Abstract

The higher order orthogonal iteration (HOOI) is used for a single-frame and multi-frame space-variant blind deconvolution (BD) performed by factorization of the tensor of blurred multi-spectral image (MSI). This is achieved by conversion of BD into blind source separation (BSS), whereupon sources represent the original image and its spatial derivatives. The HOOI-based factorization enables an essentially unique solution of the related BSS problem with orthogonality constraints imposed on factors and the core tensor of the Tucker3 model of the image tensor. In contrast, the matrix factorization-based unique solution of the same BSS problem demands sources to be statistically independent or sparse which is not true. The consequence of such an approach to BD is that it virtually does not require a priori information about the possibly space-variant point spread function (PSF): neither its model nor size of its support. For the space-variant BD problem, MSI is divided into blocks whereupon the PSF is assumed to be a space-invariant within the blocks. The success of proposed concept is demonstrated in experimentally degraded images: defocused single-frame gray scale and red-green-blue (RGB) images, single-frame gray scale and RGB images blurred by atmospheric turbulence, and a single-frame RGB image blurred by a grating (photon sieve). A comparable or better performance is demonstrated in relation to the blind Richardson-Lucy algorithm which, however, requires a priori information about parametric model of the blur.

© 2010 OSA

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References

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2010 (1)

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

2009 (4)

I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. 34(18), 2835–2837 (2009).
[Crossref] [PubMed]

F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009).
[Crossref]

H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 649–660 (2009).
[Crossref] [PubMed]

S. Cho and S. Lee, “Fast Motion Deblurring,” ACM Trans. Graph. 28(5), 1 (2009).
[Crossref]

2008 (2)

M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. 17(2), 105–116 (2008).
[Crossref] [PubMed]

Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008).
[Crossref]

2007 (3)

2006 (2)

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “A computational method for the restoration of images with an unknown, spatially-varying blur,” Opt. Express 14(5), 1767–1782 (2006).
[Crossref] [PubMed]

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

2005 (2)

2004 (1)

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. 233(1-3), 7–14 (2004).
[Crossref]

2001 (1)

S. Umeyama, “Blind deconvolution of blurred images by use of ICA,” Electron Commun. Jpn 84(12), 1–9 (2001).

2000 (3)

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000).
[Crossref]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000).
[Crossref]

H. A. L. Kiers, “Towards a standardized notation and terminology in multiway analysis,” J. Chemometr. 14(3), 105–122 (2000).
[Crossref]

1997 (2)

D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. 36(8), 1766–1775 (1997).
[Crossref] [PubMed]

M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[Crossref]

1996 (1)

D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996).
[Crossref]

1995 (1)

1993 (1)

1988 (1)

J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. 36(7), 1169–1179 (1988).
[Crossref]

1966 (1)

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31(3), 279–311 (1966).
[Crossref] [PubMed]

Agard, D. A.

Agarwala, A.

Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008).
[Crossref]

Andrews, M.

Banham, M. R.

M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[Crossref]

Bardsley, J.

Biggs, D. S. C.

Brinicombe, A. M.

Cho, S.

S. Cho and S. Lee, “Fast Motion Deblurring,” ACM Trans. Graph. 28(5), 1 (2009).
[Crossref]

Daugman, J. G.

J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. 36(7), 1169–1179 (1988).
[Crossref]

Davies, M. E.

M. E. Davies and C. J. James, “Source separation using single channel ICA,” Signal Process. 87(8), 1819–1832 (2007).
[Crossref]

De Lathauwer, L.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000).
[Crossref]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000).
[Crossref]

De Moor, B.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000).
[Crossref]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000).
[Crossref]

Du, Q.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. 233(1-3), 7–14 (2004).
[Crossref]

Fergus, R.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Fermüller, C.

H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 649–660 (2009).
[Crossref] [PubMed]

Fish, D. A.

Flusser, J.

M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. 17(2), 105–116 (2008).
[Crossref] [PubMed]

Fraser, D.

F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009).
[Crossref]

Freeman, W. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Haase, S.

Hatzinakos, D.

D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996).
[Crossref]

Hertzmann, A.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Hom, E. F. Y.

James, C. J.

M. E. Davies and C. J. James, “Source separation using single channel ICA,” Signal Process. 87(8), 1819–1832 (2007).
[Crossref]

Jefferies, S.

Ji, H.

H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 649–660 (2009).
[Crossref] [PubMed]

Jia, J.

Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008).
[Crossref]

Jia, X.

F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009).
[Crossref]

Jiang, Q. B.

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Katsaggelos, A. K.

M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[Crossref]

Kiers, H. A. L.

H. A. L. Kiers, “Towards a standardized notation and terminology in multiway analysis,” J. Chemometr. 14(3), 105–122 (2000).
[Crossref]

Kopriva, I.

Kundur, D.

D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996).
[Crossref]

Lee, S.

S. Cho and S. Lee, “Fast Motion Deblurring,” ACM Trans. Graph. 28(5), 1 (2009).
[Crossref]

Lee, T. K.

Li, F.

F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009).
[Crossref]

Lin, J.

J. Lin and A. Zhang, “Fault feature separation using wavelet-ICA filter,” NDT Int. 38(6), 421–427 (2005).
[Crossref]

Liu, G.

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Liu, Z. Q.

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Ma, H. G.

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Ma, Z. Y.

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Marchis, F.

Nagy, J.

Pike, E. R.

Plemmons, R.

Roweis, S. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Schulz, T. J.

Sedat, J. W.

Shan, Q.

Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008).
[Crossref]

Singh, B.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Šorel, M.

M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. 17(2), 105–116 (2008).
[Crossref] [PubMed]

Szu, H.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. 233(1-3), 7–14 (2004).
[Crossref]

Tucker, L. R.

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31(3), 279–311 (1966).
[Crossref] [PubMed]

Umeyama, S.

S. Umeyama, “Blind deconvolution of blurred images by use of ICA,” Electron Commun. Jpn 84(12), 1–9 (2001).

Vandewalle, J.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000).
[Crossref]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000).
[Crossref]

Walker, J. G.

Wasylkiwskyj, W.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. 233(1-3), 7–14 (2004).
[Crossref]

Zhang, A.

J. Lin and A. Zhang, “Fault feature separation using wavelet-ICA filter,” NDT Int. 38(6), 421–427 (2005).
[Crossref]

ACM Trans. Graph. (3)

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. 25(3), 787–794 (2006).
[Crossref]

Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008).
[Crossref]

S. Cho and S. Lee, “Fast Motion Deblurring,” ACM Trans. Graph. 28(5), 1 (2009).
[Crossref]

Appl. Opt. (1)

Electron Commun. Jpn (1)

S. Umeyama, “Blind deconvolution of blurred images by use of ICA,” Electron Commun. Jpn 84(12), 1–9 (2001).

IEEE Geosci. Remote Sens. Lett. (1)

F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009).
[Crossref]

IEEE Signal Process. Mag. (2)

M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[Crossref]

D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. 36(7), 1169–1179 (1988).
[Crossref]

IEEE Trans. Image Process. (1)

M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. 17(2), 105–116 (2008).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 649–660 (2009).
[Crossref] [PubMed]

J. Chemometr. (1)

H. A. L. Kiers, “Towards a standardized notation and terminology in multiway analysis,” J. Chemometr. 14(3), 105–122 (2000).
[Crossref]

J. Opt. Soc. Am. A (4)

NDT Int. (1)

J. Lin and A. Zhang, “Fault feature separation using wavelet-ICA filter,” NDT Int. 38(6), 421–427 (2005).
[Crossref]

Opt. Commun. (1)

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. 233(1-3), 7–14 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Psychometrika (1)

L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31(3), 279–311 (1966).
[Crossref] [PubMed]

SIAM J. Matrix Anal. Appl. (2)

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000).
[Crossref]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000).
[Crossref]

Signal Process. (2)

M. E. Davies and C. J. James, “Source separation using single channel ICA,” Signal Process. 87(8), 1819–1832 (2007).
[Crossref]

H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. 90(12), 3232–3241 (2010).
[Crossref]

Other (7)

R. L. Lagendijk, and J. Biemond, Iterative Identification and Restoration of Images (KAP, 1991).

A. Cichocki, R. Zdunek, A. H. Phan, and S. I. Amari, Nonnegative Matrix and Tensor Factorization (John Wiley & Sons, 2009).

A. Cichocki, and S. Amari, Adaptive Blind Signal and Image Processing (John Wiley, New York, 2002).

A. Cichocki, and A. H. Phan. Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations. IEICE Trans Fundamentals 2009; E92-A(3): 708–721.

B. W. Bader, and T. G. Kolda, MATLAB Tensor Toolbox version 2.2. http://csmr.ca.sandia.gov/~tkolda/TensorToolbox .

P. Campisi, and K. Egiazarian, eds., Blind Image Deconvolution (CRC Press, Boca Raton, 2007).

J. Miskin, and D. J. C. MacKay, “Ensemble Learning for Blind Image Separation and Deconvolution,” in: Advances in Independent Component Analysis (Springer, London, 2000).

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Figures (10)

Fig. 1
Fig. 1

RGB experimental image obtained by digital camera in manually defocused mode.

Fig. 6
Fig. 6

a) Four frames of experimental multi-frame RGB image degraded by atmospheric turbulence. b) left: average of the four frames from 6a; right: edges extracted from gray scale version of averaged image with Canny's algorithm and threshold set to 0.21.

Fig. 10
Fig. 10

From left to right: RGB image degraded by a grating (photon sieve); image restored by 5D tensor factorization space-variant blind deconvolution; image restored by 4D tensor factorization space-invariant blind deconvolution; image restored by blind R-L algorithm after 10 iterations and radius of the circular blur equal to 2 pixels.

Fig. 4
Fig. 4

Experimental RGB image restored by blind R-L algorithm after 5 iterations and radius of the circular blur equal to 2 pixels (left) and 3 pixels (right) .

Fig. 9
Fig. 9

Images restored from averaged gray scale version of four blurred frames shown in Fig. 6a by blind R-L algorithm and Gaussian PSF with kernel width of 18 pixels and a) sigma = 1.3 pixels; b) sigma = 1.9 pixels. Shown edge maps were obtained with Canny's algorithm and threshold set to 0.21.

Fig. 2
Fig. 2

Experimental gray scale version of RGB image restored by left: space-variant single-frame blind deconvolution 4D tensor factorization algorithm; right: single-frame blind deconvolution algorithm where each block is restored through one 3D tensor factorization.

Fig. 3
Fig. 3

Experimental RGB image restored by left: space-invariant single-frame blind deconvolution 4D tensor factorization algorithm; right: single-frame blind deconvolution where each spectral channel is restored through one 3D tensor factorization problem.

Fig. 5
Fig. 5

Experimental RGB image restored by left: 5D tensor factorization space-variant single-frame blind deconvolution; right: 4D tensor factorization single-frame blind deconvolution, where each block at each spectral channel is restored through a 3D tensor factorization.

Fig. 7
Fig. 7

a) Time evolution of RGB source image extracted by 5D tensor factorization space-invariant multi-frame blind deconvolution algorithm from experimental dynamic RGB shown in Fig. 6a. b) left: average of the four source images from 7a; right: edges extracted from the gray scale version of averaged image with Canny's algorithm and threshold set to 0.21.

Fig. 8
Fig. 8

a) Time evolution of gray scale source image restored by 4D tensor factorization space-invariant multi-frame blind deconvolution algorithm from gray scale version of the RGB shown in Fig. 6a. b) left: average of the four source images from 8a; right: edges extracted from averaged image with Canny's algorithm and threshold set to 0.21.

Tables (1)

Tables Icon

Table 1 Physical interpretation of modes of the multichannel image tensors associated with the various blind image deconvolution problems

Equations (14)

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G ¯ R ¯ × 1 A ( 1 ) × 2 A ( 2 ) ... × N A ( N )
g i 1 i 2 ... i N j 1 = 1 J 1 j 2 = 1 J 2 ... j N = 1 J N r j 1 j 2 ... j N a i 1 j 1 ( 1 ) a i 2 j 2 ( 2 ) ... a i N j N ( N )
G ( N ) A ( N ) R ( N ) [ A ( N 1 ) ... A ( 2 ) A ( 1 ) ] T
G ( n ) A F
A A ( N ) F R ( N ) [ A ( N 1 ) ... A ( 2 ) A ( 1 ) ] T
F ¯ ^ R ¯ × 1 A ( 1 ) × 2 A ( 2 ) ... × N 1 A ( N 1 ) = G ¯ × N ( A ( N ) )
D [ G ¯ G ¯ ^ ] = G ¯ R ¯ × 1 A ( 1 ) × 2 A ( 2 ) ... × N A ( N ) 2 2
R ¯ G ¯ × 1 A ( 1 ) T × 2 A ( 2 ) T ... × N A ( N ) T
D ( A ( 1 ) , A ( 2 ) , ... , A ( N ) ) = G ¯ × 1 A ( 1 ) T × 2 A ( 2 ) T ... × N A ( N ) T 2 2
G ( i 1 , i 2 ) = s = M M t = M M H ( s , t ) F ( i 1 s , i 2 t )
F ( i 1 s , i 2 t ) = F ( i 1 , i 2 ) s F i 1 ( i 1 , i 2 ) t F i 2 ( i 1 , i 2 )                                                               + 1 2 s 2 F i 1 i 1 ( i 1 , i 2 ) + 1 2 t 2 F i 2 i 2 ( i 1 , i 2 ) + 1 2 s t F i 1 i 2 ( i 1 , i 2 ) ...
G ( i 1 , i 2 ) = a 1 F ( i 1 , i 2 ) a 2 F i 1 ( i 1 , i 2 ) a 3 F i 2 ( i 1 , i 2 ) +                                         a 4 F i 1 i 1 ( i 1 , i 2 ) + a 5 F i 2 i 2 ( i 1 , i 2 ) + a 6 F i 1 i 2 ( i 1 , i 2 ) ...
G i 3 ( i 1 , i 2 ) = a i 3 1 F ( i 1 , i 2 )     a i 3 2 F i 1 ( i 1 , i 2 ) a i 3 3 F i 2 ( i 1 , i 2 ) +                                                                     a i 3 4 F i 1 i 1 ( i 1 , i 2 )     + a i 3 5 F i 2 i 2 ( i 1 , i 2 )     + a i 3 6 F i 1 i 2 ( i 1 , i 2 ) ...
j = 1 + p n ( i p 1 ) J p ,             with       J p = { 1 ,                 if     p = 1       or     p = 2       and     n = 1 m n p 1 I m ,               otherwise .

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