Abstract

Multifocus color image fusion is an active research area in image processing, and many fusion algorithms have been developed. However, the existing techniques can hardly deal with the problem of image blur. This study present a novel fusion approach that integrates the quaternion with traditional curvelet transform to overcome the above disadvantage. The proposed method uses a multiresolution analysis procedure based on the quaternion curvelet transform. Experimental results show that the proposed method is promising, and it does significantly improve the fusion quality compared to the existing fusion methods.

© 2012 OSA

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  1. X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
    [Crossref]
  2. S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
    [Crossref]
  3. Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
    [Crossref]
  4. Q. Zhang and B. Guo, “Multifocus image fusion using the nonsubsampled contourlet transform,” Signal Process. 89(7), 1334–1346 (2009).
    [Crossref]
  5. W. Huang and Z. L. Jing, “Multifocus image fusion using pulse coupled neutral network,” Pattern Recogn. lett. 28(9), 1123–1132 (2007).
    [Crossref]
  6. N. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion algorithm based on shearlets,” Chin. Opt. Lett. 9(4), 041001 (2011).
  7. N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
    [Crossref]
  8. W. Yajie and X. Xinhe, “A multifocus image fusion new method based on multidecision,” Proc. SPIE 6357, 63570G (2006).
    [Crossref]
  9. R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
    [Crossref]
  10. I. De and B. Chanda, “A simple and efficient algorithm for multifocus image fusion using morphological wavelets,” Signal Process. 86(5), 924–936 (2006).
    [Crossref]
  11. S. Li and B. Yang, “Multifocus image fusion using region segmentation and spatial frequency,” Image Vis. Comput. 26(7), 971–979 (2008).
    [Crossref]
  12. H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
    [Crossref]
  13. Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
    [Crossref]
  14. S. Gabarda and G. Cristóbal, “Multifocus image fusion through pseudo-Wigner distribution,” Opt. Eng. 44(4), 047001 (2005).
    [Crossref]
  15. P. L. Lin and P. Y. Huang, “Fusion methods based on dynamic-segmented morphological wavelet or cut and paste for multifocus images,” Signal Process. 88(6), 1511–1527 (2008).
    [Crossref]
  16. Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
    [Crossref]
  17. R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
    [Crossref]
  18. F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
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  19. A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
    [Crossref]
  20. Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
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  21. H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
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  22. W. Huang and Z. Jing, “Evaluation of focus measures in multi-focus image fusion,” Pattern Recogn. Lett. 28(9), 493–500 (2007).
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  23. R. Maruthi, “Spatial Domain Method for Fusing Multi-Focus Images using Measure of Fuzziness,” Int. J. Comput. Appl. 20(7), 48–57 (2011).
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  25. Y. Chen, L. Wang, Z. Sun, Y. Jiang, and G. Zhai, “Fusion of color microscopic images based on bidimensional empirical mode decomposition,” Opt. Express 18(21), 21757–21769 (2010).
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  27. Z. Zhang and R. S. Blum, “A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application,” Proc. IEEE.  87(8), 1315–1326 (1999).
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  28. K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
    [Crossref]
  29. S. J. Sangwine, “Fourier transforms of colour images using quaternion, or hypercomplex numbers,” Electron. Lett. 32(1), 1979–1980 (1996).
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  30. S. C. Pei and C. M. Cheng, “Color image processing by using binary quaternion-moment-preserving thresholding technique,” IEEE Trans. Signal Process. 8(5), 614–628 (1999).
  31. T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Trans. Image Process. 16(1), 22–35 (2007).
    [Crossref] [PubMed]
  32. D. S. Alexiadis and G. D. Sergiadis, “Estimation of motions in color image sequences using hypercomplex Fourier transforms,” IEEE Trans. Sig. Process. 18(1), 168–186 (2009).
  33. S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
    [Crossref]
  34. L. Q. Guo and M. Zhu, “Quaternion Fourier-Mellin moments for color images,” Pattern Recogn. 44(2), 187–195 (2011).
    [Crossref]
  35. B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
    [Crossref]
  36. S. Sangwine and N. L. Bihan, “Quaternion toolbox for Matlab,” http://qtfm.sourceforge.net
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    [Crossref]
  38. E. J. Candès and D. L. Donoho, “Continuous curvelet transform II. Discretization and frames,” Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005).
    [Crossref]
  39. E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “The curvelet transform website,” http://www.curvelet.org
  40. E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
    [Crossref]
  41. Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
    [Crossref]
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  43. Helicon Soft, “Helicon Focus Sample images,” http://www.heliconsoft.com/focus_samples.html

2012 (4)

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
[Crossref]

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

2011 (9)

S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
[Crossref]

L. Q. Guo and M. Zhu, “Quaternion Fourier-Mellin moments for color images,” Pattern Recogn. 44(2), 187–195 (2011).
[Crossref]

Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
[Crossref]

R. Maruthi, “Spatial Domain Method for Fusing Multi-Focus Images using Measure of Fuzziness,” Int. J. Comput. Appl. 20(7), 48–57 (2011).

F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
[Crossref]

Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
[Crossref]

N. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion algorithm based on shearlets,” Chin. Opt. Lett. 9(4), 041001 (2011).

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

2010 (3)

X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
[Crossref]

Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
[Crossref]

Y. Chen, L. Wang, Z. Sun, Y. Jiang, and G. Zhai, “Fusion of color microscopic images based on bidimensional empirical mode decomposition,” Opt. Express 18(21), 21757–21769 (2010).
[Crossref] [PubMed]

2009 (3)

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

D. S. Alexiadis and G. D. Sergiadis, “Estimation of motions in color image sequences using hypercomplex Fourier transforms,” IEEE Trans. Sig. Process. 18(1), 168–186 (2009).

Q. Zhang and B. Guo, “Multifocus image fusion using the nonsubsampled contourlet transform,” Signal Process. 89(7), 1334–1346 (2009).
[Crossref]

2008 (4)

R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
[Crossref]

S. Li and B. Yang, “Multifocus image fusion using region segmentation and spatial frequency,” Image Vis. Comput. 26(7), 971–979 (2008).
[Crossref]

P. L. Lin and P. Y. Huang, “Fusion methods based on dynamic-segmented morphological wavelet or cut and paste for multifocus images,” Signal Process. 88(6), 1511–1527 (2008).
[Crossref]

H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
[Crossref]

2007 (4)

W. Huang and Z. Jing, “Evaluation of focus measures in multi-focus image fusion,” Pattern Recogn. Lett. 28(9), 493–500 (2007).
[Crossref]

W. Huang and Z. L. Jing, “Multifocus image fusion using pulse coupled neutral network,” Pattern Recogn. lett. 28(9), 1123–1132 (2007).
[Crossref]

T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Trans. Image Process. 16(1), 22–35 (2007).
[Crossref] [PubMed]

K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
[Crossref]

2006 (3)

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

I. De and B. Chanda, “A simple and efficient algorithm for multifocus image fusion using morphological wavelets,” Signal Process. 86(5), 924–936 (2006).
[Crossref]

W. Yajie and X. Xinhe, “A multifocus image fusion new method based on multidecision,” Proc. SPIE 6357, 63570G (2006).
[Crossref]

2005 (3)

S. Gabarda and G. Cristóbal, “Multifocus image fusion through pseudo-Wigner distribution,” Opt. Eng. 44(4), 047001 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal. 19(2), 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform II. Discretization and frames,” Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005).
[Crossref]

2002 (1)

S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
[Crossref]

1999 (2)

S. C. Pei and C. M. Cheng, “Color image processing by using binary quaternion-moment-preserving thresholding technique,” IEEE Trans. Signal Process. 8(5), 614–628 (1999).

Z. Zhang and R. S. Blum, “A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application,” Proc. IEEE.  87(8), 1315–1326 (1999).
[Crossref]

1996 (1)

S. J. Sangwine, “Fourier transforms of colour images using quaternion, or hypercomplex numbers,” Electron. Lett. 32(1), 1979–1980 (1996).
[Crossref]

1995 (1)

H. Li, B. S. Manjunath, and S. K. Mitra, “Multisensor image fusion using the wavelet transform,” Graphical Models & Image Process. 57(3), 235–245 (1995).
[Crossref] [PubMed]

Ahmadi, M.

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

Alexiadis, D. S.

D. S. Alexiadis and G. D. Sergiadis, “Estimation of motions in color image sequences using hypercomplex Fourier transforms,” IEEE Trans. Sig. Process. 18(1), 168–186 (2009).

Amolius, K.

K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
[Crossref]

Baradarani, A.

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

Bihan, N. L.

S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
[Crossref]

Blum, R. S.

Z. Zhang and R. S. Blum, “A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application,” Proc. IEEE.  87(8), 1315–1326 (1999).
[Crossref]

Candès, E. J.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal. 19(2), 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform II. Discretization and frames,” Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005).
[Crossref]

Chai, Y.

Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
[Crossref]

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
[Crossref]

Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
[Crossref]

Chanda, B.

I. De and B. Chanda, “A simple and efficient algorithm for multifocus image fusion using morphological wavelets,” Signal Process. 86(5), 924–936 (2006).
[Crossref]

Chang, B.

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

Chen, B. J.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Chen, G.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Chen, Y.

Cheng, C. M.

S. C. Pei and C. M. Cheng, “Color image processing by using binary quaternion-moment-preserving thresholding technique,” IEEE Trans. Signal Process. 8(5), 614–628 (1999).

Cristóbal, G.

R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
[Crossref]

S. Gabarda and G. Cristóbal, “Multifocus image fusion through pseudo-Wigner distribution,” Opt. Eng. 44(4), 047001 (2005).
[Crossref]

Dare, P.

K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
[Crossref]

De, I.

I. De and B. Chanda, “A simple and efficient algorithm for multifocus image fusion using morphological wavelets,” Signal Process. 86(5), 924–936 (2006).
[Crossref]

Demanet, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Dillenseger, J. L.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Donoho, D. L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal. 19(2), 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform II. Discretization and frames,” Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005).
[Crossref]

Ell, T. A.

S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
[Crossref]

T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Trans. Image Process. 16(1), 22–35 (2007).
[Crossref] [PubMed]

Fang, M.

H. Shi and M. Fang, “Multi-focus Color Image Fusion Based on SWT and IHS,” in Proceedings of IEEE Conference on Fuzzy Systems and Knowledge Discovery (IEEE2007), 461–465.
[Crossref]

Feng, H.

H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
[Crossref]

Fischer, S.

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

Gabarda, S.

S. Gabarda and G. Cristóbal, “Multifocus image fusion through pseudo-Wigner distribution,” Opt. Eng. 44(4), 047001 (2005).
[Crossref]

Gristóbal, G.

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

Gu, J.

N. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion algorithm based on shearlets,” Chin. Opt. Lett. 9(4), 041001 (2011).

Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
[Crossref]

Guo, B.

Q. Zhang and B. Guo, “Multifocus image fusion using the nonsubsampled contourlet transform,” Signal Process. 89(7), 1334–1346 (2009).
[Crossref]

Guo, L. Q.

L. Q. Guo and M. Zhu, “Quaternion Fourier-Mellin moments for color images,” Pattern Recogn. 44(2), 187–195 (2011).
[Crossref]

Guo, M. Y.

Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
[Crossref]

Han, Y.

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

He, M.

X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
[Crossref]

Huang, P. Y.

P. L. Lin and P. Y. Huang, “Fusion methods based on dynamic-segmented morphological wavelet or cut and paste for multifocus images,” Signal Process. 88(6), 1511–1527 (2008).
[Crossref]

Huang, W.

W. Huang and Z. Jing, “Evaluation of focus measures in multi-focus image fusion,” Pattern Recogn. Lett. 28(9), 493–500 (2007).
[Crossref]

W. Huang and Z. L. Jing, “Multifocus image fusion using pulse coupled neutral network,” Pattern Recogn. lett. 28(9), 1123–1132 (2007).
[Crossref]

Jiang, Y.

Jing, Z.

W. Huang and Z. Jing, “Evaluation of focus measures in multi-focus image fusion,” Pattern Recogn. Lett. 28(9), 493–500 (2007).
[Crossref]

Jing, Z. L.

W. Huang and Z. L. Jing, “Multifocus image fusion using pulse coupled neutral network,” Pattern Recogn. lett. 28(9), 1123–1132 (2007).
[Crossref]

Kwok, J. T.

S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
[Crossref]

Li, H.

Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
[Crossref]

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
[Crossref]

H. Li, B. S. Manjunath, and S. K. Mitra, “Multisensor image fusion using the wavelet transform,” Graphical Models & Image Process. 57(3), 235–245 (1995).
[Crossref] [PubMed]

Li, H. F.

Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
[Crossref]

Li, Q.

H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
[Crossref]

Li, S.

S. Li and B. Yang, “Multifocus image fusion using region segmentation and spatial frequency,” Image Vis. Comput. 26(7), 971–979 (2008).
[Crossref]

S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
[Crossref]

Li, X.

X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
[Crossref]

Li, Z.

Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
[Crossref]

Liang, M.

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

Lin, P. L.

P. L. Lin and P. Y. Huang, “Fusion methods based on dynamic-segmented morphological wavelet or cut and paste for multifocus images,” Signal Process. 88(6), 1511–1527 (2008).
[Crossref]

Liu, G.

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

Lu, B.

F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
[Crossref]

Luo, F.

F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
[Crossref]

Luo, L.

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

Luo, L. M.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Ma, N.

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

Ma, Y.

N. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion algorithm based on shearlets,” Chin. Opt. Lett. 9(4), 041001 (2011).

Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
[Crossref]

Manjunath, B. S.

H. Li, B. S. Manjunath, and S. K. Mitra, “Multisensor image fusion using the wavelet transform,” Graphical Models & Image Process. 57(3), 235–245 (1995).
[Crossref] [PubMed]

Maruthi, R.

R. Maruthi, “Spatial Domain Method for Fusing Multi-Focus Images using Measure of Fuzziness,” Int. J. Comput. Appl. 20(7), 48–57 (2011).

Mendapara, P.

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

Miao, C.

F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
[Crossref]

Mitra, S. K.

H. Li, B. S. Manjunath, and S. K. Mitra, “Multisensor image fusion using the wavelet transform,” Graphical Models & Image Process. 57(3), 235–245 (1995).
[Crossref] [PubMed]

Nava, R.

R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
[Crossref]

Pei, S. C.

S. C. Pei and C. M. Cheng, “Color image processing by using binary quaternion-moment-preserving thresholding technique,” IEEE Trans. Signal Process. 8(5), 614–628 (1999).

Ramírez, B. E.

R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
[Crossref]

Redonodo, R.

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

Roux, M.

X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
[Crossref]

S?roubek, F.

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

Sangwine, S. J.

S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
[Crossref]

T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Trans. Image Process. 16(1), 22–35 (2007).
[Crossref] [PubMed]

S. J. Sangwine, “Fourier transforms of colour images using quaternion, or hypercomplex numbers,” Electron. Lett. 32(1), 1979–1980 (1996).
[Crossref]

Sergiadis, G. D.

D. S. Alexiadis and G. D. Sergiadis, “Estimation of motions in color image sequences using hypercomplex Fourier transforms,” IEEE Trans. Sig. Process. 18(1), 168–186 (2009).

Shi, H.

H. Shi and M. Fang, “Multi-focus Color Image Fusion Based on SWT and IHS,” in Proceedings of IEEE Conference on Fuzzy Systems and Knowledge Discovery (IEEE2007), 461–465.
[Crossref]

Shu, H. Z.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Sun, Z.

Toumoulin, C.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Wang, L.

Wang, N.

Wang, Y.

S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
[Crossref]

Wang, Z.

Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
[Crossref]

Wu, Q. M. J.

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

Xinhe, X.

W. Yajie and X. Xinhe, “A multifocus image fusion new method based on multidecision,” Proc. SPIE 6357, 63570G (2006).
[Crossref]

Yajie, W.

W. Yajie and X. Xinhe, “A multifocus image fusion new method based on multidecision,” Proc. SPIE 6357, 63570G (2006).
[Crossref]

Yang, B.

S. Li and B. Yang, “Multifocus image fusion using region segmentation and spatial frequency,” Image Vis. Comput. 26(7), 971–979 (2008).
[Crossref]

Yin, H.

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

Ying, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Yuan, Y.

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

Zhai, G.

Zhang, H.

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Zhang, J.

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

Zhang, Q.

Q. Zhang and B. Guo, “Multifocus image fusion using the nonsubsampled contourlet transform,” Signal Process. 89(7), 1334–1346 (2009).
[Crossref]

Zhang, X.

Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
[Crossref]

Zhang, Y.

K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
[Crossref]

Zhang, Z.

Z. Zhang and R. S. Blum, “A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application,” Proc. IEEE.  87(8), 1315–1326 (1999).
[Crossref]

Zhao, H.

H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
[Crossref]

Zhou, Z.

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

Zhu, M.

L. Q. Guo and M. Zhu, “Quaternion Fourier-Mellin moments for color images,” Pattern Recogn. 44(2), 187–195 (2011).
[Crossref]

Adv. Appl. Clifford Algebras (1)

S. J. Sangwine, T. A. Ell, and N. L. Bihan, “Fundamental representations and algebraic properties of biquater-nions or complexified quaternions,” Adv. Appl. Clifford Algebras 21(3), 607–636 (2011).
[Crossref]

Appl. Comput. Harmon. Anal. (2)

E. J. Candès and D. L. Donoho, “Continuous curvelet transform I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal. 19(2), 162–197 (2005).
[Crossref]

E. J. Candès and D. L. Donoho, “Continuous curvelet transform II. Discretization and frames,” Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005).
[Crossref]

Chin. Opt. Lett. (1)

Electron. Lett. (1)

S. J. Sangwine, “Fourier transforms of colour images using quaternion, or hypercomplex numbers,” Electron. Lett. 32(1), 1979–1980 (1996).
[Crossref]

Graphical Models & Image Process. (1)

H. Li, B. S. Manjunath, and S. K. Mitra, “Multisensor image fusion using the wavelet transform,” Graphical Models & Image Process. 57(3), 235–245 (1995).
[Crossref] [PubMed]

IEEE Trans. Image Process. (1)

T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Trans. Image Process. 16(1), 22–35 (2007).
[Crossref] [PubMed]

IEEE Trans. Sig. Process. (1)

D. S. Alexiadis and G. D. Sergiadis, “Estimation of motions in color image sequences using hypercomplex Fourier transforms,” IEEE Trans. Sig. Process. 18(1), 168–186 (2009).

IEEE Trans. Signal Process. (1)

S. C. Pei and C. M. Cheng, “Color image processing by using binary quaternion-moment-preserving thresholding technique,” IEEE Trans. Signal Process. 8(5), 614–628 (1999).

IET Image Process. (1)

X. Li, M. He, and M. Roux, “Multifocus image fusion based on redundant wavelet transform,” IET Image Process. 4(4), 283–293 (2010).
[Crossref]

Image Vis. Comput. (2)

S. Li and B. Yang, “Multifocus image fusion using region segmentation and spatial frequency,” Image Vis. Comput. 26(7), 971–979 (2008).
[Crossref]

H. Zhao, Q. Li, and H. Feng, “Multi-focus color image fusion in the HSI space using the sum-modified-laplacian and the coarse edge map,” Image Vis. Comput. 26(9), 1285–1295 (2008).
[Crossref]

Inform. Fusion (1)

R. Redonodo, F. S̆roubek, S. Fischer, and G. Gristóbal, “Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique,” Inform. Fusion 10(2), 163–171 (2009).
[Crossref]

Int. J. Comput. Appl. (1)

R. Maruthi, “Spatial Domain Method for Fusing Multi-Focus Images using Measure of Fuzziness,” Int. J. Comput. Appl. 20(7), 48–57 (2011).

Multiscale Model. Simul. (1)

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transorms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Opt. Commun. (3)

Y. Chai, H. F. Li, and M. Y. Guo, “Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain,” Opt. Commun. 284(5), 1146–1158 (2011).
[Crossref]

H. Li, Y. Chai, H. Yin, and G. Liu, “Multifocus image fusion and denoising scheme based on homogeneity similarity,” Opt. Commun. 285(2), 91–100 (2012).
[Crossref]

Y. Chai, H. Li, and Z. Li, “Multifocus image fusion scheme using focused region detection and multiresolution,” Opt. Commun. 284(19), 4376–4389 (2011).
[Crossref]

Opt. Eng. (2)

S. Gabarda and G. Cristóbal, “Multifocus image fusion through pseudo-Wigner distribution,” Opt. Eng. 44(4), 047001 (2005).
[Crossref]

Y. Yuan, J. Zhang, B. Chang, and Y. Han, “Objective quality evaluation of visible and infrared color fusion image,” Opt. Eng. 50(3), 033202 (2011).
[Crossref]

Opt. Express (1)

Optik (1)

Y. Chai, H. Li, and X. Zhang, “Multifocus image fusion based on features contrast of multiscale products in nonsubsampled contourlet transform domain,” Optik 123(7), 569–581 (2012).
[Crossref]

Pattern Recogn. (3)

A. Baradarani, Q. M. J. Wu, M. Ahmadi, and P. Mendapara, “Tunable halfband-pair wavelet filter banks and application to multifocus image fusion,” Pattern Recogn. 45(2), 657–671 (2012).
[Crossref]

L. Q. Guo and M. Zhu, “Quaternion Fourier-Mellin moments for color images,” Pattern Recogn. 44(2), 187–195 (2011).
[Crossref]

Z. Wang, Y. Ma, and J. Gu, “Multi-focus image fusion using PCNN,” Pattern Recogn. 43(6), 2003–2016 (2010).
[Crossref]

Pattern Recogn. Lett. (2)

S. Li, J. T. Kwok, and Y. Wang, “Multifocus image fusion using artificial neutral networks,” Pattern Recogn. Lett. 23(8), 985–997 (2002).
[Crossref]

W. Huang and Z. L. Jing, “Multifocus image fusion using pulse coupled neutral network,” Pattern Recogn. lett. 28(9), 1123–1132 (2007).
[Crossref]

W. Huang and Z. Jing, “Evaluation of focus measures in multi-focus image fusion,” Pattern Recogn. Lett. 28(9), 493–500 (2007).
[Crossref]

Photogramm. Eng. Remote Sens. (1)

K. Amolius, Y. Zhang, and P. Dare, “Wavelet based image fusion techniques–An introduction, review and comparison,” Photogramm. Eng. Remote Sens. 62(1), 249–263 (2007).
[Crossref]

Proc. IEEE (1)

Z. Zhang and R. S. Blum, “A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application,” Proc. IEEE.  87(8), 1315–1326 (1999).
[Crossref]

Proc. SPIE (4)

F. Luo, B. Lu, and C. Miao, “Multifocus image fusion with trace-based structure tensor,” Proc. SPIE 8200, 82001G (2011).
[Crossref]

N. Ma, L. Luo, Z. Zhou, and M. Liang, “A Multifocus image fusion in nonsubsampled contourlet domain with variational fusion stategy,” Proc. SPIE 8004, 800411 (2011).
[Crossref]

W. Yajie and X. Xinhe, “A multifocus image fusion new method based on multidecision,” Proc. SPIE 6357, 63570G (2006).
[Crossref]

R. Nava, B. E. Ramírez, and G. Cristóbal, “A novel multi-focus image fusion algorithm based on feature extraction and wavelets,” Proc. SPIE 7000, 700028 (2008).
[Crossref]

Signal Process. (4)

I. De and B. Chanda, “A simple and efficient algorithm for multifocus image fusion using morphological wavelets,” Signal Process. 86(5), 924–936 (2006).
[Crossref]

Q. Zhang and B. Guo, “Multifocus image fusion using the nonsubsampled contourlet transform,” Signal Process. 89(7), 1334–1346 (2009).
[Crossref]

P. L. Lin and P. Y. Huang, “Fusion methods based on dynamic-segmented morphological wavelet or cut and paste for multifocus images,” Signal Process. 88(6), 1511–1527 (2008).
[Crossref]

B. J. Chen, H. Z. Shu, H. Zhang, G. Chen, C. Toumoulin, J. L. Dillenseger, and L. M. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Process. 92(2), 308–318 (2012).
[Crossref]

Other (5)

S. Sangwine and N. L. Bihan, “Quaternion toolbox for Matlab,” http://qtfm.sourceforge.net

H. Shi and M. Fang, “Multi-focus Color Image Fusion Based on SWT and IHS,” in Proceedings of IEEE Conference on Fuzzy Systems and Knowledge Discovery (IEEE2007), 461–465.
[Crossref]

M. Douze, “Blur image data,” http://lear.inrialpes.fr/people/vandeweijer/data.html

Helicon Soft, “Helicon Focus Sample images,” http://www.heliconsoft.com/focus_samples.html

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “The curvelet transform website,” http://www.curvelet.org

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Figures (14)

Fig. 1
Fig. 1

Spatial domain based multifocus color image fusion scheme.

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Fig. 2
Fig. 2

Spatial and frequency domains based multifocus color image fusion scheme.

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Fig. 3
Fig. 3

Block diagram of the forward and inverse quaternion curvelet transform.

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Fig. 4
Fig. 4

Block diagram of QCT based multifocus color image fusion scheme.

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Fig. 5
Fig. 5

The source multifocus color images.

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Fig. 6
Fig. 6

Fusion results of different methods. (a) IHS and SWT based fusion result; (b) Fuzzy based fusion result; (c) SML based fusion result; (d) BEMD based fusion result; (e) DWT based fusion results; (f) proposed QCT based fusion result.

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7

Subimages taken from Fig. 6. (a) IHS and SWT based fusion result; (b) Fuzzy based fusion result; (c) SML based fusion result; (d) BEMD based fusion result; (e) DWT based fusion results; (f) proposed QCT based fusion result.

Download Full Size | PPT Slide | PDF

Fig. 8
Fig. 8

Objective evaluation of image using ICM measure in Fig. 6.

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Fig. 9
Fig. 9

The source multifocus color images.

Download Full Size | PPT Slide | PDF

Fig. 10
Fig. 10

Fusion results of different methods. (a) IHS and SWT based fusion result; (b) Fuzzy based fusion result; (c) SML based fusion result; (d) BEMD based fusion result; (e) DWT based fusion results; (f) proposed QCT based fusion result.

Download Full Size | PPT Slide | PDF

Fig. 11
Fig. 11

Subimages taken from Fig. 10. (a) IHS and SWT based fusion result; (b) Fuzzy based fusion result; (c) SML based fusion result; (d) BEMD based fusion result; (e) DWT based fusion results; (f) proposed QCT based fusion result.

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Fig. 12
Fig. 12

Objective evaluation of image using ICM measure in Fig. 10.

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Fig. 13
Fig. 13

Source multifocus color image for Multiple image fusion.

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Fig. 14
Fig. 14

Multiple image fusion results. (a) BEMD based fusion result; (b) DWT based fusion result; (c) QCT based fusion result; (d) Objective evaluation of (a), (b) and (c).

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Equations (16)

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

ℍ = { q = q r + q i i + q j j + q k k | q r , q i , q j , q k ∈ ℝ }
i 2 = j 2 = k 2 = − 1
i j = − j i = k , j k = − k j = i , k i = − i k = j
‖ q ‖ = q ⋅ q ¯ = q r 2 + q i 2 + q j 2 + q k 2
f ( m , n ) = f R ( m , n ) i + f G ( m , n ) j + f B ( m , n ) k
ϕ j , k , l ( x ) = ϕ j ( R θ l ( x − x k ( j , l ) ) )
f ( x 1 , x 2 ) = f ( 2 2 j x 1 , 2 j x 2 ) ,             j = 0 , 1 , 2 , …
( x 1 ′ x 2 ′ ) = ( cos θ l − sin θ l sin θ l cos θ l ) ( x 1 x 2 )
x k ( j , l ) = R θ l − 1 ( k 1 2 − j , k 2 2 − j / 2 )
f ( x ) = ∑ j ∑ k ∑ l < f ( x ) , ϕ j , k , l ( x ) > ϕ j , k , l ( x )
< f ( x ) , ϕ j , k , l ( x ) > ≜ ∫ ℝ 2 f ( x ) ϕ j , k , l ( x ) ¯ d x
C F 1 ( i , j ) = { C A 1 ( i , j ) , ‖ C A 1 ( i , j ) ‖ ≥ ‖ C B 1 ( i , j ) ‖ C B 1 ( i , j ) ,   otherwise
C F 2 ( i , j ) = { C A 2 ( i , j ) , ‖ C A 2 ( i , j ) ‖ ≤ ‖ C B 2 ( i , j ) ‖ C B 2 ( i , j ) ,   otherwise
C g = α I ∑ k = 0 N I − 1 I k N I P ( I k )
C c = α L ∑ k = 0 N L − 1 L k N L P ( L k )
I C M = ( 0.5 × C g 2 + 0.5 × C c 2 ) 1 2

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