Abstract

Recent advances in image processing for atmospheric propagation have provided a foundation for tackling the similar but perhaps more complex problem of underwater imaging, which is impaired by scattering and optical turbulence. As a result of these impairments underwater imagery suffers from excessive noise, blur, and distortion. Underwater turbulence impact on light propagation becomes critical at longer distances as well as near thermocline and mixing layers. In this work, we demonstrate a method for restoration of underwater images that are severely degraded by underwater turbulence. The key element of the approach is derivation of a structure tensor oriented image quality metric, which is subsequently incorporated into a lucky patch image processing framework. The utility of the proposed image quality measure guided by local edge strength and orientation is emphasized by comparing the restoration results to an unsuccessful restoration obtained with equivalent processing utilizing a standard isotropic metric. Advantages of the proposed approach versus three other state-of-the-art image restoration techniques are demonstrated using the data obtained in the laboratory water tank and in a natural environment underwater experiment. Quantitative comparison of the restoration results is performed via structural similarity index measure and normalized mutual information metric.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2014 (3)

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014).
[Crossref] [PubMed]

2012 (5)

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

D. D. Baumgartner and B. J. Schachter, “Improving FLIR ATR performance in a turbulent atmosphere with a moving platform,” Proc. SPIE 8391, 839103 (2012).

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Image Process. 21(4), 1687–1700 (2012).
[Crossref] [PubMed]

S. Gładysz, R. B. Gallé, R. Johnson, and L. Kann, “Image reconstruction of extended objects: demonstration with the Starfire optical range 3.5m telescope,” Proc. SPIE 8535, 85350 (2012).

2011 (1)

2010 (1)

2009 (4)

L. Mullen, A. Laux, and B. Cochenour, “Propagation of modulated light in water: implications for imaging and communications systems,” Appl. Opt. 48(14), 2607–2612 (2009).
[Crossref] [PubMed]

W. Hou, “A simple underwater imaging model,” Opt. Lett. 34(17), 2688–2690 (2009).
[Crossref] [PubMed]

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

M. Aubailly, M. A. Vorontsov, G. W. Carhartb, and M. T. Valley, “Automated video enhancement from a stream of atmospherically distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630 (2009).

2008 (1)

K. Dabov, A. Foi, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

2001 (1)

1999 (2)

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

M. A. Vorontsov, “Parallel image processing based on an evolution equation with anisotropic gain: integrated optoelectronic architectures,” J. Opt. Soc. Am. 16(7), 1623–1637 (1999).

1996 (1)

E. P. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Process. 5(9), 1377–1382 (1996).
[Crossref] [PubMed]

1983 (1)

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier-analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Aubailly, M.

M. Aubailly, M. A. Vorontsov, G. W. Carhartb, and M. T. Valley, “Automated video enhancement from a stream of atmospherically distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630 (2009).

Baumgartner, D. D.

D. D. Baumgartner and B. J. Schachter, “Improving FLIR ATR performance in a turbulent atmosphere with a moving platform,” Proc. SPIE 8391, 839103 (2012).

Baykal, Y.

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Britton, W.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Caimi, F. M.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Carhart, G. W.

Carhartb, G. W.

M. Aubailly, M. A. Vorontsov, G. W. Carhartb, and M. T. Valley, “Automated video enhancement from a stream of atmospherically distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630 (2009).

Cochenour, B.

Cremers, D.

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

Dabov, K.

K. Dabov, A. Foi, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).

Dalgleish, F. R.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Egiazarian, K.

K. Dabov, A. Foi, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).

Farid, H.

E. P. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Process. 5(9), 1377–1382 (1996).
[Crossref] [PubMed]

Foi, A.

K. Dabov, A. Foi, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).

Font, C.

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

Fraser, D.

Gallé, R. B.

S. Gładysz, R. B. Gallé, R. Johnson, and L. Kann, “Image reconstruction of extended objects: demonstration with the Starfire optical range 3.5m telescope,” Proc. SPIE 8535, 85350 (2012).

Giddings, T. E.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Gladysz, S.

S. Gładysz, R. B. Gallé, R. Johnson, and L. Kann, “Image reconstruction of extended objects: demonstration with the Starfire optical range 3.5m telescope,” Proc. SPIE 8535, 85350 (2012).

Goode, W.

Hawkes, D. J.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Hayes, C.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Hill, D. L. G.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Hou, W.

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

W. Hou, “A simple underwater imaging model,” Opt. Lett. 34(17), 2688–2690 (2009).
[Crossref] [PubMed]

Jarosz, E.

Ji, X.

Johnson, R.

S. Gładysz, R. B. Gallé, R. Johnson, and L. Kann, “Image reconstruction of extended objects: demonstration with the Starfire optical range 3.5m telescope,” Proc. SPIE 8535, 85350 (2012).

Kanaev, A. V.

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

Kann, L.

S. Gładysz, R. B. Gallé, R. Johnson, and L. Kann, “Image reconstruction of extended objects: demonstration with the Starfire optical range 3.5m telescope,” Proc. SPIE 8535, 85350 (2012).

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier-analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lambert, A.

Laux, A.

Leach, M. O.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Li, H.

Lohmann, A. W.

Lu, L.

Matt, S.

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

Milanfar, P.

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Image Process. 21(4), 1687–1700 (2012).
[Crossref] [PubMed]

Mitzel, D.

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

Mullen, L.

Muth, J.

Nootz, G.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Oreifej, O.

O. Oreifej, G. Shu, T. Pace, and M. Shah, “A two-stage reconstruction approach for seeing through water,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2011), pp. 1153–1160.
[Crossref]

Ouyang, B.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Pace, T.

O. Oreifej, G. Shu, T. Pace, and M. Shah, “A two-stage reconstruction approach for seeing through water,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2011), pp. 1153–1160.
[Crossref]

Pock, T.

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

Restaino, S. R.

S. R. Restaino, W. Hou, A. V. Kanaev, S. Matt, and C. Font, “Adaptive optics correction of a laser beam propagating under-water,” Proc. SPIE 9083, 90830 (2014).

Rueckert, D.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Schachter, B. J.

D. D. Baumgartner and B. J. Schachter, “Improving FLIR ATR performance in a turbulent atmosphere with a moving platform,” Proc. SPIE 8391, 839103 (2012).

Schoenemann, T.

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

Shah, M.

O. Oreifej, G. Shu, T. Pace, and M. Shah, “A two-stage reconstruction approach for seeing through water,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2011), pp. 1153–1160.
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Shu, G.

O. Oreifej, G. Shu, T. Pace, and M. Shah, “A two-stage reconstruction approach for seeing through water,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2011), pp. 1153–1160.
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

E. P. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Process. 5(9), 1377–1382 (1996).
[Crossref] [PubMed]

Smith, L. N.

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

Sonoda, L. I.

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

Sroubek, F.

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Image Process. 21(4), 1687–1700 (2012).
[Crossref] [PubMed]

Valley, M. T.

M. Aubailly, M. A. Vorontsov, G. W. Carhartb, and M. T. Valley, “Automated video enhancement from a stream of atmospherically distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630 (2009).

Vorontsov, M. A.

M. Aubailly, M. A. Vorontsov, G. W. Carhartb, and M. T. Valley, “Automated video enhancement from a stream of atmospherically distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630 (2009).

M. A. Vorontsov and G. W. Carhart, “Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images,” J. Opt. Soc. Am. A 18(6), 1312–1324 (2001).
[Crossref] [PubMed]

M. A. Vorontsov, “Parallel image processing based on an evolution equation with anisotropic gain: integrated optoelectronic architectures,” J. Opt. Soc. Am. 16(7), 1623–1637 (1999).

Vuorenkoski, A. K.

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Weidemann, A.

Weigelt, G.

Wen, Z.

Wirnitzer, B.

Woods, S.

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

Appl. Opt. (5)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier-analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

IEEE Trans. Image Process. (3)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

E. P. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Process. 5(9), 1377–1382 (1996).
[Crossref] [PubMed]

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Image Process. 21(4), 1687–1700 (2012).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill, M. O. Leach, and D. J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), 712–721 (1999).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

M. A. Vorontsov, “Parallel image processing based on an evolution equation with anisotropic gain: integrated optoelectronic architectures,” J. Opt. Soc. Am. 16(7), 1623–1637 (1999).

J. Opt. Soc. Am. A (1)

Lect. Notes Comput. Sci. (1)

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).

Opt. Eng. (2)

B. Ouyang, F. R. Dalgleish, F. M. Caimi, T. E. Giddings, W. Britton, A. K. Vuorenkoski, and G. Nootz, “Compressive line sensing underwater imaging system,” Opt. Eng. 53(5), 051409 (2014).
[Crossref]

A. V. Kanaev, W. Hou, S. Woods, and L. N. Smith, “Restoration of turbulence degraded underwater images,” Opt. Eng. 51(5), 057007 (2012).
[Crossref]

Opt. Express (1)

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Supplementary Material (3)

NameDescription
» Visualization 1: MPG (12808 KB)     
» Visualization 2: AVI (934 KB)     
» Visualization 3: AVI (2190 KB)     

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Figures (8)

Fig. 1
Fig. 1

Imaging through underwater turbulence in the laboratory tank (Visualization 1).

Fig. 2
Fig. 2

Imaging through underwater turbulence in natural environment (Visualization 2).

Fig. 3
Fig. 3

Fragment of image sequence collected through underwater turbulence in the laboratory tank (Visualization 3).

Fig. 4
Fig. 4

IQ estimation using the sequence mean of line-pattern image fragment (a) and employing (b) STOIQ algorithm with adaptive kernel (two example kernels and their locations are shown on the left of the image); (c) isotropic Gaussian with σ = 0.4; (d) isotropic Gaussian with σ = 0.8; (d) isotropic Gaussian with σ = 1.5 (shapes of the kernels are shown above the images).

Fig. 5
Fig. 5

Restoration results of the line-pattern sequence (a) the mean of the sequence; restored image using (b) bispectrum technique; (c) standard lucky patch algorithm; (d) lucky patch algorithm employing STOIQ.

Fig. 6
Fig. 6

Vertical power spectra of restored images, the mean image and the ground-truth image.

Fig. 7
Fig. 7

Restoration results of line-pattern fragment sequence: (a) underwater image collected without turbulence; (b) the mean of the sequence; and restored image using (c) standard lucky patch algorithm; (d) two-stage reconstruction algorithm for seeing through water; (e) MC blind deconvolution via fast alternating minimization; (f) bispectrum technique; (g) lucky patch algorithm employing STOIQ.

Fig. 8
Fig. 8

Restoration results of USAF sequence: (a) underwater image collected with minimal underwater turbulence; (b) the mean of the sequence; and restored image using (c) standard lucky patch algorithm; (d) two-stage reconstruction algorithm for seeing through water; (e) MC blind deconvolution via fast alternating minimization; (f) bispectrum technique; (g) lucky patch algorithm employing STOIQ.

Tables (2)

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Table 1 Line-pattern Results

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Table 2 USAF Resolution Chart Results

Equations (12)

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J= | n I( r ) | m G( r r )d r , m=1,2 n=1,2
S=I I T =( I x 2 I x I y I x I y I y 2 ),
G φ,ρ =exp{ ( xcosφ+ysinφ ) 2 / 2 ρ 2 2 ( xsinφycosφ ) 2 / 2 ρ 1 2 },
I n+1 r = I n r γ n ( I n r I n ), n=1...N
γ n ( r )=K{ J n J n r , if J n > J n r 0 , if J n < J n r
| I( u ) | 2 N = | O( u ) | 2 | H( u ) | 2 N ,
B I ( u , v )=I( u )I( v ) I ( u + v ),
B I ( u , v ) N =O( u )O( v ) O * ( u + v ) B H ( u , v ) N ,
O * ( u + v )= B I ( u , v ) N O( u )O( v ) B H ( u , v ) N .
arg[ O( u + v ) ]=arg[ O( u ) ]+arg[ O( v ) ]arg[ B I ( u , v ) ]
SSI M P i,j = ( 2 μ i μ j +a )( 2 σ i,j +b ) ( μ i 2 + μ j 2 +a )( σ i 2 + σ j 2 +b ) ,
NM I 1,2 = H 1 + H 2 H ,

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