectral and colorimetric accuracy in reflectance reconstruction, these two error terms can be merged into a single error term

*δ*.

*δ*can then be expressed as

*c*≤1 is used to adjust the proportions of spectral and colorimetric errors,

**E**

_{r}and

**E**

_{e}denote the

*K*×1 vectors of the average spectral and colorimetric errors, respectively, and

**w**denotes the

*K*×1 weighting vector. Accordingly, the standard deviation of

*δ*can be expressed as

**∑**

_{r}and ∑

_{e}denote the

*K*×

*K*covariance matrix of the spectral and colorimetric error, respectively. Considering that the spectral and colorimetric errors are always of different magnitudes,

**E**

_{r},

**E**

_{e},

**E**

_{r}, and

**∑**

_{e}are normalized in the calculation of weighting

**w**.

**w**can be obtained by solving the following objective function:

*K*techniques sum to 1, while the second constraint forces positive contribution of each technique.

## 5. Experimental results and discussion

*L*=16 multispectral images of CDC were taken using the filters with center wavelengths at 400 nm, 420 nm, 440 nm, …, 700nm, under an approximate D65 lighting condition. The spatial non-uniformity of the lighting field was corrected by using a white paper [4

**22**, 1231–1240 (2005). [CrossRef]

*w*

_{1}=

*w*

_{2}=

*w*

_{3}=1/3. Nonparametric statistical test indicates that the color difference errors of the proposed methods and the averaging method are smaller than those of the other three methods at a significant level

*p*=0.05. The performance of the averaging method is close to that of the proposed method 1 using objective function (16). The advantage of the proposed method 1 over the averaging method lies that the former can balance the colorimetric accuracy and spectral accuracy by adjusting the

*c*value. The reconstructed spectral reflectance curves with minimum and maximum color difference errors are shown in Fig.2. The proposed method 2 using objective function (17) does not yield better performance, indicating that it may be not feasible to minimize both average color error and standard deviation. Actually, minimizing standard deviation will also decrease the average color error to a certain extent.

## 6. Conclusion

## Acknowledgments

## References and links

1. | J. Y. Hardeberg, “Acquisition and reproduction of color images: colorimetric and multispectral approaches,” Ph.D. dissertation (Ecole Nationale Superieure des Telecommunications, 1999). |

2. | H. L. Shen and J. H. Xin, “Spectral characterization of a color scanner by adaptive estimation,” J. Opt. Soc. Am. A |

3. | H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the spectral reflectance of art paintings,” Appl. Opt. |

4. | V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, “Characterization of trichromatic color cameras by using a new multispectral imaging technique,” J. Opt. Soc. Am. A |

5. | M. Shi and G. Healey, “Using reflectance models for color scanner calibration,” J. Opt. Soc. Am. A |

6. | F. H. Imai and R. S. Berns, “Spectral estimation using trichromaitc digital cameras,” in Proceedings of the International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives (Society of Multispectral Imaging of Japan, Chiba, Japan, 1999). |

7. | M. Soriano, W. Oblefias, and C. Saloma, “Fluorescence spectrum estimation using multiple color images and minimum negativity constraint,” Opt. Express , |

8. | V. Cardei and B. Funt, “Committee-based color constancy,” in Proc. IS&T/SID Seventh Color Imaging Conf: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Virginia, 1999), pp.311–313. |

9. | H. Stokman and T. Gevers, “Selection and fusion of color models for image feature detection,” IEEE Trans. Pattern Anal. Mach. Intell. |

10. | K. Barnard and B. Funt, “Camera characterization for color research,” Color Res. Appl. |

11. | W. K. Pratt, |

12. | L. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A |

13. | R. McDonald and K. J. Smith, “CIE94 - a new colour difference formaula,” J. Soc. Dyers Colour. |

**OCIS Codes**

(110.4190) Imaging systems : Multiple imaging

(330.1710) Vision, color, and visual optics : Color, measurement

(330.1730) Vision, color, and visual optics : Colorimetry

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 5, 2007

Revised Manuscript: March 13, 2007

Manuscript Accepted: April 9, 2007

Published: April 20, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Hui-Liang Shen, John H. Xin, and Si-Jie Shao, "Improved reflectance reconstruction for multispectral imaging by combining different techniques," Opt. Express **15**, 5531-5536 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-9-5531

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### References

- J. Y. Hardeberg, "Acquisition and reproduction of color images: colorimetric and multispectral approaches," Ph.D. dissertation (Ecole Nationale Superieure des Telecommunications, 1999).
- H. L. Shen and J. H. Xin, "Spectral characterization of a color scanner by adaptive estimation," J. Opt. Soc. Am. A 21, 1125-1130 (2004). [CrossRef]
- H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, "System design for accurately estimating the spectral reflectance of art paintings," Appl. Opt. 39, 6621-6632 (2000). [CrossRef]
- V. Cheung, S. Westland, C. Li, J. Hardeberg, and D. Connah, "Characterization of trichromatic color cameras by using a new multispectral imaging technique," J. Opt. Soc. Am. A 22, 1231-1240 (2005). [CrossRef]
- M. Shi and G. Healey, "Using reflectance models for color scanner calibration," J. Opt. Soc. Am. A 19, 645-656 (2002). [CrossRef]
- F. H. Imai and R. S. Berns, "Spectral estimation using trichromaitc digital cameras," in Proceedings of the International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives (Society of Multispectral Imaging of Japan, Chiba, Japan, 1999).
- M. Soriano, W. Oblefias, and C. Saloma, "Fluorescence spectrum estimation using multiple color images and minimum negativity constraint," Opt. Express 16, 1458-1464 (2002).
- V. Cardei and B. Funt, "Committee-based color constancy," in Proc. IS&T/SID Seventh Color Imaging Conf.: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Virginia, 1999), pp. 311-313.
- H. Stokman and T. Gevers, "Selection and fusion of color models for image feature detection," IEEE Trans. Pattern Anal. Mach. Intell. 29, 371-381 (2007). [CrossRef] [PubMed]
- K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 152-163 (2002). [CrossRef]
- W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
- L. Maloney, "Evaluation of linear models of surface spectral reflectance with small numbers of parameters," J. Opt. Soc. Am. A 3, 1673-1683 (1986). [CrossRef] [PubMed]
- R. McDonald and K. J. Smith, "CIE94 - a new colour difference formaula," J. Soc. Dyers Colour. 111, 376-379 (1995). [CrossRef]

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