Abstract

In ground-based astronomy, images of objects in outer space are acquired via ground-based telescopes. However, the imaging system is generally interfered by atmospheric turbulence, and hence images so acquired are blurred with unknown point-spread function (PSF). To restore the observed images, the wavefront of light at the telescope’s aperture is utilized to derive the PSF. A model with the Tikhonov regularization has been proposed to find the high-resolution phase gradients by solving a least-squares system. Here we propose the l1-lp (p=1, 2) model for reconstructing the phase gradients. This model can provide sharper edges in the gradients while removing noise. The minimization models can easily be solved by the Douglas–Rachford alternating direction method of a multiplier, and the convergence rate is readily established. Numerical results are given to illustrate that the model can give better phase gradients and hence a more accurate PSF. As a result, the restored images are much more accurate when compared to the traditional Tikhonov regularization model.

© 2012 Optical Society of America

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2012

B. S. He and X. M. Yuan, “On the O(1/n) convergence rate of Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[CrossRef]

Y. W. Wen and R. H. Chan, “Parameter selection for total variation based image restoration using discrepancy principle,” IEEE Trans. Image Process. 21, 1770–1781 (2012).
[CrossRef]

B. S. He and X. M. Yuan, “Convergence analysis of primal-dual algorithms for total variation image restoration,” SIAM J. Imaging Sci. 5, 119–149 (2012).
[CrossRef]

2011

S. M. Jefferies and M. Hart, “Deconvolution from wave front sensing using the frozen flow hypothesis,” Opt. Express 19, 1975–1984 (2011).
[CrossRef]

R. H. Chan, J. F. Yang, and X. M. Yuan, “Alternating direction method for image inpainting in wavelet domain,” SIAM J. Imaging Sci. 4, 807–826 (2011).
[CrossRef]

2010

M. Ng, P. A. Weiss, and X. M. Yuan, “Solving constrained total-variation problems via alternating direction methods,” SIAM J. Sci. Comput. 32, 2710–2736 (2010).
[CrossRef]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

2008

J. M. Bardsley, “Wavefront reconstruction methods for adaptive optics systems on ground-based telescopes,” SIAM J. Matrix Anal. Appl. 30, 67–83 (2008).
[CrossRef]

2007

R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007).
[CrossRef]

2004

M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging. Vis. 20, 99–120 (2004).
[CrossRef]

2003

R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

2002

B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002).
[CrossRef]

S. M. Jefferies, M. L. Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. 41, 2095–2102 (2002).
[CrossRef]

2001

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

2000

E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A. 17, 1177–1184 (2000).
[CrossRef]

B. S. He, H. Yang, and S. L. Wang, “Alternating direction method with self-Adaptive penalty parameters for monotone variational inequalities,” J. Opt. Theory Appl. 106, 337–356 (2000).
[CrossRef]

1999

1998

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE. Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

N. K. Bose and K. Boo, “High-resolution image reconstruction with multisensors,” Int. J. Imaging Syst. Technol. 9, 294–304 (1998).
[CrossRef]

1997

S. Aliney, “A property of the minimum vectors of a regularizing functional defined by means of the absolute norm,” IEEE Trans. Signal Process. 45, 913–917 (1997).
[CrossRef]

1996

1995

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41, 613–627 (1995).
[CrossRef]

1992

L. Rudin, S. Osher, and F. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

1979

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

1977

1976

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

1975

R. Glowinski and A. Marocco, “Approximation par éléments finis d’ordre un et résolution par pénalisation dualité d’une classe de problèmes non linéaires,” RAIRO R2, 41–76 (1975).

Afonso, M.

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

Aliney, S.

S. Aliney, “A property of the minimum vectors of a regularizing functional defined by means of the absolute norm,” IEEE Trans. Signal Process. 45, 913–917 (1997).
[CrossRef]

Arsenin, V.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems(V. H. Winston, 1997).

Bardsley, J. M.

J. M. Bardsley, “Wavefront reconstruction methods for adaptive optics systems on ground-based telescopes,” SIAM J. Matrix Anal. Appl. 30, 67–83 (2008).
[CrossRef]

Bioucas-Dias, J.

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

Boo, K.

N. K. Bose and K. Boo, “High-resolution image reconstruction with multisensors,” Int. J. Imaging Syst. Technol. 9, 294–304 (1998).
[CrossRef]

Bose, N. K.

N. K. Bose and K. Boo, “High-resolution image reconstruction with multisensors,” Int. J. Imaging Syst. Technol. 9, 294–304 (1998).
[CrossRef]

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Chan, R. H.

Y. W. Wen and R. H. Chan, “Parameter selection for total variation based image restoration using discrepancy principle,” IEEE Trans. Image Process. 21, 1770–1781 (2012).
[CrossRef]

R. H. Chan, J. F. Yang, and X. M. Yuan, “Alternating direction method for image inpainting in wavelet domain,” SIAM J. Imaging Sci. 4, 807–826 (2011).
[CrossRef]

R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007).
[CrossRef]

R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Chan, T. F.

R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE. Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods (SIAM, 2005).

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Chu, Q.

J. Nagy, S. Jefferies, and Q. Chu, “Fast PSF reconstruction using the frozen flow hypothesis,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference (2010).

Conan, J. M.

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

Donoho, D.

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41, 613–627 (1995).
[CrossRef]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Fatemi, F.

L. Rudin, S. Osher, and F. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Figueiredo, M.

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

Fried, D. L.

Gabay, D.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

Georges, J.

Glowinski, R.

R. Glowinski and A. Marocco, “Approximation par éléments finis d’ordre un et résolution par pénalisation dualité d’une classe de problèmes non linéaires,” RAIRO R2, 41–76 (1975).

R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-Verlag, 1984).

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

Goodman, J. W.

E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A. 17, 1177–1184 (2000).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Han, D. R.

B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002).
[CrossRef]

Harding, C. M.

Hart, M.

Hart, M. L.

He, B. S.

B. S. He and X. M. Yuan, “On the O(1/n) convergence rate of Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[CrossRef]

B. S. He and X. M. Yuan, “Convergence analysis of primal-dual algorithms for total variation image restoration,” SIAM J. Imaging Sci. 5, 119–149 (2012).
[CrossRef]

B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002).
[CrossRef]

B. S. He, H. Yang, and S. L. Wang, “Alternating direction method with self-Adaptive penalty parameters for monotone variational inequalities,” J. Opt. Theory Appl. 106, 337–356 (2000).
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Hege, E. K.

Huang, T. S.

R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (1984), Vol. 1, pp. 317–339.

Hudgin, R. H.

Jakobsson, H.

Jefferies, S.

J. Nagy, S. Jefferies, and Q. Chu, “Fast PSF reconstruction using the frozen flow hypothesis,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference (2010).

Jefferies, S. M.

Johnston, R. A.

Lam, E. Y.

E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A. 17, 1177–1184 (2000).
[CrossRef]

Lane, R. G.

Liao, L. Z.

B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002).
[CrossRef]

Marocco, A.

R. Glowinski and A. Marocco, “Approximation par éléments finis d’ordre un et résolution par pénalisation dualité d’une classe de problèmes non linéaires,” RAIRO R2, 41–76 (1975).

Mercier, B.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

Michau, V.

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

Morozov, V. A.

V. A. Morozov, Methods for Solving Incorrectly Posed Problems. (Springer-Verlag, 1984).

Mugnier, L. M.

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

Nagy, J.

J. Nagy, S. Jefferies, and Q. Chu, “Fast PSF reconstruction using the frozen flow hypothesis,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference (2010).

Ng, M.

M. Ng, P. A. Weiss, and X. M. Yuan, “Solving constrained total-variation problems via alternating direction methods,” SIAM J. Sci. Comput. 32, 2710–2736 (2010).
[CrossRef]

Nikolova, M.

M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging. Vis. 20, 99–120 (2004).
[CrossRef]

Nocedaland, J.

J. Nocedaland and S. J. Wright, Numerical Optimization(Springer-Verlag, 2006).

Osher, S.

L. Rudin, S. Osher, and F. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Robert, C.

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and F. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Salem, S.

L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001).
[CrossRef]

Shen, J.

T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods (SIAM, 2005).

Shen, L. X.

R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Shen, Z. W.

R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007).
[CrossRef]

R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Tikhonov, A.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems(V. H. Winston, 1997).

Tsai, R. Y.

R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (1984), Vol. 1, pp. 317–339.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

Wahba, G.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Wang, S. L.

B. S. He, H. Yang, and S. L. Wang, “Alternating direction method with self-Adaptive penalty parameters for monotone variational inequalities,” J. Opt. Theory Appl. 106, 337–356 (2000).
[CrossRef]

Weiss, P. A.

M. Ng, P. A. Weiss, and X. M. Yuan, “Solving constrained total-variation problems via alternating direction methods,” SIAM J. Sci. Comput. 32, 2710–2736 (2010).
[CrossRef]

Wen, Y. W.

Y. W. Wen and R. H. Chan, “Parameter selection for total variation based image restoration using discrepancy principle,” IEEE Trans. Image Process. 21, 1770–1781 (2012).
[CrossRef]

Wong, C. K.

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE. Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

Wright, S. J.

J. Nocedaland and S. J. Wright, Numerical Optimization(Springer-Verlag, 2006).

Xia, T.

R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007).
[CrossRef]

Yang, H.

B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002).
[CrossRef]

B. S. He, H. Yang, and S. L. Wang, “Alternating direction method with self-Adaptive penalty parameters for monotone variational inequalities,” J. Opt. Theory Appl. 106, 337–356 (2000).
[CrossRef]

Yang, J. F.

R. H. Chan, J. F. Yang, and X. M. Yuan, “Alternating direction method for image inpainting in wavelet domain,” SIAM J. Imaging Sci. 4, 807–826 (2011).
[CrossRef]

Yuan, X. M.

B. S. He and X. M. Yuan, “On the O(1/n) convergence rate of Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[CrossRef]

B. S. He and X. M. Yuan, “Convergence analysis of primal-dual algorithms for total variation image restoration,” SIAM J. Imaging Sci. 5, 119–149 (2012).
[CrossRef]

R. H. Chan, J. F. Yang, and X. M. Yuan, “Alternating direction method for image inpainting in wavelet domain,” SIAM J. Imaging Sci. 4, 807–826 (2011).
[CrossRef]

M. Ng, P. A. Weiss, and X. M. Yuan, “Solving constrained total-variation problems via alternating direction methods,” SIAM J. Sci. Comput. 32, 2710–2736 (2010).
[CrossRef]

Appl. Comput. Harmon. Anal.

R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007).
[CrossRef]

Appl. Opt.

Comput. Math. Appl.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

Found. Trends Mach. Learn.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

True phase and its gradients.

Fig. 2.
Fig. 2.

First frame high-resolution (256-by-256) and low-resolution (32-by-32) phase gradients within the telescope aperture.

Fig. 3.
Fig. 3.

Reconstructed horizontal (top row), vertical (center row) direction phase gradients and the PSF from the eighth frame (bottom row).

Fig. 4.
Fig. 4.

Relative errors for all reconstructed PSFs.

Fig. 5.
Fig. 5.

Image restoration using PSF reconstructed by different models.

Fig. 6.
Fig. 6.

Reconstructed horizontal (top row) and vertical (center row) direction phase gradients and the PSF (bottom row).

Fig. 7.
Fig. 7.

Relative errors for all reconstructed PSFs.

Fig. 8.
Fig. 8.

Image restoration using PSF reconstructed by different models.

Fig. 9.
Fig. 9.

Reconstructed horizontal (top row) and vertical (center row) direction phase gradients and the PSF (bottom row).

Fig. 10.
Fig. 10.

Image restoration using PSF reconstructed by different models.

Fig. 11.
Fig. 11.

Performances of l 1 - l 2 model with different parameter values. Left: for different α with β = 10 3 . Right: for different β with α = 10 4 .

Tables (1)

Tables Icon

Table 1. Wavefront Reconstruction with Different Seeing Conditions

Equations (24)

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g ( x , y ) = k ( x , y ) * f ( x , y ) + ε g ( x , y ) ,
k ( x , y ) = | F 1 { P ( x , y ) exp [ ι ( 1 ω ( x , y ) ) ] } | 2 = | F 1 { P ( x , y ) exp [ ι ϕ ( x , y ) ] } | 2 ,
k ( x , y ) = | F 1 { P ( x , y ) } | 2 ,
[ s x s y ] = [ D x D y ] ϕ + ε s .
s x i = R W A i s x + n x i and s y i = R W A i s y + n y i , i = 1 , 2 , , m ,
min s α s 2 2 + 1 2 i = 1 m R W A i s d i 2 2 ,
min s α s 1 + 1 p i = 1 m R W A i s d i p p , p = 1 , 2 .
min θ 1 ( x ) + θ 2 ( y ) s.t. B x + C y = b , x X , y Y ,
L ( x , y , λ ) = θ 1 ( x ) + θ 2 ( y ) λ , B x + C y b + β 2 B x + C y b 2 2 ,
min s α s 1 + 1 p RWAs d p p , p = 1 , 2 .
min α s 1 + 1 2 R W A v d 2 2 s.t. s = v .
L ( s , v , λ ) = α s 1 + 1 2 R W A v d 2 2 λ , s v + β 2 s v 2 2 ,
s k + 1 = arg min s α s 1 + β 2 s v k λ k β 2 2 ,
v k + 1 = arg min v 1 2 R W A v d 2 2 λ k , s k + 1 v + β 2 s k + 1 v 2 2 ,
[ β I + ( RWA ) T RWA ] v k + 1 = ( β s k + 1 λ k ) + ( RWA ) T d .
min α u 1 + v 1 s.t. s u = 0 and RWAs v = d .
B ( I RWA ) , C ( I 0 0 I ) and b ( 0 d ) ,
L ( s , u , v , λ 1 , λ 2 ) = α u 1 + v 1 + β 1 2 u s λ 1 β 1 2 2 + β 2 2 RWAs v d λ 2 β 2 2 2 ,
s k + 1 = arg min s β 1 s u k + λ 1 k β 1 2 2 + β 2 RWAs v k d λ 2 k β 2 2 2 ,
[ β 1 I + β 2 ( RWA ) T RWA ] s k + 1 = ( β 1 u k λ 1 k ) + ( RWA ) T [ β 2 ( v k + d ) + λ 2 k ] .
( u k + 1 , v k + 1 ) = arg min u , v α u 1 + v 1 + β 1 2 u s k + 1 λ 1 k β 1 2 2 + β 2 2 v RWA s k + 1 + d + λ 2 k β 2 2 2 ,
u k + 1 = arg min u α u 1 + β 1 2 u s k + 1 λ 1 k β 1 2 2 .
v k + 1 = arg min v v 1 + β 2 2 v RWA s k + 1 + d + λ 2 k β 2 2 2 ,
v k + 1 = shrink 1 β 2 ( RWAs k + 1 d λ 2 k β 2 ) .

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