Abstract

Measuring an array of variables is central to many systems, including imagers (array of pixels), spectrometers (array of spectral bands) and lighting systems. Each of the measurements, however, is prone to noise and potential sensor saturation. It is recognized by a growing number of methods that such problems can be reduced by multiplexing the measured variables. In each measurement, multiple variables (radiation channels) are mixed (multiplexed) by a code. Then, after data acquisition, the variables are decoupled computationally in post processing. Potential benefits of the use of multiplexing include increased signal-to-noise ratio and accommodation of scene dynamic range. However, existing multiplexing schemes, including Hadamard-based codes, are inhibited by fundamental limits set by sensor saturation and Poisson distributed photon noise, which is scene dependent. There is thus a need to find optimal codes that best increase the signal to noise ratio, while accounting for these effects. Hence, this paper deals with the pursuit of such optimal measurements that avoid saturation and account for the signal dependency of noise. The paper derives lower bounds on the mean square error of demultiplexed variables. This is useful for assessing the optimality of numerically-searched multiplexing codes, thus expediting the numerical search. Furthermore, the paper states the necessary conditions for attaining the lower bounds by a general code. We show that graph theory can be harnessed for finding such ideal codes, by the use of strongly regular graphs.

© 2007 Optical Society of America

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2007 (3)

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang, "Polarization multiplexing and demultiplexing for appearancebased modeling," IEEE Trans. PAMI 29,362-367 (2007).
[CrossRef]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, "Multiplexing for optimal lighting," IEEE Trans. PAMI 29,1339-1354 (2007).
[CrossRef]

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. "Longwave infrared (LWIR) coded aperture dispersive spectrometer," Opt. Express 15,5742-5753 (2007).
[CrossRef] [PubMed]

2006 (1)

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

2005 (3)

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

K. C. Lee, J. Ho, and D. J. Kriegman, "Acquiring linear subspaces for face recognition under variable lighting," IEEE Trans. PAMI 27,684-698 (2005).
[CrossRef]

A. Wuttig, "Optimal transformations for optical multiplex measurements in the presence of photon noise," Appl. Opt. 44,2710-2719 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

2003 (1)

G. Nitzsche and R. Riesenberg. "Noise, fluctuation and HADAMARD-transform-spectrometry," In Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

2002 (2)

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. "Spectrally resolved fluorescence lifetime imaging microscopy," Appl. Spectrosc. 56, 63-84 (2002).
[CrossRef]

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

2000 (2)

Y. A. Shutova, "Optimization of binary masks for Hadamard-transform optical spectrometers," J. Opt. Technol. 67,50-53 (2000).
[CrossRef]

M. T. Chu, "A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values," SIAM J. on Numerical Analysis 37,1004-1020 (2000).
[CrossRef]

1997 (2)

M. Alicacute, B. Mond, J. Pecbreve aricacute and V. Volenec, "The arithmetic-geometric-harmonic-mean and related matrix inequalities," Numer. Linear Algebra Appl. 264,55-62 (1997).
[CrossRef]

A. Busboom, H. D. Schotten, and H. Elders-Boll, "Coded aperture imaging with multiple measurements," J. Opt. Soc. Am. A 14,1058-1065 (1997).
[CrossRef]

1996 (1)

J. F. Turner and P. J. Treado. "Adaptive filtering and hadamard transform imaging spectroscopy with an acoustooptic tunable filter (AOTF)," Proc. SPIE 2599, 285-293 (1996).
[CrossRef]

1992 (1)

V. P. Kozlov and E. V. Sedunov, "Optimization of multiplex measuring systems in the presence of statistical signal fluctuations," Cybern. Syst. Anal. 28,830-839 (1992).
[CrossRef]

1988 (1)

G. K. Skinner. "X-ray imaging with coded masks," Sci. Am. 259,84-89 (1988).
[CrossRef] [PubMed]

1979 (1)

R. J. Proctor, G. K. Skinner, and A. P. Willmore, "The design of optimum coded mask X-ray telescopes," Royal Astron. Soc. Monthly Notices 187,633-643 (1979).

1978 (2)

1974 (1)

T. M. Palmieri, "Multiplex methods and advantages in X-ray astronomy," Astrophys. Space Sci. 28,277-287 (1974).
[CrossRef]

1968 (1)

J. J. Seidel, "Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3," Numer. Linear Algebra Appl. 1,281-289 (1968).
[CrossRef]

Alicacute, M.

M. Alicacute, B. Mond, J. Pecbreve aricacute and V. Volenec, "The arithmetic-geometric-harmonic-mean and related matrix inequalities," Numer. Linear Algebra Appl. 264,55-62 (1997).
[CrossRef]

Arndt-Jovin, D. J.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. "Spectrally resolved fluorescence lifetime imaging microscopy," Appl. Spectrosc. 56, 63-84 (2002).
[CrossRef]

Belhumeur, P. N.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, "Multiplexing for optimal lighting," IEEE Trans. PAMI 29,1339-1354 (2007).
[CrossRef]

Bolas, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

Brady, D. J.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. "Longwave infrared (LWIR) coded aperture dispersive spectrometer," Opt. Express 15,5742-5753 (2007).
[CrossRef] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

Busboom, A.

Cannon, T. M.

Chen, B.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

Chu, M. T.

M. T. Chu, "A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values," SIAM J. on Numerical Analysis 37,1004-1020 (2000).
[CrossRef]

Coifman, R. R.

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

Cula, O. G.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang, "Polarization multiplexing and demultiplexing for appearancebased modeling," IEEE Trans. PAMI 29,362-367 (2007).
[CrossRef]

Dana, K. J.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang, "Polarization multiplexing and demultiplexing for appearancebased modeling," IEEE Trans. PAMI 29,362-367 (2007).
[CrossRef]

Debevec, P.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

DeVerse, R. A.

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

Elders-Boll, H.

Fateley, W. G.

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

Fenimore, E. E.

Fernandez, C.

Gardner, A.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

Gehm, M. E.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. "Longwave infrared (LWIR) coded aperture dispersive spectrometer," Opt. Express 15,5742-5753 (2007).
[CrossRef] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

Geshwind, F. B.

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

Guenther, B. D.

Hammaker, R. M.

W. G. Fateley, R. M. Hammaker, R. A. DeVerse, R. R. Coifman, and F. B. Geshwind. "The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph," Vib. Spectrosc. 29,163-170 (2002).
[CrossRef]

Hanley, Q. S.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. "Spectrally resolved fluorescence lifetime imaging microscopy," Appl. Spectrosc. 56, 63-84 (2002).
[CrossRef]

Hawkins, T.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

Ho, J.

K. C. Lee, J. Ho, and D. J. Kriegman, "Acquiring linear subspaces for face recognition under variable lighting," IEEE Trans. PAMI 27,684-698 (2005).
[CrossRef]

Horowitz, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

Jovin, T. M.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. "Spectrally resolved fluorescence lifetime imaging microscopy," Appl. Spectrosc. 56, 63-84 (2002).
[CrossRef]

Kozlov, V. P.

V. P. Kozlov and E. V. Sedunov, "Optimization of multiplex measuring systems in the presence of statistical signal fluctuations," Cybern. Syst. Anal. 28,830-839 (1992).
[CrossRef]

Kriegman, D. J.

K. C. Lee, J. Ho, and D. J. Kriegman, "Acquiring linear subspaces for face recognition under variable lighting," IEEE Trans. PAMI 27,684-698 (2005).
[CrossRef]

Lee, K. C.

K. C. Lee, J. Ho, and D. J. Kriegman, "Acquiring linear subspaces for face recognition under variable lighting," IEEE Trans. PAMI 27,684-698 (2005).
[CrossRef]

Levoy, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

McCain, S. T.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

McDowall, I.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

Mond, B.

M. Alicacute, B. Mond, J. Pecbreve aricacute and V. Volenec, "The arithmetic-geometric-harmonic-mean and related matrix inequalities," Numer. Linear Algebra Appl. 264,55-62 (1997).
[CrossRef]

Nayar, S. K.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, "Multiplexing for optimal lighting," IEEE Trans. PAMI 29,1339-1354 (2007).
[CrossRef]

Nitzsche, G.

G. Nitzsche and R. Riesenberg. "Noise, fluctuation and HADAMARD-transform-spectrometry," In Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

Pai, D. K.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang, "Polarization multiplexing and demultiplexing for appearancebased modeling," IEEE Trans. PAMI 29,362-367 (2007).
[CrossRef]

Palmieri, T. M.

T. M. Palmieri, "Multiplex methods and advantages in X-ray astronomy," Astrophys. Space Sci. 28,277-287 (1974).
[CrossRef]

Pitsianis, N. P.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

Potuluri, P.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

Proctor, R. J.

R. J. Proctor, G. K. Skinner, and A. P. Willmore, "The design of optimum coded mask X-ray telescopes," Royal Astron. Soc. Monthly Notices 187,633-643 (1979).

Riesenberg, R.

G. Nitzsche and R. Riesenberg. "Noise, fluctuation and HADAMARD-transform-spectrometry," In Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

Schechner, Y. Y.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, "Multiplexing for optimal lighting," IEEE Trans. PAMI 29,1339-1354 (2007).
[CrossRef]

Schotten, H. D.

Sedunov, E. V.

V. P. Kozlov and E. V. Sedunov, "Optimization of multiplex measuring systems in the presence of statistical signal fluctuations," Cybern. Syst. Anal. 28,830-839 (1992).
[CrossRef]

Seidel, J. J.

J. J. Seidel, "Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3," Numer. Linear Algebra Appl. 1,281-289 (1968).
[CrossRef]

Shutova, Y. A.

Skinner, G. K.

G. K. Skinner. "X-ray imaging with coded masks," Sci. Am. 259,84-89 (1988).
[CrossRef] [PubMed]

R. J. Proctor, G. K. Skinner, and A. P. Willmore, "The design of optimum coded mask X-ray telescopes," Royal Astron. Soc. Monthly Notices 187,633-643 (1979).

Sullivan, M. E.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. "Longwave infrared (LWIR) coded aperture dispersive spectrometer," Opt. Express 15,5742-5753 (2007).
[CrossRef] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 43,2965-2974 (2006).
[CrossRef]

Tchou, C.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

Treado, P. J.

J. F. Turner and P. J. Treado. "Adaptive filtering and hadamard transform imaging spectroscopy with an acoustooptic tunable filter (AOTF)," Proc. SPIE 2599, 285-293 (1996).
[CrossRef]

Turner, J. F.

J. F. Turner and P. J. Treado. "Adaptive filtering and hadamard transform imaging spectroscopy with an acoustooptic tunable filter (AOTF)," Proc. SPIE 2599, 285-293 (1996).
[CrossRef]

Unger, J.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec, "Performance relighting and reflectance transformation with time-multiplexed illumination," ACM TOG 24,756-764 (2005).

Vaish, V.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, "Synthetic aperture confocal imaging," ACM TOG 23,825-834 (2004).

Wang, D.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang, "Polarization multiplexing and demultiplexing for appearancebased modeling," IEEE Trans. PAMI 29,362-367 (2007).
[CrossRef]

Wenger, A.

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Figures (7)

Fig. 1.
Fig. 1.

An example of a strongly regular graph (Peterson) [31]. This graph has the parameters (N=10;k=3;α=0;β=1).

Fig. 2.
Fig. 2.

MSE ˜ as a function of {µf } N f =1. Each green dot marks the vector μ that minimizes MSE ˜ , when S is fixed. The highlighted line marks the ideal value of S. The green dot along this line marks the global minimum of MSE ˜ . This global minimum is derived in closed form and is thus unaffected by local minima.

Fig. 3.
Fig. 3.

The curve 1/µ represents elements summed in Eq. (34). The black dashed lines mark a state of the SVs of W. If the largest squared-SV, µ N , is reduces by Δµ N and in return the smallest squared-SV, µ 1, is increased by Δµ N , the sum in Eq. (34) is reduced.

Fig. 4.
Fig. 4.

The bound B min(C), for N=63. Here C varies from 1 to 63. The minimum of B min(C) is at C opt. There may exist C sat, above which saturation occurs, inhibiting multiplexing.

Fig. 5.
Fig. 5.

A semi-logarithmic plot of Eq. (62).

Fig. 6.
Fig. 6.

An example for an adjacency matrix of an SRG with parameters (45; 12; 3;3) developed by Ref. [35]. Here 1s and 0s are represented by white and black squares, respectively. Here α=β=3, hence this graph satisfies the conditions of Theorem 7.

Fig. 7.
Fig. 7.

(a) Graylevel spectral radiance of light transmitted through the atmosphere [38]. It is used to simulate the ground truth i. (b) Absolute error values |î-i| of simulated estimates based either on trivial sensing (red plot) or on SRG-based multiplexing (green plot).

Equations (89)

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a = W i + υ .
i ̂ = W 1 a .
MSE i ̂ = σ a 2 N trace [ ( W t W ) 1 ] .
trace ( RQ ) = trace ( QR ) .
trace ( W ) = f = 1 N λ f .
ξ f = μ f .
ξ m = λ m m .
m = f N ξ m m = f N λ m f { 2 , , N }
m=1 N ξ m = m=1 N | λ m |.
ω p , q = { 1 if p and q are neighbors 0 otherwise
ω ̅ p , q = { 1 if ω p , q = 0 and p q 0 otherwise .
q = 1 N ω p , q = k p .
k ( k α 1 ) = ( N k 1 ) β .
λ 1 Ω = ( α β ) + Δ 2
λ 2 Ω = ( α β ) Δ 2
λ 3 Ω = k ,
Δ ( α β ) 2 + 4 ( k β ) .
ξ f Ω = λ f Ω f .
C = C Had N + 1 2 .
s = 1 N w m , s = C m { 1 , 2 , , N } .
0 w m , s 1 m , s { 1 , 2 , , N } .
MSE ˜ MSE i ̂ σ a 2 = 1 N trace [ ( W t W ) 1 ] .
min W MSE ˜ min W 1 N trace [ ( W t W ) 1 ]
s . t . s = 1 N w m , s C = 0 m { 1 , , N }
w m , s 0 m , s { 1 , , N }
w m , s 1 0 m , s { 1 , , N } .
MSE ˜ 1 N trace [ ( W t W ) 1 ] = 1 N f = 1 N 1 μ f ,
μ f ξ f 2 f { 1 , , N } .
ξ N λ N .
λ N = C .
μ N ξ N 2 C 2 .
MSE ˜ = 1 N μ N + 1 N f = 1 N 1 1 μ f .
S f = 1 N μ f
min μ N { 1 N μ N + 1 N f = 1 N 1 1 μ f } = 1 N C 2 + 1 N f = 1 N 1 1 μ f .
Δ MSE ˜ = MSE ˜ μ f Δ μ N = 1 N μ f 2 Δ μ N < 0 .
Δ MSE ˜ = Δ μ N N ( 1 μ 1 2 1 μ N 2 ) < 0 .
ξ N = C
μ N = C 2 .
1 N f = 1 N 1 1 μ f N 1 N ( 1 N 1 f = 1 N 1 1 μ f ) .
1 N 1 f = 1 N 1 1 μ f N 1 f = 1 N 1 μ f .
1 N 1 f = 1 N 1 1 μ f N 1 S C 2 .
MSE ˜ B ,
B = 1 N C 2 + ( N 1 ) 2 N ( S C 2 ) .
trace ( W t W ) = trace ( W W t ) .
( W W t ) m , m = s = 1 N ( w m , s ) 2 .
( w m , s ) 2 w m , s .
( W W t ) m , m s = 1 N w m , s = C .
S = k = 1 N μ k = trace ( W t W ) .
S = trace ( W W t ) = m = 1 N ( W W t ) m , m N C .
0 S N C .
S ideal = N C .
B B min
B min ( C ) = [ 1 N C 2 + ( N 1 ) 2 N ( N C C 2 ) ] .
MSE i ̂ σ a 2 B min ( C ) .
Ψ { [ 1 , N ] + } .
C opt free arg min C Ψ B min ( C ) .
B min C 2 N C 3 ( N 1 ) 2 ( N 2 C ) N ( N C C 2 ) 2 = 0 .
C opt free = N 2 2 N 3 ± ( N 1 ) ( N 2 2 N + 9 ) 4 N 8 .
C opt int ROUND ( C opt free )
Ψ int { [ 1 , N ] + } .
C opt int = arg min C Ψ int B min ( C ) .
C Had C opt free = N 1 N 2 . N + 1 ( N 1 ) 2 + 8 4 .
lim N ( C Had C opt free ) = 0.5 .
C Had > C opt free > C Had 0.5 .
C opt int = C Had ,
Ψ sat = { Ψ [ 1 , C sat ] } ,
C opt sat = arg min C Ψ sat B min ( C ) .
C opt sat = { C opt free if C opt free C sat C sat otherwise .
ξ 1 Ω = C β
ξ 2 Ω = C .
α = k ( k 1 ) ( N 1 ) = C ( C 1 ) ( N 1 ) .
VAR ( n electr photo ) = ( n electr photo ) ,
a = n electr photo Q electr .
( n electr photo ) Q electr 2 = a Q electr .
σ a 2 = κ gray 2 + a Q electr .
a i s C .
σ a 2 = κ gray 2 + C η 2 .
MSE i ̂ = σ a 2 MSE ˜ .
MSE i ̂ = ( κ gray 2 + C η 2 ) MSE ˜ .
C opt final = arg min C MSE i ̂ ( C ) .
W offset W C I .
s = 1 N w m , s offset = 0 m { 1 , 2 , , N } .
max m { 1 , , N } u m f = 1 .
u m f = 1
w m , s u m f 1 m , s { 1 , , N } .
( W u f ) m = s = 1 N w m , s u s f s = 1 N w m , s u s f C .
( W u f ) m = λ f u f m = λ f u m f .
( W u f ) m = λ f .
λ f C .

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