Abstract

We propose a novel image-recovery method using the covariance matrix of the red–green–blue (R–G–B) color histogram and tensor theories. The image-recovery method is called the color histogram normalization algorithm. It is known that the color histograms of an image taken under varied illuminations are related by a general affine transformation of the R–G–B coordinates when the illumination is changed. We propose a simplified affine model for application with illumination variation. This simplified affine model considers the effects of only three basic forms of distortion: translation, scaling, and rotation. According to this principle, we can estimate the affine transformation matrix necessary to recover images whose color distributions are varied as a result of illumination changes. We compare the normalized color histogram of the standard image with that of the tested image. By performing some operations of simple linear algebra, we can estimate the matrix of the affine transformation between two images under different illuminations. To demonstrate the performance of the proposed algorithm, we divide the experiments into two parts: computer-simulated images and real images corresponding to illumination changes. Simulation results show that the proposed algorithm is effective for both types of images. We also explain the noise-sensitive skew-rotation estimation that exists in the general affine model and demonstrate that the proposed simplified affine model without the use of skew rotation is better than the general affine model for such applications.

© 2001 Optical Society of America

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References

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  1. M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
    [CrossRef]
  18. D. Paulus, L. Csink, H. Niemann, “Color cluster rotation,” in Proceedings of the 1998 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 161–165.
  19. S. C. Pei, C. N. Lin, “Image normalization for pattern recognition,” Image Vision Comput. 13, 711–723 (1995).
    [CrossRef]
  20. D. Cyganski, J. A. Orr, “Application of tensor theory to object recognition and orientation determination,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 662–673 (1985).
    [CrossRef]
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  23. S. C. Pei, C. L. Tseng, “Illuminant-invariant color image recognition using color histogram normalization and block average feature,” manuscript available from the authors.
  24. G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.
  25. C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.
  26. A. Z. Kouzani, F. He, K. Sammut, “Quadtree principal component analysis and its application to facial expression classification,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. 6, pp. 835–839.
  27. T. Kawatani, “Handwritten kanji recognition using combined complementary classifiers in a cascade arrangement,” in Proceedings of the Fifth International Conference on Document Analysis and Recognition (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 503–506.
  28. S. C. Pei, C. C. Wu, “Normalization of 3-D objects under general affine transformations,” manuscript available from the authors.
  29. C. C. Wu, “2-D Images and 3-D objects normalization using moment-based covariance matrices” (Master’s Thesis, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, 2000).

1997 (2)

D. Slater, G. Healey, “The illumination-invariant matching of deterministic local structure in color images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1146–1151 (1997).
[CrossRef]

G. Healey, D. Slater, “Computing illumination-invariant descriptors of spatially filtered color image regions,” IEEE Trans. Image Process. 6, 1002–1013 (1997).
[CrossRef] [PubMed]

1996 (2)

I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
[CrossRef]

D. Slater, G. Healey, “The illumination-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

1995 (2)

B. Funt, G. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

S. C. Pei, C. N. Lin, “Image normalization for pattern recognition,” Image Vision Comput. 13, 711–723 (1995).
[CrossRef]

1994 (1)

1991 (2)

G. Healey, “Estimating spectral reflectance using highlights,” Image Vision Comput. 9, 333–337 (1991).
[CrossRef]

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

1989 (1)

1987 (1)

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

1986 (3)

1985 (1)

D. Cyganski, J. A. Orr, “Application of tensor theory to object recognition and orientation determination,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 662–673 (1985).
[CrossRef]

1971 (1)

1964 (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Ballard, D.

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

Bebis, G.

G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Coleman, T.

C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.

Csink, L.

D. Paulus, L. Csink, H. Niemann, “Color cluster rotation,” in Proceedings of the 1998 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 161–165.

Cyganski, D.

D. Cyganski, J. A. Orr, “Application of tensor theory to object recognition and orientation determination,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 662–673 (1985).
[CrossRef]

D’Zmura, M.

da Vitoria Lobo, N.

G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.

Fahsi, A.

C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.

Finlayson, G.

B. Funt, G. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

Funt, B.

B. Funt, G. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

Georgiopoulos, M.

G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.

Gose, E.

E. Gose, R. Johnsonbaugh, S. Jost, Pattern Recognition and Image Analysis (Prentice-Hall PTR, Upper Saddle River, N.J., 1996).

Hallikainen, J.

He, F.

A. Z. Kouzani, F. He, K. Sammut, “Quadtree principal component analysis and its application to facial expression classification,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. 6, pp. 835–839.

Healey, G.

G. Healey, D. Slater, “Computing illumination-invariant descriptors of spatially filtered color image regions,” IEEE Trans. Image Process. 6, 1002–1013 (1997).
[CrossRef] [PubMed]

D. Slater, G. Healey, “The illumination-invariant matching of deterministic local structure in color images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1146–1151 (1997).
[CrossRef]

D. Slater, G. Healey, “The illumination-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

G. Healey, D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003–3010 (1994).
[CrossRef]

G. Healey, “Estimating spectral reflectance using highlights,” Image Vision Comput. 9, 333–337 (1991).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1987).

Hung, C. C.

C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.

Jaaskelainen, T.

Johnsonbaugh, R.

E. Gose, R. Johnsonbaugh, S. Jost, Pattern Recognition and Image Analysis (Prentice-Hall PTR, Upper Saddle River, N.J., 1996).

Jost, S.

E. Gose, R. Johnsonbaugh, S. Jost, Pattern Recognition and Image Analysis (Prentice-Hall PTR, Upper Saddle River, N.J., 1996).

Kawatani, T.

T. Kawatani, “Handwritten kanji recognition using combined complementary classifiers in a cascade arrangement,” in Proceedings of the Fifth International Conference on Document Analysis and Recognition (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 503–506.

Kouzani, A. Z.

A. Z. Kouzani, F. He, K. Sammut, “Quadtree principal component analysis and its application to facial expression classification,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. 6, pp. 835–839.

Land, E. H.

Lee, S. W.

S. Lin, S. W. Lee, “Using chromaticity distributions and eigenspace analysis for pose-, illumination-, and specularity-invariant recognition of 3D objects,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 426–431.

Lennie, P.

Lin, C. N.

S. C. Pei, C. N. Lin, “Image normalization for pattern recognition,” Image Vision Comput. 13, 711–723 (1995).
[CrossRef]

Lin, S.

S. Lin, S. W. Lee, “Using chromaticity distributions and eigenspace analysis for pose-, illumination-, and specularity-invariant recognition of 3D objects,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 426–431.

Maloney, L.

McCann, J. J.

Niemann, H.

D. Paulus, L. Csink, H. Niemann, “Color cluster rotation,” in Proceedings of the 1998 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 161–165.

Oja, E.

E. Oja, Subspace Methods for Pattern Recognition (Research Studies Press, Letchworth, Hertfordshire, England, 1983).

Orr, J. A.

D. Cyganski, J. A. Orr, “Application of tensor theory to object recognition and orientation determination,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 662–673 (1985).
[CrossRef]

Parkkinen, J. P. S.

Paulus, D.

D. Paulus, L. Csink, H. Niemann, “Color cluster rotation,” in Proceedings of the 1998 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 161–165.

Pei, S. C.

S. C. Pei, C. N. Lin, “Image normalization for pattern recognition,” Image Vision Comput. 13, 711–723 (1995).
[CrossRef]

S. C. Pei, C. L. Tseng, “Illuminant-invariant color image recognition using color histogram normalization and block average feature,” manuscript available from the authors.

S. C. Pei, C. C. Wu, “Normalization of 3-D objects under general affine transformations,” manuscript available from the authors.

Rothe, I.

I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
[CrossRef]

Sammut, K.

A. Z. Kouzani, F. He, K. Sammut, “Quadtree principal component analysis and its application to facial expression classification,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. 6, pp. 835–839.

Shah, M.

G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.

Slater, D.

G. Healey, D. Slater, “Computing illumination-invariant descriptors of spatially filtered color image regions,” IEEE Trans. Image Process. 6, 1002–1013 (1997).
[CrossRef] [PubMed]

D. Slater, G. Healey, “The illumination-invariant matching of deterministic local structure in color images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1146–1151 (1997).
[CrossRef]

D. Slater, G. Healey, “The illumination-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

G. Healey, D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003–3010 (1994).
[CrossRef]

Suesse, H.

I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
[CrossRef]

Swain, M.

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

Tadesse, W.

C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.

Tseng, C. L.

S. C. Pei, C. L. Tseng, “Illuminant-invariant color image recognition using color histogram normalization and block average feature,” manuscript available from the authors.

Voss, K.

I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
[CrossRef]

Wandell, B.

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

L. Maloney, B. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

Wu, C. C.

S. C. Pei, C. C. Wu, “Normalization of 3-D objects under general affine transformations,” manuscript available from the authors.

C. C. Wu, “2-D Images and 3-D objects normalization using moment-based covariance matrices” (Master’s Thesis, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, 2000).

IEEE Trans. Image Process. (1)

G. Healey, D. Slater, “Computing illumination-invariant descriptors of spatially filtered color image regions,” IEEE Trans. Image Process. 6, 1002–1013 (1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (6)

D. Slater, G. Healey, “The illumination-invariant matching of deterministic local structure in color images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1146–1151 (1997).
[CrossRef]

B. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

I. Rothe, K. Voss, H. Suesse, “The method of normalization to determine invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 366–376 (1996).
[CrossRef]

B. Funt, G. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

D. Cyganski, J. A. Orr, “Application of tensor theory to object recognition and orientation determination,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 662–673 (1985).
[CrossRef]

D. Slater, G. Healey, “The illumination-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

Image Vision Comput. (2)

S. C. Pei, C. N. Lin, “Image normalization for pattern recognition,” Image Vision Comput. 13, 711–723 (1995).
[CrossRef]

G. Healey, “Estimating spectral reflectance using highlights,” Image Vision Comput. 9, 333–337 (1991).
[CrossRef]

Int. J. Comput. Vision (1)

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Psychon. Sci. (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Other (12)

D. Paulus, L. Csink, H. Niemann, “Color cluster rotation,” in Proceedings of the 1998 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 161–165.

S. Lin, S. W. Lee, “Using chromaticity distributions and eigenspace analysis for pose-, illumination-, and specularity-invariant recognition of 3D objects,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 426–431.

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1987).

E. Oja, Subspace Methods for Pattern Recognition (Research Studies Press, Letchworth, Hertfordshire, England, 1983).

E. Gose, R. Johnsonbaugh, S. Jost, Pattern Recognition and Image Analysis (Prentice-Hall PTR, Upper Saddle River, N.J., 1996).

S. C. Pei, C. L. Tseng, “Illuminant-invariant color image recognition using color histogram normalization and block average feature,” manuscript available from the authors.

G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, M. Shah, “Learning affine transformations of the plane for model-based object recognition,” in Proceedings of the 13th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 4, pp. 60–64.

C. C. Hung, A. Fahsi, W. Tadesse, T. Coleman, “A comparative study of remotely sensed data classification using principal components analysis and divergence,” in Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 3, pp. 2444–2449.

A. Z. Kouzani, F. He, K. Sammut, “Quadtree principal component analysis and its application to facial expression classification,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. 6, pp. 835–839.

T. Kawatani, “Handwritten kanji recognition using combined complementary classifiers in a cascade arrangement,” in Proceedings of the Fifth International Conference on Document Analysis and Recognition (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 503–506.

S. C. Pei, C. C. Wu, “Normalization of 3-D objects under general affine transformations,” manuscript available from the authors.

C. C. Wu, “2-D Images and 3-D objects normalization using moment-based covariance matrices” (Master’s Thesis, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, 2000).

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Figures (8)

Fig. 1
Fig. 1

Diagram of color histogram normalization.

Fig. 2
Fig. 2

Basket of fruit, one original (real) image (top left) taken under standard illumination and three computer-simulated images taken under yellow (top right), green (bottom left), and blue (bottom right) illumination. In each set: (b) color histogram of (a), (c) normalized histogram of (b), (d) recovered image by use of the color histogram normalization algorithm.

Fig. 3
Fig. 3

(a) “Suonatore di Differaro,” one original (real) image (top left) taken under standard illumination and three computer-simulated images taken under darker (top right), red (bottom left), and magenta (bottom left) illumination. In each set: (b) color histogram of (a), (c) normalized histogram of (b), (d) recovered image by use of the color histogram normalization algorithm.

Fig. 4
Fig. 4

Sketched map of the obtaining of the real images taken under different illuminations.

Fig. 5
Fig. 5

(a) Basket of fruit, four real images taken under lighter (top left), yellow (top right), green (bottom left), and blue (bottom right) illumination. In each set: (b) color histogram of (a), (c) normalized histogram of (b), (d) recovered image by use of the color histogram normalization algorithm.

Fig. 6
Fig. 6

(a) “Suonatore di Differaro,” four real images taken under lighter (top left), darker (top right), red (bottom left), and magenta (bottom right) illumination. In each set: (b) color histogram of (a), (c) normalized histogram of (b), (d) recovered image by use of the color histogram normalization algorithm.

Fig. 7
Fig. 7

Comparison of the color histogram normalization algorithm with two different affine models: (a) original image, (b) real image (under yellow illumination), (c) color histogram of (a),(d) color histogram of (b),(e) color histogram of the recovered image whose algorithm is considered in the proposed simplified affine model, (f) color histogram of the recovered image whose algorithm is considered in the general affine model, (g) color histogram of the companison between the original image and the recovered image by use of the proposed simplified affine model, (h) color histogram of the comparison between the original image and the recovered image by use of the general affine model. The PSNR¯ of (e) is ∼9 dB higher than that of (f).

Fig. 8
Fig. 8

Comparison of the difference in the estimation of the affine transformation coefficient by the simplified affine model and the general affine model. It can be seen that the former is a little more accurate than the latter. So the estimation of the coefficient by using either simplified affine model or general affine model is feasible for noise-free case of such illuminant-changing application. Note that PSNR¯ of recovered image by using color histogram normalization algorithm considered in the simplified affine model is 47.8546, and PSNR¯ of recovered image considered in the general affine model is 47.7566; the difference 0.1 dB maybe is due to the quantized truncation error during computer calculation.

Tables (9)

Tables Icon

Table 1 PSNR Values of The Basket of Fruit with Computer-Simulated Images (Unit: dB)

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Table 2 PSNR Values of “Suonatore di Differaro” with Computer-Simulated Images (Unit: dB)

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Table 3 PSNR Values of “Vaso di Fiori” with Computer-Simulated Images (Unit: dB)

Tables Icon

Table 4 PSNR Values of The Basket of Fruit with Real Images (Unit: dB)

Tables Icon

Table 5 PSNR Values of “Suonatore di Differaro” with Real Images (Unit: dB)

Tables Icon

Table 6 PSNR Values of “Vaso di Fiori” with Real Images (Unit: dB)

Tables Icon

Table 7 Comparison of PSNR¯ Values of “The Basket of Fruit” with Two Different Affine Models (Real Image)

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Table 8 Comparison of PSNR¯ Values of “Suonatore di Differaro” with Two Different Affine Models (Real Image)

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Table 9 Comparison of PSNR¯ Values of “Vaso di Fiori” with Two Different Affine Models (Real Image)

Equations (66)

Equations on this page are rendered with MathJax. Learn more.

ρi(x, y)=λl(λ)s(x, y, λ)fi(λ)dλ,1in,
s(x, y, λ)=j=1mσj(x, y)Sj(λ),
ρ(x, y)=Aσ(x, y),
Aij=l(λ)Sj(λ)fi(λ)dλ.
ρ(x, y)=Aσ(x, y),ρ˜(x, y)=A˜σ(x, y).
ρ˜(x, y)=Mρ(x, y),
H(ρ˜)=H(Mρ).
p(r, g, b)=1ifthecolorexists0ifthecolordoesnotexist.
f(r, g, b)=p(r, g, b)Ωp(r, g, b)dV,
c=[CrCgCb]T,
Cr=Ωrf(r, g, b)dV,
Cg=Ωgf(r, g, b)dV,
Cb=Ωbf(r, g, b)dV,
μijk=E[(r-Cr)i(g-Cg)j(b-Cb)k]=Ω(r-Cr)i(g-Cg)j(b-Cb)kf(r, g, b)dV.
M=μ200μ110μ101μ110μ020μ011μ101μ011μ002.
uvw=a11a12a13a21a22a23a31a32a33rgb+b1b2b3,
u=Ar+b.
M=AMAT
A=a11a12a13a21a22a23a31a32a33.
e3=e1×e2.
(e1 e2 e3),(e1-e2-e3), (-e1 e2-e3),(-e1-e2 e3).
M=AMAT=a2M.
AT=A-1
M=AMAT=AMA-1,
E=A-1E or E=-A-1E,
E=e1Te2Te3T,E=e1Te2Te3T.
E1=e1Te2Te3T=e1re1ge1be2re2ge2be3re3ge3b,
E2=e1T-e2T-e3T=e1re1ge1b-e2r-e2g-e2b-e3r-e3g-e3b,
E3=-e1Te2T-e3T=-e1r-e1g-e1be2re2ge2b-e3r-e3g-e3b,
E4=-e1T-e2Te3T=-e1r-e1g-e1b-e2r-e2g-e2be3re3ge3b.
rgb=E1r-Crg-Cgb-Cb,iin{1, 2, 3, 4}.
u-Cuv-Cvw-Cw=a11a12a13a21a22a23a31a32a33r-Crg-Cgb-Cb.
y1y2y3=A11A21A31A12A22A32A13A23A33x1x2x3,
y1y2y3=u-Cuv-Cvw-Cw,x1x2x3=r-Crg-Cgb-Cb, A11A21A31A12A22A32A13A23A33=a11a12a13a21a22a23a31a32a33.
yi=Ajixjfori=1, 2, 3.
xi=ajiyjfori=1, 2, 3,
Ajiakj=1ifi=k0ifik=δik.
Tijk=Ωxixjxk  f(x1, x2)dx1dx2,
Tijk=Ωxixjxk  f(x1, x2)dx1dx2,
T¯ijk=yiyjyk  f(y1, y2)dy1dy2,
yiyjyk =AliAmjAnk  xlxmxn 
T¯iT¯jT¯k =AliAmjAnk  TlTmTn .
T¯ijk(A)=AliAmjAnkTlmn,
T¯111(E1)=T¯111(E2)=-T¯111(E3)=-T¯111(E4),
T¯222(E1)=-T¯222(E2)=T¯222(E3)=-T¯222(E4).
Choose E in{E1 E2 E3 E4}suchthat T¯111>0andT¯222>0simultaneously.
rgb=Er-Crg-Cgb-Cb.
λi=a2λi,i=1, 2, 3.
rgb=cλ Er-Crg-Cgb-Cb.
If(T¯111<0),then e1=-e1.
If(T¯222<0),then e2=-e2.
e3=e1×e2, where×denotescrossproduct.
rgb=cλ Er-Crg-Cgb-Cb.
rgb=a11a12a13a21a22a23a31a32a33rgb+b1b2b3.
uvw=cλo Eor-Crg-Cgb-Cb.
uvw=cλa Ear-Crg-Cgb-Cb
r-Crg-Cgb-Cb=λaλo EaTEor-Crg-Cgb-Cb.
G=λaλo EaTEo.
rgb=Grgb+CrCgCb-GCrCgCb.
a11a12a13a21a22a23a31a32a33=G,
b1b2b3=CrCgCb-GCrCgCb.
rgb=G-1rgb+CrCgCb-G-1CrCgCb.
PSNR_R=i=1128j=1128[R(i, j)-R(i, j)]2/(128×128×max_R2),
PSNR_G=i=1128j=1128[G(i, j)-G(i, j)]2/(128×128×max_G2),
PSNR_B=i=1128j=1128[B(i, j)-B(i, j)]2/(128×128×max_B2),
PSNR¯=(PSNR_R+PSNR_G+PSNR_B)/3,

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