Abstract

Our goal is to optimize regularized image reconstruction for emission tomography with respect to lesion detectability in the reconstructed images. We consider model observers whose decision variable is the maximum value of a local test statistic within a search area. Previous approaches have used simulations to evaluate the performance of such observers. We propose an alternative approach, where approximations of tail probabilities for the maximum of correlated Gaussian random fields facilitate analytical evaluation of detection performance. We illustrate how these approximations, which are reasonably accurate at low probability of false alarm operating points, can be used to optimize regularization with respect to lesion detectability.

© 2007 Optical Society of America

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  1. P. C. Hansen and D. P. O'Leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 1487-1506 (1993).
  2. C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996).
    [CrossRef]
  3. L. B. Lusted, "Signal detectability and medical decision-making," Science 171, 1217-1219 (1971).
    [CrossRef] [PubMed]
  4. C. E. Metz, "Basic principles of ROC analysis," Semin Nucl. Med. 8, 283-298 (1978).
    [CrossRef] [PubMed]
  5. M. S. Chesters, "Human visual perception and ROC methodology in medical imaging," Phys. Med. Biol. 37, 1433-1484 (1992).
    [CrossRef] [PubMed]
  6. R. G. Swensson, "Unified measurement of observer performance in detecting and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
    [CrossRef] [PubMed]
  7. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2003).
  8. J. Qi and R. H. Huesman, "Theoretical study of lesion detectability of MAP reconstruction using computer observers," IEEE Trans. Med. Imaging 20, 815-822 (2001).
    [CrossRef] [PubMed]
  9. Y. Xing, I.-T. Hsiao, and G. Gindi, "Rapid calculation of detectability in Bayesian single photon emission computed tomography," Phys. Med. Biol. 48, 3755-3774 (2003).
    [CrossRef] [PubMed]
  10. J. Qi, "Analysis of lesion detectability in Bayesian emission reconstruction with nonstationary object variability," IEEE Trans. Med. Imaging 23, 321-329 (2004).
    [CrossRef] [PubMed]
  11. A. Yendiki and J. A. Fessler, "Analysis of observer performance in known-location tasks for tomographic image reconstruction," IEEE Trans. Med. Imaging 25, 28-41 (2006).
    [CrossRef] [PubMed]
  12. J. Qi and R. H. Huesman, "Penalized maximum-likelihood image reconstruction for lesion detection," Phys. Med. Biol. 51, 4017-4030 (2006).
    [CrossRef]
  13. K. J. Myers and H. H. Barrett, "Addition of a channel mechanism to the ideal-observer model," J. Opt. Soc. Am. A 4, 2447-2457 (1987).
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  15. S. D. Wollenweber, B. M. W. Tsui, D. S. Lalush, E. C. Frey, K. J. LaCroix, and G. T. Gullberg, "Comparison of Hotelling observer models and human observers in defect detection from myocardial SPECT imaging," IEEE Trans. Nucl. Sci. 46, 2098-2103 (1999).
    [CrossRef]
  16. H. C. Gifford, M. A. King, D. J. de Vries, and E. J. Soares, "Channelized Hotelling and human observer correlation for lesion detection in hepatic SPECT imaging," J. Nucl. Med. 41, 514-521 (2000).
    [PubMed]
  17. C. K. Abbey and H. H. Barrett, "Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability," J. Opt. Soc. Am. A 18, 473-488 (2001).
    [CrossRef]
  18. S. Sankaran, E. C. Frey, K. L. Gilland, and B. M. Tsui, "Optimum compensation method and filter cutoff frequency in myocardial spect: a human observer study," J. Nucl. Med. 43, 432-438 (2002).
    [PubMed]
  19. A. Yendiki, "Analysis of signal detectability in statistically reconstructed tomographic images," Ph. D. dissertation (University of Michigan, 2005).
  20. H. C. Gifford, P. H. Pretorius, and M. A. King, "Comparison of human- and model-observer LROC studies," Proc. SPIE 5034, 112-122 (2003).
    [CrossRef]
  21. H. C. Gifford, P. E. Kinahan, C. Lartizien, and M. A. King, "Evaluation of multiclass model observers in PET LROC studies," in Proceedings of IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2004), Vol. 7, pp. 4068-4071.
  22. P. K. Khurd and G. R. Gindi, "LROC model observers for emission tomographic reconstruction," Proc. SPIE 5372, 509-520 (2004).
    [CrossRef]
  23. J. Qi and R. H. Huesman, "Fast approach to evaluate MAP reconstruction for lesion detection and localization," Proc. SPIE 5372, 273-282 (2004).
    [CrossRef]
  24. R. J. Adler, "On excursion sets, tube formulas and maxima of random fields," Ann. Appl. Probab. 10, 1-74 (2000).
    [CrossRef]
  25. D. O. Siegmund and K. J. Worsley, "Testing for a signal with unknown location and scale in a stationary Gaussian random field," Ann. Stat. 23, 608-639 (1995).
    [CrossRef]
  26. P. Khurd and G. Gindi, "Rapid computation of LROC figures of merit using numerical observers (for SPECT/PET reconstruction)," IEEE Trans. Nucl. Sci. 52, 618-626 (2005).
    [CrossRef]
  27. R. J. Adler, The Geometry of Random Fields (Wiley, 1981).
  28. K. J. Worsley, S. Marrett, P. Neelin, A. C. Vandal, K. J. Friston, and A. C. Evans, "A unified statistical approach for determining significant signals in images of cerebral activation," Hum. Brain Mapp 4, 58-73 (1996).
    [CrossRef] [PubMed]
  29. K. J. Worsley, "Detecting activation in fMRI data," Stat. Methods Med. Res. 12, 401-418 (2003).
    [CrossRef] [PubMed]
  30. A. Yendiki and J. A. Fessler, "Analysis of observer performance in detecting signals with location uncertainty for regularized tomographic image reconstruction," in Proceedings of IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2004), Vol. 4, pp. 2620-2624.
  31. J. A. Fessler, "ASPIRE 3.0 user's guide: a sparse iterative reconstruction library," Tech. Rep. 293, Commnications and Signal Processing Laboratory, Department of EECS, University of Michigan, Ann Arbor, MI, 48109-2122 (1995). Available at http://www.eecs.umich.edu/~fessler.
  32. G. Zubal, G. Gindi, M. Lee, C. Harrell, and E. Smith, "High resolution anthropomorphic phantom for Monte Carlo analysis of internal radiation sources," in IEEE Symposium on Computer-Based Medical Systems (IEEE, 1990), pp. 540-547.
  33. J. A. Fessler and W. L. Rogers, "Spatial resolution properties of penalized-likelihood image reconstruction methods: space-invariant tomographs," IEEE Trans. Image Process. 5, 1346-1358 (1996).
    [CrossRef] [PubMed]

2006

A. Yendiki and J. A. Fessler, "Analysis of observer performance in known-location tasks for tomographic image reconstruction," IEEE Trans. Med. Imaging 25, 28-41 (2006).
[CrossRef] [PubMed]

J. Qi and R. H. Huesman, "Penalized maximum-likelihood image reconstruction for lesion detection," Phys. Med. Biol. 51, 4017-4030 (2006).
[CrossRef]

2005

P. Khurd and G. Gindi, "Rapid computation of LROC figures of merit using numerical observers (for SPECT/PET reconstruction)," IEEE Trans. Nucl. Sci. 52, 618-626 (2005).
[CrossRef]

2004

P. K. Khurd and G. R. Gindi, "LROC model observers for emission tomographic reconstruction," Proc. SPIE 5372, 509-520 (2004).
[CrossRef]

J. Qi and R. H. Huesman, "Fast approach to evaluate MAP reconstruction for lesion detection and localization," Proc. SPIE 5372, 273-282 (2004).
[CrossRef]

J. Qi, "Analysis of lesion detectability in Bayesian emission reconstruction with nonstationary object variability," IEEE Trans. Med. Imaging 23, 321-329 (2004).
[CrossRef] [PubMed]

2003

Y. Xing, I.-T. Hsiao, and G. Gindi, "Rapid calculation of detectability in Bayesian single photon emission computed tomography," Phys. Med. Biol. 48, 3755-3774 (2003).
[CrossRef] [PubMed]

K. J. Worsley, "Detecting activation in fMRI data," Stat. Methods Med. Res. 12, 401-418 (2003).
[CrossRef] [PubMed]

H. C. Gifford, P. H. Pretorius, and M. A. King, "Comparison of human- and model-observer LROC studies," Proc. SPIE 5034, 112-122 (2003).
[CrossRef]

2002

S. Sankaran, E. C. Frey, K. L. Gilland, and B. M. Tsui, "Optimum compensation method and filter cutoff frequency in myocardial spect: a human observer study," J. Nucl. Med. 43, 432-438 (2002).
[PubMed]

2001

C. K. Abbey and H. H. Barrett, "Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability," J. Opt. Soc. Am. A 18, 473-488 (2001).
[CrossRef]

J. Qi and R. H. Huesman, "Theoretical study of lesion detectability of MAP reconstruction using computer observers," IEEE Trans. Med. Imaging 20, 815-822 (2001).
[CrossRef] [PubMed]

2000

H. C. Gifford, M. A. King, D. J. de Vries, and E. J. Soares, "Channelized Hotelling and human observer correlation for lesion detection in hepatic SPECT imaging," J. Nucl. Med. 41, 514-521 (2000).
[PubMed]

R. J. Adler, "On excursion sets, tube formulas and maxima of random fields," Ann. Appl. Probab. 10, 1-74 (2000).
[CrossRef]

1999

S. D. Wollenweber, B. M. W. Tsui, D. S. Lalush, E. C. Frey, K. J. LaCroix, and G. T. Gullberg, "Comparison of Hotelling observer models and human observers in defect detection from myocardial SPECT imaging," IEEE Trans. Nucl. Sci. 46, 2098-2103 (1999).
[CrossRef]

1996

R. G. Swensson, "Unified measurement of observer performance in detecting and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
[CrossRef] [PubMed]

C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996).
[CrossRef]

J. A. Fessler and W. L. Rogers, "Spatial resolution properties of penalized-likelihood image reconstruction methods: space-invariant tomographs," IEEE Trans. Image Process. 5, 1346-1358 (1996).
[CrossRef] [PubMed]

K. J. Worsley, S. Marrett, P. Neelin, A. C. Vandal, K. J. Friston, and A. C. Evans, "A unified statistical approach for determining significant signals in images of cerebral activation," Hum. Brain Mapp 4, 58-73 (1996).
[CrossRef] [PubMed]

1995

D. O. Siegmund and K. J. Worsley, "Testing for a signal with unknown location and scale in a stationary Gaussian random field," Ann. Stat. 23, 608-639 (1995).
[CrossRef]

1993

P. C. Hansen and D. P. O'Leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 1487-1506 (1993).

1992

1987

1978

C. E. Metz, "Basic principles of ROC analysis," Semin Nucl. Med. 8, 283-298 (1978).
[CrossRef] [PubMed]

1971

L. B. Lusted, "Signal detectability and medical decision-making," Science 171, 1217-1219 (1971).
[CrossRef] [PubMed]

Ann. Appl. Probab.

R. J. Adler, "On excursion sets, tube formulas and maxima of random fields," Ann. Appl. Probab. 10, 1-74 (2000).
[CrossRef]

Ann. Stat.

D. O. Siegmund and K. J. Worsley, "Testing for a signal with unknown location and scale in a stationary Gaussian random field," Ann. Stat. 23, 608-639 (1995).
[CrossRef]

Hum. Brain Mapp

K. J. Worsley, S. Marrett, P. Neelin, A. C. Vandal, K. J. Friston, and A. C. Evans, "A unified statistical approach for determining significant signals in images of cerebral activation," Hum. Brain Mapp 4, 58-73 (1996).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

J. A. Fessler and W. L. Rogers, "Spatial resolution properties of penalized-likelihood image reconstruction methods: space-invariant tomographs," IEEE Trans. Image Process. 5, 1346-1358 (1996).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

J. Qi and R. H. Huesman, "Theoretical study of lesion detectability of MAP reconstruction using computer observers," IEEE Trans. Med. Imaging 20, 815-822 (2001).
[CrossRef] [PubMed]

J. Qi, "Analysis of lesion detectability in Bayesian emission reconstruction with nonstationary object variability," IEEE Trans. Med. Imaging 23, 321-329 (2004).
[CrossRef] [PubMed]

A. Yendiki and J. A. Fessler, "Analysis of observer performance in known-location tasks for tomographic image reconstruction," IEEE Trans. Med. Imaging 25, 28-41 (2006).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

S. D. Wollenweber, B. M. W. Tsui, D. S. Lalush, E. C. Frey, K. J. LaCroix, and G. T. Gullberg, "Comparison of Hotelling observer models and human observers in defect detection from myocardial SPECT imaging," IEEE Trans. Nucl. Sci. 46, 2098-2103 (1999).
[CrossRef]

P. Khurd and G. Gindi, "Rapid computation of LROC figures of merit using numerical observers (for SPECT/PET reconstruction)," IEEE Trans. Nucl. Sci. 52, 618-626 (2005).
[CrossRef]

Inverse Probl.

C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996).
[CrossRef]

J. Nucl. Med.

H. C. Gifford, M. A. King, D. J. de Vries, and E. J. Soares, "Channelized Hotelling and human observer correlation for lesion detection in hepatic SPECT imaging," J. Nucl. Med. 41, 514-521 (2000).
[PubMed]

S. Sankaran, E. C. Frey, K. L. Gilland, and B. M. Tsui, "Optimum compensation method and filter cutoff frequency in myocardial spect: a human observer study," J. Nucl. Med. 43, 432-438 (2002).
[PubMed]

J. Opt. Soc. Am. A

Med. Phys.

R. G. Swensson, "Unified measurement of observer performance in detecting and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
[CrossRef] [PubMed]

Phys. Med. Biol.

Y. Xing, I.-T. Hsiao, and G. Gindi, "Rapid calculation of detectability in Bayesian single photon emission computed tomography," Phys. Med. Biol. 48, 3755-3774 (2003).
[CrossRef] [PubMed]

M. S. Chesters, "Human visual perception and ROC methodology in medical imaging," Phys. Med. Biol. 37, 1433-1484 (1992).
[CrossRef] [PubMed]

J. Qi and R. H. Huesman, "Penalized maximum-likelihood image reconstruction for lesion detection," Phys. Med. Biol. 51, 4017-4030 (2006).
[CrossRef]

Proc. SPIE

H. C. Gifford, P. H. Pretorius, and M. A. King, "Comparison of human- and model-observer LROC studies," Proc. SPIE 5034, 112-122 (2003).
[CrossRef]

P. K. Khurd and G. R. Gindi, "LROC model observers for emission tomographic reconstruction," Proc. SPIE 5372, 509-520 (2004).
[CrossRef]

J. Qi and R. H. Huesman, "Fast approach to evaluate MAP reconstruction for lesion detection and localization," Proc. SPIE 5372, 273-282 (2004).
[CrossRef]

Science

L. B. Lusted, "Signal detectability and medical decision-making," Science 171, 1217-1219 (1971).
[CrossRef] [PubMed]

Semin Nucl. Med.

C. E. Metz, "Basic principles of ROC analysis," Semin Nucl. Med. 8, 283-298 (1978).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.

P. C. Hansen and D. P. O'Leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 1487-1506 (1993).

Stat. Methods Med. Res.

K. J. Worsley, "Detecting activation in fMRI data," Stat. Methods Med. Res. 12, 401-418 (2003).
[CrossRef] [PubMed]

Other

A. Yendiki and J. A. Fessler, "Analysis of observer performance in detecting signals with location uncertainty for regularized tomographic image reconstruction," in Proceedings of IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2004), Vol. 4, pp. 2620-2624.

J. A. Fessler, "ASPIRE 3.0 user's guide: a sparse iterative reconstruction library," Tech. Rep. 293, Commnications and Signal Processing Laboratory, Department of EECS, University of Michigan, Ann Arbor, MI, 48109-2122 (1995). Available at http://www.eecs.umich.edu/~fessler.

G. Zubal, G. Gindi, M. Lee, C. Harrell, and E. Smith, "High resolution anthropomorphic phantom for Monte Carlo analysis of internal radiation sources," in IEEE Symposium on Computer-Based Medical Systems (IEEE, 1990), pp. 540-547.

H. C. Gifford, P. E. Kinahan, C. Lartizien, and M. A. King, "Evaluation of multiclass model observers in PET LROC studies," in Proceedings of IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2004), Vol. 7, pp. 4068-4071.

R. J. Adler, The Geometry of Random Fields (Wiley, 1981).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2003).

A. Yendiki, "Analysis of signal detectability in statistically reconstructed tomographic images," Ph. D. dissertation (University of Michigan, 2005).

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Figures (5)

Fig. 1
Fig. 1

PSfrag replacements. Mean background f ¯ b with the target signal f s superimposed (a) and a profile through them (b). The largest of the search areas considered (diameter of 23   pixels ) is indicated as a black circle in (a).

Fig. 2
Fig. 2

Detection performance of MaCPPW observers versus QPWLS reconstruction resolution: P D obtained analytically (a), P D obtained empirically (b), and AUC obtained empirically (c). Results are shown for five different degrees of prewhitening accuracy. The search area is a disk with a diameter of 9   pixels .

Fig. 3
Fig. 3

QPWLS resolution that maximizes the P D (obtained analytically and empirically) or AUC (obtained empirically) versus search area diameter (a). AUC improvement with optimally regularized QPWLS over unregularized WLS versus search area diameter (b). Results are shown for the MaCNPW observer.

Fig. 4
Fig. 4

QPWLS reconstructions of a noisy Poisson data set with a resolution of 3   pixels (a), 4   pixels (b), or 5   pixels (c), and profiles through the three images (d). The signal is present in the center of the search area.

Fig. 5
Fig. 5

Empirical and analytical probabilities of false alarm and detection for the MaCNPW observer with a search area diameter of 7   pixels [(a)–(c)] and 15   pixels [(d)–(f)]. The plots show the P FA versus detection threshold [(a),(d)], the P D versus detection threshold [(b),(e)], and the P D (with error bars) versus QPWLS resolution for a fixed P FA = 0.02 [(c),(f)]. (The error bars are small enough to fit in the plot markers.)

Equations (44)

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

H 0 : f = f b ( signal absent )
H l : f = f b + f s , l ( signal present at location l , l = 1 , , n L ) .
E [ y f ] = A f + r ,
Cov { y f } = diag { A f + r } ,
f ̂ ( y ) = Z y
Z = ( F + R ) 1 A Π 1 ,
f R f = β j = 1 n p k N j ( f j f k ) 2 ,
t max max l = 1 , , n L t l ,
t l = w l c ̂ l , c ̂ l = C l ( f ̂ E [ f ̂ b ] ) + ε l ,
w l [ ( 1 γ ) I + γ ( 1 2 Cov { c ̂ H l } + 1 2 Cov { c ̂ H 0 } ) ] ( E [ c ̂ H l ] E [ c ̂ H 0 ] ) ,
P D , l ( τ ) P { t max τ H l } , for some l { 1 , , n L } ,
P FA ( τ ) P { t max τ H 0 } .
P { t max τ } d = 0 2 R d ( S ) ρ d ( τ ) ,
ρ 0 ( τ ; σ T , Λ T ) 1 Φ ( τ σ T ) ,
ρ 1 ( τ ; σ T , Λ T ) det Λ T 1 4 2 π σ T e τ 2 2 σ T 2 ,
ρ 2 ( τ ; σ T , Λ T ) det Λ T 1 2 ( 2 π ) 3 2 σ T 3 τ e τ 2 2 σ T 2 ,
P { t max τ H l } = P { T ( x l ) τ H l } + P { t max τ , T ( x l ) < τ H l } = 1 Φ ( τ μ T ( x l ) σ T ) + 0 P { t max τ T ( x l ) = τ s , H l } ϕ ( τ s μ T ( x l ) σ T ) d s ,
P { t max τ T ( x l ) = τ s , H l } P { T ̇ ( x l ) E [ T ̈ ( x l ) T ( x l ) = τ s ] 1 T ̇ ( x l ) 2 s } ,
E [ T ̈ ( x ) T ( x ) ] = E [ T ̈ ( x ) ] + Cov { T ̈ ( x ) , T ( x ) } T ( x ) E [ T ( x ) ] Var { T ( x ) }
P { t max > τ H l } 1 Φ ( τ μ T ( x l ) σ T ) + ϕ ( τ μ T ( x l ) σ T ) 1 σ T [ 2 R T ( 0 ) 2 x i ] [ 2 μ T ( x l ) 2 x i ] .
μ t H 0 = 0 ,
μ t H l = V Z A f ¯ s , l , l = 1 , , n L ,
Π t H l = V Z Π l Z V + diag { w l Π ε l w l , l = 1 , , n L } , l = 0 , , n L ,
Π 0 = diag { A f ¯ b + r } + A K b A ,
Π l = Π 0 + diag { A f ¯ s , l } + A K s , l A , l = 1 , , n L .
U e 0 = 1 n p 1 ,
f ¯ s , l = U 1 E l X 0 ,
C l = U 1 E l T 0 ,
F U 1 Λ U ,
R U 1 Ω U ,
K b U 1 N 0 U ,
K b + K s , l U 1 N l U , l = 1 , , n L ,
w l [ ( 1 γ ) I + γ ( T 0 [ Λ + Ω ] 2 [ Λ + Λ 2 N ̌ l ] T 0 + Π ε l ) ] T 0 ( Λ + Ω ) 1 Λ X 0 ,
V n p U 1 diag { V 0 } U I SA ,
μ t H l n p I SA U 1 Φ U e l , l = 1 , , n L ,
Π t H l n p I SA U 1 Ψ U I SA + ( w l c Π ε l c w l c ) I ,
l = 0 , , n L ,
Φ diag { V k * X k λ k λ k + ω k , k = 1 , , n p } ,
Ψ diag { V k 2 λ k ( 1 + λ k ν k l ) ( λ k + ω k ) 2 , k = 1 , , n p } ,
P FA ( τ ) d = 0 2 R d ( S ) ρ d ( τ ; σ T 0 , Λ T 0 ) ,
σ T 0 2 k = 1 n p V k 2 λ k ( 1 + λ k ν k 0 ) ( λ k + ω k ) 2 + w l c Π ε l c w l c .
P D , l c ( τ ) 1 Φ ( τ μ T l c σ T l c ) + ϕ ( τ μ T l c σ T l c ) 1 σ T l c [ 2 R T l c ( 0 ) 2 x i ] [ 2 μ T l c ( x l c ) 2 x i ] .
μ T l c k = 1 n p V k * X k λ k ( λ k + ω k ) ,
σ T l c 2 k = 1 n p V k 2 λ k ( 1 + λ k ν k l c ) ( λ k + ω k ) 2 + w l c Π ε l c w l c .

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