Abstract

This paper presents a third-order diffusion equations-based optical image reconstruction algorithm. The algorithm has been implemented using finite element discretizations coupled with a hybrid regularization that combines both Marquardt and Tikhonov schemes. Numerical examples are used to compare between the third- and first-order reconstructions. The results show that the third-order reconstruction codes are more stable than the first-order codes, and are capable of reconstructing void-like regions. From the examples given, it has also been shown that the first-order codes fail to both qualitatively and quantitatively reconstruct the void-like regions.

© 1999 Optical Society of America

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  1. R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.
  2. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [Crossref] [PubMed]
  3. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [Crossref]
  4. X. D. Li, T Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction Tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
    [Crossref] [PubMed]
  5. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Optics Express 1, 6–12 (1997).
    [Crossref] [PubMed]
  6. S. A. Walker, S. Fantini, and E. Gratton, “Image reconstruction by backprojection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
    [Crossref] [PubMed]
  7. S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
    [Crossref] [PubMed]
  8. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
    [Crossref] [PubMed]
  9. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [Crossref]
  10. K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
    [Crossref] [PubMed]
  11. H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
    [Crossref] [PubMed]
  12. D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B 352, 649–659 (1997).
    [Crossref]
  13. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
    [Crossref] [PubMed]
  14. M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
    [Crossref]
  15. In Mathematics and Physics of Emerging Biomedical Imaging (National Academy Press, Washington, D.C., 1996).
  16. H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE 2389, 608–614 (1995).
    [Crossref]
  17. D. A. Boas et al, ”Photon Migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE 2389, pp. 240–247 (1995).
    [Crossref]
  18. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Optics Express 1, 404–413 (1997), http://epubs.osa.org/oearchive/source/2831.htm
    [Crossref] [PubMed]

1998 (2)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

1997 (7)

D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B 352, 649–659 (1997).
[Crossref]

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Optics Express 1, 404–413 (1997), http://epubs.osa.org/oearchive/source/2831.htm
[Crossref] [PubMed]

X. D. Li, T Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction Tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[Crossref] [PubMed]

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Optics Express 1, 6–12 (1997).
[Crossref] [PubMed]

S. A. Walker, S. Fantini, and E. Gratton, “Image reconstruction by backprojection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
[Crossref] [PubMed]

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
[Crossref] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

1996 (3)

1995 (4)

H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE 2389, 608–614 (1995).
[Crossref]

D. A. Boas et al, ”Photon Migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE 2389, pp. 240–247 (1995).
[Crossref]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[Crossref]

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[Crossref] [PubMed]

’t Hooft, G. W.

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[Crossref] [PubMed]

Aronson, R.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

Arridge, S.

M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
[Crossref]

Arridge, S. R.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

Asten, N. A. A. J. van

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[Crossref] [PubMed]

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

Boas, D. A.

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Optics Express 1, 404–413 (1997), http://epubs.osa.org/oearchive/source/2831.htm
[Crossref] [PubMed]

D. A. Boas et al, ”Photon Migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE 2389, pp. 240–247 (1995).
[Crossref]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[Crossref]

Chance, B.

Chang, J. H.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

Colak, S. B.

Cope, M.

D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B 352, 649–659 (1997).
[Crossref]

Delpy, D.

M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
[Crossref]

Delpy, D. T.

D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B 352, 649–659 (1997).
[Crossref]

Durduran, T

Fantini, S.

Firbank, M.

M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
[Crossref]

Graber, H. L.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

Gratton, E.

Hebden, J. C.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[Crossref] [PubMed]

Jiang, H.

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[Crossref]

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
[Crossref] [PubMed]

H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE 2389, 608–614 (1995).
[Crossref]

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[Crossref] [PubMed]

Li, X. D.

Mark, M. B. van der

Matson, C. L.

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Optics Express 1, 6–12 (1997).
[Crossref] [PubMed]

Melissen, J. B. M.

O’Leary, M. A.

Osterberg, U. L.

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[Crossref]

Paasschens, J. C. J.

Papaioannou, D. G.

Pattanayak, D. N.

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[Crossref]

Paulsen, K. D.

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[Crossref]

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
[Crossref] [PubMed]

H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE 2389, 608–614 (1995).
[Crossref]

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[Crossref] [PubMed]

Pogue, B. W.

Schomberg, H.

Schweiger, M.

M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
[Crossref]

Walker, S. A.

Wang, Y.

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

Yodh, A. G.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Med. Phys. (2)

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[Crossref] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, and M. S. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998).
[Crossref] [PubMed]

Opt. Lett. (2)

Optics Express (2)

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Optics Express 1, 6–12 (1997).
[Crossref] [PubMed]

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Optics Express 1, 404–413 (1997), http://epubs.osa.org/oearchive/source/2831.htm
[Crossref] [PubMed]

Phil. Trans. R. Soc. Lond. B (1)

D. T. Delpy and M. Cope, “Quantification in tissue near-infrared spectroscopy,” Phil. Trans. R. Soc. Lond. B 352, 649–659 (1997).
[Crossref]

Phys. Med. and Biol. (1)

M. Firbank, S. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non- scattering regions,” Phys. Med. and Biol. 41767–783 (1996).
[Crossref]

Phys. Med. Biol. (2)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[Crossref] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

Proc. SPIE (2)

H. Jiang and K. D. Paulsen, “A finite element based higher-order diffusion approximation of light propagation in tissues,” Proc. SPIE 2389, 608–614 (1995).
[Crossref]

D. A. Boas et al, ”Photon Migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, Proc. SPIE 2389, pp. 240–247 (1995).
[Crossref]

Other (2)

In Mathematics and Physics of Emerging Biomedical Imaging (National Academy Press, Washington, D.C., 1996).

R. L. Barbour, H. L. Graber, Y. Wang, J. H. Chang, and R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography, G. Miller ed., SPIE Institute for Advanced Optical Technologies (SPIE Optical Engineering Press Vol. IS11, 1993), 87–120.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Geometry of the test case under study; (b) reconstructed D image for the first test case; (c) reconstructed μa image for the first test case.

Fig. 2. (a)
Fig. 2. (a)

Recovered D image for the second case using the third-order codes; (b) recovered μa image for the second case using the third-order codes; (c) recovered D image for the second case using the first-order codes; (b) recovered μa image for the second case using the firs-order codes;

Equations (14)

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· D ( r ) Φ ( 1 ) ( r ) μ a ( r ) Φ ( 1 ) ( r ) · D ( r ) Φ ( 2 ) ( r ) + 6 · D ( r ) 1 Φ ( 3 ) ( r ) + 6 · D ( r ) 2 Φ ( 4 ) ( r ) = S ( r )
· D ( r ) Φ ( 1 ) ( r ) + 25 7 · D ( r ) Φ ( 2 ) ( r ) 5 μ t ( r ) Φ ( 2 ) ( r ) 60 7 · D ( r ) 1 Φ ( 3 ) ( r ) 60 7 · D ( r ) 2 Φ ( 4 ) ( r ) = 0
· D ( r ) 1 Φ ( 1 ) ( r ) 10 7 · D ( r ) 1 Φ ( 2 ) ( r ) + 90 7 · D ( r ) Φ ( 3 ) ( r ) 10 μ t ( r ) Φ ( 3 ) ( r ) = 0
1 2 · D ( r ) 2 Φ ( 1 ) ( r ) 5 7 · D ( r ) 2 Φ ( 2 ) ( r ) + 45 7 · D ( r ) Φ ( 4 ) ( r ) 5 μ t ( r ) Φ ( 4 ) ( r ) = 0
j = 1 N { Φ j ( 1 ) [ p = 1 P D p ϕ p ϕ j · ϕ i q = 1 Q μ p ϕ p ϕ j ϕ i ] + Φ j ( 2 ) p = 1 P D p ϕ p ϕ j · ϕ i 6 Φ j ( 3 ) p = 1 P D p ϕ p 1 ϕ j · ϕ i 6 Φ j ( 4 ) p = 1 P D p ϕ p 2 ϕ j · ϕ i }
= S ϕ i D n ̂ · Φ ( 1 ) ϕ i ds + D n ̂ · Φ ( 2 ) ϕ i ds 6 D n ̂ · 1 Φ ( 3 ) ϕ i ds 6 D n ̂ · 2 Φ ( 4 ) ϕ i ds
j = 1 N { Φ j ( 1 ) p = 1 P D p ϕ p ϕ j · ϕ i 25 7 Φ j ( 2 ) [ p = 1 P D p ϕ p ϕ j · ϕ i + 5 3 p = 1 Q D p 1 ϕ q ϕ j ϕ i ] + 60 7 Φ j ( 3 ) p = 1 P D p ϕ p 1 ϕ j · ϕ i 60 7 Φ j ( 4 ) p = 1 P D p ϕ p 2 ϕ j · ϕ i }
= D n ̂ · Φ ( 1 ) ϕ i ds + 25 7 D n ̂ · Φ ( 2 ) ϕ i ds 60 7 D n ̂ · 1 Φ ( 3 ) ϕ i ds 60 7 D n ̂ · 2 Φ ( 4 ) ϕ i ds
j = 1 N { Φ j ( 1 ) p = 1 P D p ϕ p 1 ϕ j · ϕ i + 10 7 Φ j ( 2 ) p = 1 P D p ϕ p 1 ϕ j · ϕ i 90 7 Φ j ( 3 ) [ p = 1 P D p ϕ p ϕ j · ϕ i + 10 3 p = 1 P D p 1 ϕ p ϕ j ϕ i ] }
= D n ̂ · 1 Φ ( 1 ) ϕ i ds + 10 7 D n ̂ · 1 Φ ( 2 ) ϕ i ds 90 7 D n ̂ · Φ ( 3 ) ϕ i ds
j = 1 N { Φ j ( 1 ) 1 2 p = 1 P D p ϕ p 2 ϕ j · ϕ i + 5 7 Φ j ( 2 ) p = 1 P D p ϕ p 2 ϕ j · ϕ i 45 7 Φ j ( 4 ) [ p = 1 P D p ϕ p ϕ j · ϕ i + 5 3 p = 1 P D p 1 ϕ p ϕ j ϕ i ] }
= 1 2 D n ̂ · 2 Φ ( 1 ) ϕ i ds + 5 7 D n ̂ · 2 Φ ( 2 ) ϕ i ds 45 7 D n ̂ · Φ ( 4 ) ϕ i ds
( T + λI ) Δχ = T ( Φ o Φ c )
= [ Φ 1 ( 1 ) D 1 Φ 1 ( 1 ) D 2 Φ 1 ( 1 ) D N Φ 1 ( 1 ) μ a , 1 Φ 1 ( 1 ) μ a , 2 Φ 1 ( 1 ) μ a , N Φ 2 ( 1 ) D 1 Φ 2 ( 1 ) D 2 Φ 2 ( 1 ) D N Φ 2 ( 1 ) μ a , 1 Φ 2 ( 1 ) μ a , 2 Φ 2 ( 1 ) μ a , N Φ M ( 1 ) D 1 Φ M ( 1 ) D 2 Φ M ( 1 ) D N Φ M ( 1 ) μ a , 1 Φ M ( 1 ) μ a , 2 Φ M ( 1 ) μ a , N ]

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