Abstract

It is possible to capture images with high-fidelity color by using a spectrum-based color image reproduction method that estimates the spectral information of objects from a multispectral image along with information from illumination. When multispectral images do not have a sufficient signal-to-noise ratio (SNR), the accuracy of the spectral and color estimation will be reduced. To improve color estimation accuracy, this paper proposes a spatio-spectral Wiener estimation, which uses spatial correlation as well as spectral correlation. In order to show the effectiveness of the proposed method, computer simulations and an experiment are carried out using a six-channel video camera. As a result, it is assured that the proposed method improves color estimation accuracy and suppresses color noise.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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2007

2006

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

N. Shimano, "Recovery of spectral reflectances of objects being imaged without prior knowledge," IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

2005

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

2004

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

2002

2001

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

2000

1997

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

1993

A. K. Katsaggelos, K. T. Lay, and N. P. Galatsanos, "A general framework for frequency domain multi-channel signal processing," IEEE Trans. Image Process. 2, 417-420 (1993).
[CrossRef] [PubMed]

1992

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

1991

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

1989

N. P. Galatsanos and R. T. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

1984

B. R. Hunt and O. Kubler, "Karhunen-Loeve multispectral image restoration, part I: Theory," IEEE Trans. Acoust., Speech Signal Process. 32, 592-600 (1984).
[CrossRef]

Ajito, T.

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

Cai, P.

Chin, R. T.

N. P. Galatsanos and R. T. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

Erdem, A. T.

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

Fukuda, H.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

Galatsanos, N. P.

A. K. Katsaggelos, K. T. Lay, and N. P. Galatsanos, "A general framework for frequency domain multi-channel signal processing," IEEE Trans. Image Process. 2, 417-420 (1993).
[CrossRef] [PubMed]

N. P. Galatsanos and R. T. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

Haneishi, H.

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

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

Hasegawa, T.

Hideaki, H.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

Hill, B.

B. Hill, "Color capture, color management and the problem of metamerism," Proc. SPIE 3963, 3-14 (2000).

Honda, T.

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Hosoi, A.

Hunt, B. R.

B. R. Hunt and O. Kubler, "Karhunen-Loeve multispectral image restoration, part I: Theory," IEEE Trans. Acoust., Speech Signal Process. 32, 592-600 (1984).
[CrossRef]

Iwama, R.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Kanazawa, H.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

Katsaggelos, A. K.

A. K. Katsaggelos, K. T. Lay, and N. P. Galatsanos, "A general framework for frequency domain multi-channel signal processing," IEEE Trans. Image Process. 2, 417-420 (1993).
[CrossRef] [PubMed]

Komiya, Y.

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Koshimoto, J.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

Kubler, O.

B. R. Hunt and O. Kubler, "Karhunen-Loeve multispectral image restoration, part I: Theory," IEEE Trans. Acoust., Speech Signal Process. 32, 592-600 (1984).
[CrossRef]

Lay, K. T.

A. K. Katsaggelos, K. T. Lay, and N. P. Galatsanos, "A general framework for frequency domain multi-channel signal processing," IEEE Trans. Image Process. 2, 417-420 (1993).
[CrossRef] [PubMed]

Miyake, Y.

Motomura, H.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

Murakami, Y.

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, "Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction," Appl. Opt. 41, 4840-4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

Obi, T.

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, "Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction," Appl. Opt. 41, 4840-4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Ohsawa, K.

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

Ohya, Y.

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Ohyama, N.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, "Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction," Appl. Opt. 41, 4840-4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Ozkan, M. K.

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

Sezan, M. I.

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

Shao, S.

Shen, H.

Shimano, N.

N. Shimano, "Recovery of spectral reflectances of objects being imaged without prior knowledge," IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

Suryani, L.

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Tanaka, K.

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Tekalp, A. M.

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

Teraji, T.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

Tsuchida, M.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

Tsumura, N.

Uchiyama, T.

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

Wada, T.

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Xin, J.

Yamaguchi, M.

M. Yamaguchi, H. Hideaki, H. Fukuda, J. Koshimoto, H. Kanazawa, M. Tsuchida, R. Iwama, and N. Ohyama, "High-fidelity video and still-image communication based on spectral information: Natural Vision system and its applications," Proc. SPIE 6062, 129-140 (2006).

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, "Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction," Appl. Opt. 41, 4840-4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation," Proc. SPIE 4663, 15-26 (2001).
[CrossRef]

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

Yamaguchi, S. M.

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Yokoyama, Y.

Appl. Opt.

IEEE Trans. Acoust., Speech Signal Process.

B. R. Hunt and O. Kubler, "Karhunen-Loeve multispectral image restoration, part I: Theory," IEEE Trans. Acoust., Speech Signal Process. 32, 592-600 (1984).
[CrossRef]

N. P. Galatsanos and R. T. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

IEEE Trans. Image Process.

N. Shimano, "Recovery of spectral reflectances of objects being imaged without prior knowledge," IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

M. K. Ozkan, A. T. Erdem, M. I. Sezan, and A. M. Tekalp, "Efficient multiframe Wiener restoration of blurred and noisy image sequences," IEEE Trans. Image Process. 1, 453-476 (1992).
[CrossRef] [PubMed]

A. K. Katsaggelos, K. T. Lay, and N. P. Galatsanos, "A general framework for frequency domain multi-channel signal processing," IEEE Trans. Image Process. 2, 417-420 (1993).
[CrossRef] [PubMed]

J. Imaging Sci. Technol.

K. Ohsawa, T. Ajito, H. Fukuda, Y. Komiya, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Six-band HDTV camera system for spectrum-based color reproduction," J. Imaging Sci. Technol. 48, 85-92 (2004).

Opt. Commun.

L. Suryani, S. M. Yamaguchi, N. Ohyama, T. Honda, and K. Tanaka, "Estimation of an image sampled by a CCD sensor array using a color synthetic method," Opt. Commun. 84, 133-138 (1991).
[CrossRef]

Opt. Express

Proc. SPIE

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, "Development of 16-band multispectral image archiving system," Proc. SPIE 5667, 136-145 (2005).
[CrossRef]

B. Hill, "Color capture, color management and the problem of metamerism," Proc. SPIE 3963, 3-14 (2000).

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicine," Proc. SPIE 3031, 482-489 (1997).
[CrossRef]

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[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Block diagrams of four different approaches for multispectral-based color image reproduction with noise restoration.

Fig. 2.
Fig. 2.

Spectral sensitivity of six-band video camera used in simulations.

Fig. 3.
Fig. 3.

Spectral power distribution of D65 and F2 illuminants.

Fig. 4.
Fig. 4.

SNR of each of the six-channel images under simulation conditions.

Fig. 5.
Fig. 5.

Reconstructed peak L* value of a one-pixel white object used as a spatial resolution measure. Less resolution measure implies more resolution degradation.

Fig. 6.
Fig. 6.

Spectral reflectance images used in the simulations: Toy (left) and Scarf (right). Images displayed in sRGB.

Fig. 7.
Fig. 7.

Simulation results of comparative evaluation among four methods, 1D, 1D+2D, 2D+1D, and 3D: (a) ΔE, (b) σL , (c) σa , and (d) σb as a function of spatial resolution measure. Filters corresponding to plots in dotted squares are applied for reproduction of natural scene images.

Fig. 8.
Fig. 8.

65×65 pixel area of original and reconstructed images of Scarf. Monochrome images are B-channel of sRGB images. Random noise seen in 1D reconstruction is reduced in 1D+2D, 2D+1D, and 3D reconstructions. Details in left angel are reconstructed most clearly by 3D.

Fig. 9.
Fig. 9.

Spectral sensitivity of 6-band video camera used in experiments.

Fig. 10.
Fig. 10.

Spectral power distributions of daylight and fluorescent illuminants used in experiments.

Fig. 11.
Fig. 11.

SNR of each of the six-channel images under the experimental conditions.

Fig. 12.
Fig. 12.

Experimental results under high and low SNR cases: (a) ΔE, (b) σL , (c) σa , and (d) σb . Each filter is designed to give approximately the same spatial resolution measurements as shown in Table 2.

Fig. 13.
Fig. 13.

Examples of the reproduced images under fluorescent illumination: (a) f/5 (high SNR) by 1D, (b) f/8 (low SNR) by 1D, and (c) f/8 (low SNR) by 3D. The top image is shown as a reference. Comparing two images of low SNR in the bottom row, the color reproducibility in the image of 3D is improved compared to 1D, while the noise is not completely eliminated by 3D.

Tables (2)

Tables Icon

Table 1. Average and Maximum ΔE* ab for Natural Scene Images, Toy and Scarf

Tables Icon

Table 2. Spatial Resolution Measures and Spatial Correlation Coefficients in Experimental Settings

Equations (29)

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g = Hf + n ,
f ̂ = Ag ,
A = fg T gg T 1
= ff T H T ( H ff T H T + nn T ) 1 ,
x = ( x cent ( M 2 1 ) 2 x cent x cent + ( M 2 1 ) 2 ) , y = ( y cent ( M 2 1 ) 2 y cent y cent + ( M 2 1 ) 2 ) .
y = x + η .
x ̂ cent = b T y ,
b T = x cent y T yy T 1
= x cent x T ( xx T + η η T ) 1 .
𝓕 = ( f cent ( M 2 1 ) 2 f cent f cent + ( M 2 1 ) 2 ) , 𝓖 = ( g cent ( M 2 1 ) 2 g cent g cent + ( M 2 1 ) 2 ) , 𝓝 = ( n cent ( M 2 1 ) 2 n cent n cent + ( M 2 1 ) 2 ) ,
𝓖 = ( H H H ) 𝓕 + 𝓝 ,
= ( I M 2 H ) 𝓕 + 𝓝 ,
f ̂ cent = C 𝓖 ,
C = f cent 𝓖 T 𝓖 𝓖 T 1
= f cent 𝓕 T ( I M 2 H ) T [ ( I M 2 H ) 𝓕 𝓕 T ( I M 2 H ) T + 𝓝 𝓝 T ] 1 .
𝓕 𝓕 T = xx ¯ T ff T ,
SNR 20 log 10 ( signal of the white object standard deviation of the noise ) [ dB ] .
ff T = 1 24 i = 1 24 m i m i T ,
xx T = σ 2 [ R ( ρ 1 ) R ( ρ 2 ) ]
= σ 2 ( ρ 1 0 R ( ρ 2 ) ρ 1 1 R ( ρ 2 ) ρ 1 M 1 R ( ρ 2 ) ρ 1 1 R ( ρ 2 ) ρ 1 0 R ( ρ 2 ) ρ 1 M 2 R ( ρ 2 ) ρ 1 M 1 R ( ρ 2 ) ρ 1 M 2 R ( ρ 2 ) ρ 1 0 R ( ρ 2 ) ) ,
R ( ρ ) = ( ρ 0 ρ 1 ρ M 1 ρ 1 ρ 0 ρ M 2 ρ M 1 ρ M 2 ρ 0 ) ,
x cent = p T x ,
b T = p T xx T xx T 1 = p T ,
f cent = P 𝓕 , P = ( 0 L , , 0 L , I L , 0 L , , 0 L ) ,
C = P 𝓕 𝓕 T ( I M 2 H ) T [ ( I M 2 H ) 𝓕 𝓕 T ( I M 2 H ) T ] 1 .
C = P ( xx ¯ T ff T ) ( I M 2 H ) T [ ( I M 2 H ) ( xx ¯ T ff T ) ( I M 2 H ) T ] 1
= P [ xx ¯ T ( ff T H T ) ] [ xx ¯ T ( H ff T H T ) ] 1
= P { I M 2 [ ff T H T ( H ff T H T ) 1 ] }
= ( 0 L , , 0 L , ff T H T ( H ff T H T ) 1 , 0 L , , 0 L ) .

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