Abstract

A framework is proposed for optimal joint design of the optical and reconstruction filters in a computational imaging system. First, a technique for the design of a physically unconstrained system is proposed whose performance serves as a universal bound on any realistic computational imaging system. Increasing levels of constraints are then imposed to emulate a physically realizable optical filter. The proposed design employs a generalized Benders’ decomposition method to yield multiple globally optimal solutions to the nonconvex optimization problem. Structured, closed-form solutions for the design of observation and reconstruction filters, in terms of the system input and noise autocorrelation matrices, are presented. Numerical comparison with a state-of-the-art optical system shows the advantage of joint optimization and concurrent design.

© 2008 Optical Society of America

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References

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2006

2004

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

2003

2002

2001

1995

1971

1970

A. Geoffrion, “Elements of large-scale mathematical programming,” Manage. Sci. 16, 652-691 (1970).
[CrossRef]

Andrews, H.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Athale, R.

Baheti, P. K.

Bhakta, V.

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2003).

Bräuer, A.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Brookes, M.

M. Brookes, The Matrix Reference Manual (Imperial College, 2005), http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html#Intro.

Cathey, T.

Christensen, M.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

Dannberg, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Diamantaras, K.

K. Diamantaras and S. Kung, Principal Component Neural Networks: Theory and Applications (Wiley, 1996).

Douglas, S.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

Dowski, E.

Duparré, J.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Faraji, H.

H. Faraji and W. James MacLean, “CCD noise removal in digital images,” IEEE Trans. Image Process. 15, (2006).
[CrossRef]

Geoffrion, A.

A. Geoffrion, “Elements of large-scale mathematical programming,” Manage. Sci. 16, 652-691 (1970).
[CrossRef]

A. Geoffrion, “Generalized benders decomposition,” J. Optim. Theory Appl. 10, 237-260 (1972).

Golub, G.

G. Golub and C. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Gonzalez, R.

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).

Goodman, J.

Haney, M.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

Herzig, H.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Hunt, B.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Ichioka, Y.

Ishida, K.

Kondou, N.

Kumagai, T.

Kung, S.

K. Diamantaras and S. Kung, Principal Component Neural Networks: Theory and Applications (Wiley, 1996).

Loan, C.

G. Golub and C. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Luenberger, D.

D. Luenberger, Linear and Non-Linear Programming, 2nd ed. (Springer, 2004).

MacLean, W. James

H. Faraji and W. James MacLean, “CCD noise removal in digital images,” IEEE Trans. Image Process. 15, (2006).
[CrossRef]

Mait, J.

J. Mait, “A history of imaging: revisiting the past to chart the future,” Opt. Photon. News 17, 22-27 (2006).
[CrossRef]

J. Mait, R. Athale, and J. van der Gracht, “Evolutionary paths in imaging and recent trends,” Opt. Express 11, 2093-2101(2003).
[CrossRef] [PubMed]

Mirani, T.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Neifeld, M. A.

O'Sullivan, J. A.

J. A. O'Sullivan, “Alternating minimization algorithms: from Blahut-Arimoto to expectation-maximization,” in Codes, Curves, and Signals: Common Threads in Communications, A. Vardy, ed. (Springer, 1998), pp. 173-192.
[CrossRef]

Pelli, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Rajan, D.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

Russell, F.

Scharf, T.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Schreiber, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Strang, G.

G. Strang, Introduction to Linear Algebra, 3rd ed. (Wellesley-Cambridge Press, 2003).

Tanida, J.

van der Gracht, J.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2003).

Völkel, R.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Wood, S.

M. Christensen, V. Bhakta, D. Rajan, T. Mirani, S. Douglas, S. Wood, and M. Haney, “Adaptive flat multiresolution multiplexed computational imaging architecture utilizing micromirror arrays to steer subimager fields of view,” Appl. Opt. 45, 2884-2892 (2006).
[CrossRef] [PubMed]

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

Woods, R.

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).

Yamada, K.

Appl. Opt.

IEEE Trans. Image Process.

H. Faraji and W. James MacLean, “CCD noise removal in digital images,” IEEE Trans. Image Process. 15, (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Optim. Theory Appl.

A. Geoffrion, “Generalized benders decomposition,” J. Optim. Theory Appl. 10, 237-260 (1972).

Manage. Sci.

A. Geoffrion, “Elements of large-scale mathematical programming,” Manage. Sci. 16, 652-691 (1970).
[CrossRef]

Opt. Express

Opt. Photon. News

J. Mait, “A history of imaging: revisiting the past to chart the future,” Opt. Photon. News 17, 22-27 (2006).
[CrossRef]

Proc. SPIE

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H. Herzig, and A. Bräuer, “Artificial compound eyes--different concepts and their application to ultraflat image acquisition sensors,” Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Other

M. Christensen, M. Haney, D. Rajan, S. Wood, and S. Douglas, “Panoptes: A thin agile multi-resolution imaging sensor,” presented at Government Microcircuit Applications and Critical Technology Conference (GOMACTech-05), 4-7 April 2005.

T. Mirani, M. Christensen, S. Douglas, D. Rajan, and S. Wood, “Optimal co-design of computational imaging systems,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings (ICASSP '05) (IEEE, 2005). Vol. 2, pp. 597-600.
[CrossRef]

J. A. O'Sullivan, “Alternating minimization algorithms: from Blahut-Arimoto to expectation-maximization,” in Codes, Curves, and Signals: Common Threads in Communications, A. Vardy, ed. (Springer, 1998), pp. 173-192.
[CrossRef]

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

M. Brookes, The Matrix Reference Manual (Imperial College, 2005), http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html#Intro.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2003).

D. Luenberger, Linear and Non-Linear Programming, 2nd ed. (Springer, 2004).

K. Diamantaras and S. Kung, Principal Component Neural Networks: Theory and Applications (Wiley, 1996).

G. Strang, Introduction to Linear Algebra, 3rd ed. (Wellesley-Cambridge Press, 2003).

G. Golub and C. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).

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Figures (7)

Fig. 1
Fig. 1

Simplified block diagram.

Fig. 2
Fig. 2

Convergence of MSE for alternating minimization procedure.

Fig. 3
Fig. 3

(a) Detected Lena image. (b) Reconstructed Lena image, PSNR = 38.2337 . (c) Reconstructed Lena image, H nec , PSNR = 36.9964 . (d) Reconstructed image, H ots , spot size = 0.55 μm , PSNR = 36.4454 . (e) Reconstructed image, H ots , spot size = 1.3 μm , PSNR = 33.7098 .

Fig. 4
Fig. 4

(a) Detected airport image. (b) Reconstructed airport image, PSNR = 38.5194 . (c) Reconstructed airport image, H nec , PSNR = 36.7812 . (d) Reconstructed airport image, H ots , spot size = 0.55 μm , PSNR = 34.5474 . (e) Reconstructed airport image, H ots , spot size = 1.3 μm , PSNR = 33.8328 .

Fig. 5
Fig. 5

PSNR versus number of pixels at the detector m, λ z 2 = 1 .

Fig. 6
Fig. 6

PSNR versus number of pixels at the detector m, λ z 2 = 10 .

Fig. 7
Fig. 7

PSNR versus inverse of noise variance, m = 16 and m = 32 .

Equations (94)

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

g = Hf + z ,
f ^ = Wg = W ( Hf + z ) .
e = f f ^ .
J = E ( e t e ) = E [ Tr ( e e t ) ] ,
J = E [ Tr ( e e t ) ] = E [ Tr [ ( f Wg ) ( f Wg ) t ] ] = E [ Tr [ f f t W ( Hf f t + z f t ) ( f f t H t + f z t ) W t + W ( Hf f t H t + z f t H t + Hf z t + z z t ) W t ] ] .
J = Tr { E [ ff t ] E [ WHff t ] E [ ff t H t W t ] + E [ WHf f t H t W t ] + E [ Wz z t W t ] } .
R f = E [ f f t ] , R z = E [ z z t ] .
J = ϕ ( H , W ) = Tr [ R f 2 WH R f + WH R f H t W t + W R z W t ] .
W * = R f H t ( H R f H t + R z ) 1 .
min { H , W } Tr [ R f 2 WH R f + WH R f H t W t + W R z W t ] .
D = [ 2 J H 2 2 J H W 2 J W H 2 J W 2 ]
H ec = { H ec :     H ec j n × 1 = j m × 1 } ,
min { H ec H ec , W ec } J ec = Tr [ R f 2 W ec H ec R f + W ec H ec R f H ec t W ec t + W ec R z W ec t ] ,
H nneg = { H nneg : H nneg i j 0     i = 1 m , j = 1 n } ,
min { H H ec H nneg , W nec } Tr [ R f 2 W nec H nec R f + W nec H nec R f H nec t W nec t + W nec R z W nec t ] .
min { H , W } ϕ ( H , W ) subject to Ψ ( H , W ) 0 , H H , W W ,
min W v ( W ) subject to W W V ,
v ( W ) inf H ϕ ( H , W ) subject to Ψ ( H , W ) 0 , H H ,
V = { W :     Ψ ( H , W ) 0 for some H H } .
min { H } J = Tr [ R f 2 WH R f + WH R f H t W t + W R z W t ] .
J H = 2 W t R f + 2 W t WH R f = 0.
H * = ( W t W ) 1 W t .
min { W } Tr [ R f W ( W t W ) 1 W t R f + W R z W t ] .
W = R f W ( W t W ) 1 [ R z + ( W t W ) 1 W t R f W ( W t W ) 1 ] 1 .
W * = R f H * t ( H * R f H * t + R z ) 1 ,
H * = ( W * t W * ) 1 W * t
H = ( ( R f H t ( H R f H t + R z ) 1 ) t ( R f H t ( H R f H t + R z ) 1 ) ) 1 ( R f H t ( H R f H t + R z ) 1 ) t = ( H R f H t + R z ) ( H R f 2 H t ) 1 H R f .
H R f H t = H R f H t + R z .
U = Q z .
σ H i     i = 1 m .
V = Q f P n ,
P n = [ 0 P n m I m 0 ] ,
U Σ H V t = [ ( U Σ H V t Q f Λ f Q f t V Σ H t U t + Q z Λ z Q z t ) . ( U Σ H V t Q f Λ f 2 Q f t V Σ H t U t ) 1 ( U Σ H V t Q f Λ f Q f t ) ] .
Σ H = ( Σ H P n t Λ f P n Σ H t + Λ z ) ( Σ H P n t Λ f 2 P n Σ H t ) 1 Σ H P n t Λ f P n .
Λ f = [ Λ f n m 0 0 Λ f m ] ,
Λ f , P = P n t Λ f P n = [ Λ f m 0 0 P n m t Λ f n m P n m ] .
Λ f 2 , P = P n t Λ f 2 P n = [ Λ f m 2 0 0 P n m t Λ f n m 2 P n - m ] .
Σ H = ( Σ H Λ f , P Σ H t + Λ z ) ( Σ H Λ f 2 , P Σ H t ) 1 Σ H Λ f , P .
σ H i = ( λ f i + p σ H i 2 + λ z i ) ( λ f i + p σ H i ) λ f i + p 2 σ H i 2    i = 1 m .
λ zi σ H i λ f i + p = 0    i = 1 m ,
J = Tr [ R f R f H t ( R z + H R f H t ) 1 H R f ] .
= Tr [ R f ] Tr [ R f H t ( H R f H t ) 1 ( H R f ) ] .
J = Tr [ R f ] Tr [ X ] ,
X = R f H t ( H R f H t ) 1 ( H R f ) .
Tr [ X ] = Tr [ Q f Λ f P Σ H t ( Σ H P t Λ f P Σ H t ) 1 Σ H P t Λ f Q f t ] .
Tr [ X ] = Tr [ Q f Λ f P Σ H t ( Σ H Λ f , P Σ H t ) 1 Σ H P t Λ f Q f t ] .
Tr [ X ] = i = 1 m λ f i + p 2 σ H i 2 ( σ H i 2 λ f i + p ) 1 = i = 1 m λ f i + p ,
J = Tr [ λ f 1 0 0 0 λ f 2 0 λ f n ] Tr [ 0 0 0 0 0 0 0 0 0 0 0 0 λ f n m + 1 0 0 0 0 0 0 0 λ f n ] ,
= i = 1 n λ f i i = 1 m λ f i + p = i = 1 n m λ f i .
H ec * = ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 ,
J ec = Tr [ R f 2 W H ec R f + W H ec R f H ec t W t + W R z W t ] + λ [ H ec j n × 1 j m × 1 ] ,
J ec H ec = 2 W t R f + 2 W t W H ec R f + λ t j 1 × n = 0 ,
H ec = 1 2 ( W t W ) 1 ( 2 W t R f λ t j 1 × n ) R f 1 .
1 2 ( W t W ) 1 ( 2 W t R f λ t j 1 × n ) R f 1 j n × 1 = j m × 1 λ t = 2 α ( W t j n × 1 W t W j m × 1 ) ,
H ec * = ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 .
min W ec Tr [ R f W ec ( W ec t W ec ) 1 W ec t R f + 1 α 2 W ec ( W ec t W ec ) 1 W ec t j n × n R f 1 j n × n 2 α 2 W ec j m × n R f 1 j n × n + 1 α 2 W ec j m × n R f 1 j n × m W ec t + W ec R z W ec t ]
W ec * = R f H ec t ( H ec R f H ec t + R z ) 1 ,
H ec * = ( W ec t W ec ) 1 W ec t ( R f 1 α j n × n + 1 α W ec j m × n ) R f 1 .
U ec = Q z , V ec = Q f P n ,
H ec j n × 1 = j m × 1 U ec Σ H ec V ec t j n × 1 = j m × 1 .
Q z Σ H ec ( Q f P n ) t j n × 1 = j m × 1 Σ H ec P n t Q f t j n × 1 = Q z t j m × 1 .
σ H ec i = b i / a i     i = 1 m .
i = 1 n λ f i i = 1 m λ f i + p ( σ H ec i 2 λ f i + p σ H ec i 2 λ f i + p + λ z i ) .
J ec = Tr [ R f ] Tr [ X ] ,
X = R f H ec t ( H ec R f H ec t + R z ) 1 ( H ec R f ) .
Tr [ X ] = Tr [ Q f Λ f P n Σ H ec t ( Σ H ec P n t Λ f P n Σ H ec t + Λ z ) 1 Σ H ec P n t Λ f Q f t ] .
Tr [ X ] = Tr [ Q f Λ f P n Σ H ec t ( Σ H ec Λ f , P Σ H ec t + Λ z ) 1 Σ H ec P n t Λ f Q f t ] .
Tr [ X ] = i = 1 m λ f i + p 2 σ H ec i 2 ( σ H ec i 2 λ f i + p + λ z i ) 1 ,
J ec = Tr [ R f ] Tr [ X ] = i = 1 n λ f i i = 1 m λ f i + p 2 σ H ec i 2 ( σ H ec i 2 λ f i + p + λ z i ) 1 = i = 1 n λ f i i = 1 m λ f i + p ( σ H ec i 2 λ f i + p σ H ec i 2 λ f i + p + λ z i ) ,
J ec J = [ i = 1 n λ f i i = 1 m λ f i + p ( σ H ec i 2 λ f i + p σ H ec i 2 λ f i + p + λ z i ) ] [ i = 1 n λ f i i = 1 m λ f i + p ]
= i = 1 m λ f i + p i = 1 m λ f i + p ( σ H ec i 2 λ f i + p σ H ec i 2 λ f i + p + λ z i ) .
J alt i = Tr [ R f 2 W alt i H alt i R f + W alt i H alt i R f H alt i t W alt i t + W alt i R z W alt i t ] .
PSNR = 10 log 10 ( 255 2 / MSE ) ,
J = Tr [ R f 2 W ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) + A + W R z W t ] ,
A = W ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 R f R f 1 ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 W t .
Tr [ A ] = Tr [ W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 ] = Tr [ ( R f 1 α j n × n + 1 α W j m × n ) R f 1 ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 W t ] = Tr [ ( I 1 α j n × n R f 1 + 1 α W j m × n R f 1 ) ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 W t ] = Tr [ ( R f 1 α j n × n + 1 α W j m × n 1 α j n × n + 1 α 2 j n × n R f 1 j n × n 1 α 2 W j m × n R f 1 j n × n + 1 α j n × m W t 1 α 2 j n × n R f 1 j n × m W t + 1 α 2 W j m × n R f 1 j n × m W t ) W ( W t W ) 1 W t ] = Tr [ ( R f 2 α j n × n + 1 α W j m × n + 1 α 2 j n × n R f 1 j n × n 1 α 2 W j m × n R f 1 j n × n + 1 α j n × m W t 1 α 2 j n × n R f 1 j n × m W t + 1 α 2 W j m × n R f 1 j n × m W t ) W ( W t W ) 1 W t ] .
Tr [ A ] = Tr [ W ( W t W ) 1 W t R f 2 α W ( W t W ) 1 W t j n × n + 1 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × n + 2 α W j m × n 1 α 2 W j m × n R f 1 j n × n 1 α 2 j n × n R f 1 j n × m W t + 1 α 2 W j m × n R f 1 j n × m W t ] .
J = Tr [ R f W ( W t W ) 1 W t R f + 1 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × n 1 α 2 W j m × n R f 1 j n × n 1 α 2 j n × m W t j n × n R f 1 + 1 α 2 j n × m W t W j m × n R f 1 + W R z W t ] ,
J = Tr [ R f ] Tr [ ( W t W ) 1 W t R f W + 1 α 2 ( W t W ) 1 W t j n × n R f 1 j n × n W 2 α 2 j m × n R f 1 j n × n W + 1 α 2 W t W j m × n R f 1 j n × m + W t W R z ] .
d d X Tr [ ( X t CX ) 1 ( X t BX ) ] = 2 CX ( X t CX ) 1 X t BX ( X t CX ) 1 + 2 BX ( X t CX ) 1 .
d d X Tr [ XD X t ] = 2 XD .
J W = 2 W ( W t W ) 1 W t R f W ( W t W ) 1 2 R f W ( W t W ) 1 2 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × n W ( W t W ) 1 + 2 α 2 j n × n R f 1 j n × n W ( W t W ) 1 2 α 2 j n × n R f 1 j n × m + 2 α 2 W j m × n R f 1 j n × m + 2 W R z = 0.
W = ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 [ ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 + R z ] 1
W [ ( W t W ) 1 W t ( R f 1 α j n × n + 1 α W j m × n ) R f 1 ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 + R z ] = ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 .
W [ ( ( W t W ) 1 W t R f 1 α ( W t W ) 1 W t j n × n + 1 α j m × n ) ( W ( W t W ) 1 1 α R f 1 j n × n W ( W t W ) 1 + 1 α R f 1 j n × m ) + R z ] = ( R f 1 α j n × n + 1 α j n × m W t ) W ( W t W ) 1 .
0 = W ( W t W ) 1 W t R f W ( W t W ) 1 1 α W ( W t W ) 1 W t j n × n W ( W t W ) 1 + 1 α W j m × n W ( W t W ) 1 1 α W ( W t W ) 1 W t j n × n W ( W t W ) 1 + 1 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × n W ( W t W ) 1 1 α 2 W j m × n R f 1 j n × n W ( W t W ) 1 + 1 α W ( W t W ) 1 W t j n × m 1 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × m + 1 α 2 W j m × n R f 1 j n × m + W R z R f W ( W t W ) 1 + 1 α j n × n W ( W t W ) 1 1 α j n × m .
0 = 2 W ( W t W ) 1 W t R f W ( W t W ) 1 2 R f W ( W t W ) 1 2 α 2 W ( W t W ) 1 W t j n × n R f 1 j n × n W ( W t W ) 1 + 2 α 2 j n × n R f 1 j n × n W ( W t W ) 1 2 α 2 j n × n R f 1 j n × m + 2 α 2 W j m × n R f 1 j n × m + 2 W R z .
H = ( H R f H t + R z ) ( H R f 2 H t ) 1 H R f ( R f 1 α j n × n + 1 α R f H t ( H R f H t + R z ) 1 j m × n ) R f 1 .
U Σ H V t = ( U Σ H V t Q f Λ f Q f t V Σ H t U t + Q z Λ z Q z t ) ( U Σ H V t Q f Λ f 2 Q f t V Σ H t U t ) 1 . ( U Σ H V t Q f Λ f Q f t ) ( I 1 α j n × n Q f Λ f 1 Q f t + 1 α Q f Λ f Q f t V Σ H t U t ( U Σ H V t Q f Λ f Q f t V Σ H t U t + Q z Λ z Q z t ) 1 j m × n Q f Λ f 1 Q f t ) .
Σ H = ( Σ H Λ f , P Σ H t + Λ z ) ( Σ H Λ f 2 , P Σ H t ) 1 Σ H Λ f , P 1 α ( Σ H Λ f , P Σ H t + Λ z ) ( Σ H Λ f 2 , P Σ H t ) 1 Σ H Λ f , P a a t Λ f 1 , P + 1 α b a t Λ f 1 , P .
σ H i 2 λ f i a i 2 σ H i λ f i b i a i + λ z i a i 2 λ z i λ f i = 0 ,
σ H i 2 λ f i a i a j σ H i λ f i b i a j + λ z i a i a j = 0.
σ H i = b i / a i ,
Σ H a = b ,

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