Abstract

Near-Infrared (NIR) tomographic image reconstruction is a non-linear, ill-posed and ill-conditioned problem, and so in this study, different ways of penalizing the objective function with structural information were investigated. A simple framework to incorporate structural priors is presented, using simple weight matrices that have either Laplacian or Helmholtz-type structures. Using both MRI-derived breast geometry and phantom data, a systematic and quantitative comparison was performed with and without spatial priors. The Helmholtz-type structure can be seen as a more generalized approach for incorporating spatial priors into the reconstruction scheme. Moreover, parameter reduction (i.e. hard prior information) in the imaging field through the enforcement of spatially explicit regions may lead to erroneous results with imperfect spatial priors.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. Q. Zhu, N. G. Chen, and S. C. Kurtzman, "Imaging tumor angiogenesis by use of combined near-infrared diffusive light and ultrasound," Opt. Lett. 28, 337-339 (2003).
    [CrossRef] [PubMed]
  5. Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, "Coregistered tomographic x-ray and optical breast imaging: initial results," J Biomed. Opt. 10, 024033:1-9 (2005).
    [CrossRef] [PubMed]
  6. B. Brooksby, S. Jiang, C. Kogel, M. Doyley, H. Dehghani, J. B. Weaver, S. P. Poplack, B. W Pogue, and K. D Paulsen, "Magnetic resonance guided near infrared tomography of the breast," Rev. Sci. Instrum. 75, 5262-5270 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  9. V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, "Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement," Proc. Nat. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
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  10. G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. Boas, "Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information," Phys. Med. Biol. 50, 3941-3956 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  33. S. Jiang, B. W. Pogue, T. O. McBride, M. M. Doyley, S. P. Poplack, and K. D. Paulsen, "Near-infrared breast tomography calibration with optoelastic tissue simulating phantoms," J. Electron. Imaging 12, 613-620 (2003).
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2007

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, "Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography," Med. Phys. 34, 2085-2098 (2007).
[CrossRef] [PubMed]

2006

P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, "Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis," Opt. Express 14, 6113-6127 (2006).
[CrossRef] [PubMed]

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack and K. D. Paulsen, "Imaging breast adipose and fibroglandular tissue molecular signatures using Hybrid MRI-Guided Near-Infrared Spectral Tomography," Proc. Nat. Acad. Sci. U.S.A. 103, 8828-8833 (2006).
[CrossRef]

2005

B. Brooksby, S. Jiang, H. Dehghani, B. W. Pogue, K. D. Paulsen, J. Weaver, C. Kogel and S. P. Poplack, "Combining near infrared tomography and magnetic resonance imaging to study in vivo breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10, 051504:1-10 (2005).
[CrossRef] [PubMed]

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, "Coregistered tomographic x-ray and optical breast imaging: initial results," J Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef] [PubMed]

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. Boas, "Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information," Phys. Med. Biol. 50, 3941-3956 (2005).
[CrossRef] [PubMed]

A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, "Optimal linear inverse solution with multiple priors in diffuse optical tomography," Appl. Opt. 44, 1948-1956 (2005).
[CrossRef] [PubMed]

X. Wang, B.W. Pogue, S. Jiang, X. Song, K.D. Paulsen, C. Kogel, S.P. Poplack, andW.A.Wells, "Approximation of Mie scattering parameters in near-infrared tomography of normal breast tissue in vivo," J. Biomed. Opt. 10, 051704:1-8 (2005).
[CrossRef] [PubMed]

2004

B. Brooksby, S. Jiang, C. Kogel, M. Doyley, H. Dehghani, J. B. Weaver, S. P. Poplack, B. W Pogue, and K. D Paulsen, "Magnetic resonance guided near infrared tomography of the breast," Rev. Sci. Instrum. 75, 5262-5270 (2004).
[CrossRef]

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, "Improved quantification of small objects in near-infrared diffuse optical tomography," J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

2003

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, "Near infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities," IEEE J. Sel. Top. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack and K. D. Paulsen, "Interpreting hemoglobin and water concentration, oxygen saturation and scattering measured in Vivo by Near-Infrared Breast Tomography," Proc. Nat. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef]

S. Jiang, B. W. Pogue, T. O. McBride, M. M. Doyley, S. P. Poplack, and K. D. Paulsen, "Near-infrared breast tomography calibration with optoelastic tissue simulating phantoms," J. Electron. Imaging 12, 613-620 (2003).
[CrossRef]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, "Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results," Appl. Opt. 42, 135-145 (2003).
[CrossRef] [PubMed]

Q. Zhu, N. G. Chen, and S. C. Kurtzman, "Imaging tumor angiogenesis by use of combined near-infrared diffusive light and ultrasound," Opt. Lett. 28, 337-339 (2003).
[CrossRef] [PubMed]

H. Dehghani, B.W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, "Three-dimensional optical tomography: resolution in small-object imaging," Appl. Opt. 42, 3117-3126 (2003).
[CrossRef] [PubMed]

A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, "Tomographic optical breast imaging guided by three-dimensional mammography," Appl. Opt. 42, 5181-5190 (2003).
[CrossRef] [PubMed]

2002

M. F. Ernst and J. A. Roukema, "Diagnosis of non-palpable breast cancer: a review," The Breast 11, 13 (2002).
[CrossRef]

2001

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Sig. Proc. Mag. 18, 57-75 (2001).
[CrossRef]

A. H. Hielscher and S. Bartel, "Use of penalty terms in gradient-based iterative reconstruction schemes for optical tomography," J. Biomed. Opt. 6, 183-192 (2001).
[CrossRef] [PubMed]

2000

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, "Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement," Proc. Nat. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef]

B. W. Pogue, K. D. Paulsen, H. Kaufman, and C. Abele, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

1999

S. R. Arridge, "Optical tomography in medical imaging," Inv.Problems 15, R41-R93 (1999).
[CrossRef]

1998

V. Ntziachristos, X. H. Ma, and B. Chance, "Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography," Rev. Sci. Instrum. 69, 4221-4233 (1998).
[CrossRef]

1997

1996

1995

H. Jiang, K. D. Paulsen, U. Osterberg, B. W. Pogue, and M. S. Patterson, "Simultaneous reconstruction of optical absorption and scattering maps in turbid media from near-infrared frequency-domain data," Opt. Lett. 20, 2128- 2130 (1995).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, J. W. Chang, S. L. S. Barbour, P. C. Koo, R. Aronson, "MRI-guided optical tomography: Prospects and computation for a new imaging method," IEEE Comp. Sci. Eng. 2, 63-77 (1995).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiroaka, and D. T. Delpy, "The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions," Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

K. D. Paulsen and H. Jiang, "Spatially varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
[CrossRef] [PubMed]

1963

A. N. Tikhonov, "Regularization of mathematically incorrectly posed problems," Soviet Math. Dokl. 4, 1624- 1627 (1963).

D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
[CrossRef]

1944

K. Levenberg, "A method for the solution of certain nonlinear problems in least squares," Q. Appl. Math. 2, 164-168 (1944).

Appl. Opt.

IEEE Comp. Sci. Eng.

R. L. Barbour, H. L. Graber, J. W. Chang, S. L. S. Barbour, P. C. Koo, R. Aronson, "MRI-guided optical tomography: Prospects and computation for a new imaging method," IEEE Comp. Sci. Eng. 2, 63-77 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, "Near infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities," IEEE J. Sel. Top. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

IEEE Sig. Proc. Mag.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Sig. Proc. Mag. 18, 57-75 (2001).
[CrossRef]

J Biomed. Opt.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, "Coregistered tomographic x-ray and optical breast imaging: initial results," J Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef] [PubMed]

J. Biomed. Opt.

A. H. Hielscher and S. Bartel, "Use of penalty terms in gradient-based iterative reconstruction schemes for optical tomography," J. Biomed. Opt. 6, 183-192 (2001).
[CrossRef] [PubMed]

B. Brooksby, S. Jiang, H. Dehghani, B. W. Pogue, K. D. Paulsen, J. Weaver, C. Kogel and S. P. Poplack, "Combining near infrared tomography and magnetic resonance imaging to study in vivo breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10, 051504:1-10 (2005).
[CrossRef] [PubMed]

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, "Improved quantification of small objects in near-infrared diffuse optical tomography," J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

X. Wang, B.W. Pogue, S. Jiang, X. Song, K.D. Paulsen, C. Kogel, S.P. Poplack, andW.A.Wells, "Approximation of Mie scattering parameters in near-infrared tomography of normal breast tissue in vivo," J. Biomed. Opt. 10, 051704:1-8 (2005).
[CrossRef] [PubMed]

B. W. Pogue, K. D. Paulsen, H. Kaufman, and C. Abele, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

J. Electron. Imaging

S. Jiang, B. W. Pogue, T. O. McBride, M. M. Doyley, S. P. Poplack, and K. D. Paulsen, "Near-infrared breast tomography calibration with optoelastic tissue simulating phantoms," J. Electron. Imaging 12, 613-620 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Soc. Ind. Appl. Math.

D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Med. Phys.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, "Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography," Med. Phys. 34, 2085-2098 (2007).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiroaka, and D. T. Delpy, "The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions," Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

K. D. Paulsen and H. Jiang, "Spatially varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. Boas, "Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information," Phys. Med. Biol. 50, 3941-3956 (2005).
[CrossRef] [PubMed]

Problems

S. R. Arridge, "Optical tomography in medical imaging," Inv.Problems 15, R41-R93 (1999).
[CrossRef]

Proc. Nat. Acad. Sci. U.S.A.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, "Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement," Proc. Nat. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack and K. D. Paulsen, "Interpreting hemoglobin and water concentration, oxygen saturation and scattering measured in Vivo by Near-Infrared Breast Tomography," Proc. Nat. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef]

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack and K. D. Paulsen, "Imaging breast adipose and fibroglandular tissue molecular signatures using Hybrid MRI-Guided Near-Infrared Spectral Tomography," Proc. Nat. Acad. Sci. U.S.A. 103, 8828-8833 (2006).
[CrossRef]

Q. Appl. Math.

K. Levenberg, "A method for the solution of certain nonlinear problems in least squares," Q. Appl. Math. 2, 164-168 (1944).

Rev. Sci. Instrum.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Simulated μa and μ´ s distributions from a breast (obtained from a volunteer) are shown in the first column. Optical properties for the region labeled ‘0’ (fat) are: μa = 0.006 mm-1 and μs = 0.6 mm-1. Region ‘1’ (fibroglandular) values are: μa = 0.012 mm-1 and μ´ s = 1.2 mm-1. Region ‘2’ (tumor) values are: μa = 0.018 mm-1 and μ´ s = 1.8 mm-1. Reconstructed μa and μ´ s images from different techniques with simulated data having 1% random noise and imperfect structural information in defining region ‘1’ (7% reduction compared to the original segmentation) are shown in the rest of the columns. The middle two columns use soft prior structural information while the last column shows the result with hard prior information. In the Helmholtz case, κ = 1/8 mm-1 (BPE) was used. (b) Cross-sectional plots, along the dotted line in the actual image (see first column of (a)), of true and reconstructed μa and μ´ s distributions.

Fig. 2.
Fig. 2.

(a) Reconstructed μa and μ´ s images from different techniques with simulated data having 5% random noise and perfect structural priors (actual images are shown in the first column of Fig. 1(a)). The first column shows the reconstruction results without the use of prior information. The middle two columns use soft prior structural information while the last row shows the result with hard prior information. In the Helmholtz case, κ = 1/8 mm-1 (BPE) was used. (b) The mean values and standard deviations (plotted as error bars) in μa and μ´ s for different regions of breast geometry (labeled in actual image) with increasing noise level (1% to 10 %).

Fig. 3.
Fig. 3.

Photograph for gelatin phantom (representing the idealized two-dimensional cross-sectional geometry shown as first column in Fig. 4(a)) used in the experimental studies.

Fig. 4.
Fig. 4.

(a) Actual μa and μ´ s distributions (axial cross-section) of phantom (Fig. 3) case are shown in the first column. Optical properties for the region labeled ‘0’ are: μa = 0.0065 mm-1 and μ´ s = 0. 65 mm-1. Region ‘1’ values are: μa = 0.01 mm-1 and μ´ s = 1.0 mm-1. Region ‘2’ (tumor) values are: μa = 0.02 mm-1 and μ´ s = 1.2 mm-1. Reconstructed μa and μ´ s distribution from different techniques (discussed in Sec. 2) from the experimental phantom data. Second column of images does not use prior information. The middle rows use soft prior structural information and the last row of images were recovered with hard priors. In the Helmholtz case, κ = 1/16 mm-1 (BPE) was used. (b) Cross-sectional plots along the dotted line in the actual image (see first column of (a)) of the true and reconstructed μa and μ´ s distributions.

Fig. 5.
Fig. 5.

(a) Reconstructed μa and μ´ s images from the experimental phantom data using Helmholtz-type regularization matrix for different values of κ, which are given at the top of each column. (b) Cross-sectional plots along the dotted line of the actual images in Fig. 4(a) (first column) are shown with the data from reconstructed μa and μ´ s images in (a). The best prior estimate (BPE) case (κ = 1/16 mm-1) is also presented for comparison.

Tables (2)

Tables Icon

Table 1. Mean and standard deviation of the reconstructed (a) μa and (b) μ´ s values in different regions (labeled in first column of Fig. 1(a)) recovered with simulated data having 1% random noise and imperfect structural information defining region ‘1’ (7% reduction compared to the original segmentation). The corresponding reconstructed images are shown in Fig. 1(a)

Tables Icon

Table 2. Mean and standard deviation of the reconstructed (a) μa and (b) μ´ s values in different regions (labeled in first column of Fig. 4(a)) recovered from the experimental phantom data. The corresponding reconstructed images are shown in Fig. 4(a) and 5(a).

Equations (19)

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. D ( r ) ∇Φ ( r , ω ) + ( μ a ( r ) + c ) Φ ( r , ω ) = Q o ( r , ω )
D ( r ) = 1 3 [ μ a ( r ) + μ ' s ( r ) ]
Ω = min D , μ a { y F ( D , μ a ) 2 + λ ( D , μ a ) ( D 0 , μ a 0 ) 2 }
( J T J + λI ) ( δ D , δμ a ) = J T ( y F ( D , μ a ) ) λ [ ( D , μ a ) ( D 0 , μ a 0 ) ]
( J T J + 2 λI ) ( δ D , δμ a ) = J T ( y F ( D , μ a ) )
Ω = min D , μ a { y F ( D , μ a ) 2 + λ L [ ( D , μ a ) ( D 0 , μ a 0 ) ] 2 }
( J T J + λ L T L ) ( δ D , δμ a ) = J T ( y F ( D , μ a ) ) λ L T L [ ( D , μ a ) ( D 0 , μ a 0 ) ]
{ H D 2 H D μ a H D μ a H μ a 2 + ( λ D ) L T L 0 0 ( λ μ a ) L T L } [ δk δ μ a ] = [ ( J T ) D ( y F ( D , μ a ) ) ( J T ) μ a ( y F ( D , μ a ) ) ] [ ( λ D ) L T L ( D i 1 D 0 ) ( λ μ a ) L T L ( μ ai 1 μ a 0 ) ]
2 u ( r ) = 0
2 u ( r ) h 2 u 1 + u 2 + Nu N 2 + + u N 1 + u N = 0
u 1 N + u 2 N + + u N 2 + + u N 1 N + u N N = 0
L ij = { 0 if i and j are not in the same region 1 N if i and j are in the same region 1 if i = j
2 u ( r ) k 2 u ( r ) = 0
( 2 κ 2 ) u ( r ) h 2 u 1 + u 2 + + [ ( N + ( κh ) 2 ) ] u N 2 + + u N 1 + u N = 0
u 1 ( N + ( κh ) 2 ) + u 2 ( N + ( κh ) 2 ) + + u N 2 + + u N 1 ( N + ( κh ) 2 ) + u N ( N + ( κh ) 2 ) = 0
L ij = { 0 if i and j are not in the same region 1 N + ( κ h ) 2 if i and j are in the same region 1 if i = j
J ˜ = JR
ij = { 1 if i R L j 0 otherwise
( δ D , δμ a ) = R ( δD ˜ , δ μ a ˜ )

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