Abstract

A shape reconstruction algorithm for optical tomography is introduced that uses a level-set formulation for the shapes. Evolution laws based on gradient directions for a cost functional are derived for two different level-set functions, one describing the absorption and one the diffusion parameter, as well as for the parameter values inside these shapes. Numerical experiments are presented in 2D that show that the new method is able to simultaneously recover shapes and contrast values of absorbing and scattering objects embedded in a moderately heterogeneous background medium from simulated noisy data.

© 2006 Optical Society of America

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  1. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
    [CrossRef]
  2. B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
    [CrossRef] [PubMed]
  3. M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
    [CrossRef] [PubMed]
  4. S. R. Arridge, Inverse Probl. 15, R41 (1999).
    [CrossRef]
  5. O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
    [CrossRef]
  6. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).
  7. F. Santosa, ESAIM Control, Optim. Calcul. Var. 1, 17 (1996).
    [CrossRef]
  8. J. A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge U. Press, 1999).
  9. E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
    [CrossRef]
  10. X.-C. Tai and T. F. Chan, Int. J. Num. Anal. Mod. 1, 25 (2004).
  11. P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
    [CrossRef]
  12. S. R. Arridge, Appl. Opt. 34, 7395 (1995).
    [CrossRef] [PubMed]
  13. M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
    [CrossRef] [PubMed]
  14. P. C. Hansen and D. P. O'Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
    [CrossRef]

2005 (3)

E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
[CrossRef]

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
[CrossRef] [PubMed]

2004 (1)

X.-C. Tai and T. F. Chan, Int. J. Num. Anal. Mod. 1, 25 (2004).

2003 (1)

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

2001 (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

2000 (2)

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
[CrossRef]

1999 (2)

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

J. A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge U. Press, 1999).

1996 (1)

F. Santosa, ESAIM Control, Optim. Calcul. Var. 1, 17 (1996).
[CrossRef]

1995 (1)

1993 (1)

P. C. Hansen and D. P. O'Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

1988 (1)

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Abele, C.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Arridge, S. R.

M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
[CrossRef] [PubMed]

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

S. R. Arridge, Appl. Opt. 34, 7395 (1995).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Chan, T. F.

E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
[CrossRef]

X.-C. Tai and T. F. Chan, Int. J. Num. Anal. Mod. 1, 25 (2004).

Chung, E. T.

E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
[CrossRef]

Cope, M.

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Delpy, D. T.

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Dorn, O.

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
[CrossRef]

Fedkiw, R.

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

González-Rodriguez, P.

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

Hansen, P. C.

P. C. Hansen and D. P. O'Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

Kaufman, H.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Kindelan, M.

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
[CrossRef]

Moscoso, M.

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

Nissilä, I.

M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
[CrossRef] [PubMed]

O'Leary, D. P.

P. C. Hansen and D. P. O'Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

Osher, S.

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

Paulsen, K. D.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Pogue, B. W.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Rappaport, C.

O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
[CrossRef]

Santosa, F.

F. Santosa, ESAIM Control, Optim. Calcul. Var. 1, 17 (1996).
[CrossRef]

Schweiger, M.

M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
[CrossRef] [PubMed]

Sethian, J. A.

J. A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge U. Press, 1999).

Tai, X.-C.

E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
[CrossRef]

X.-C. Tai and T. F. Chan, Int. J. Num. Anal. Mod. 1, 25 (2004).

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Appl. Opt. (1)

ESAIM Control, Optim. Calcul. Var. (1)

F. Santosa, ESAIM Control, Optim. Calcul. Var. 1, 17 (1996).
[CrossRef]

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. 18, 57 (2001).
[CrossRef]

Int. J. Num. Anal. Mod. (1)

X.-C. Tai and T. F. Chan, Int. J. Num. Anal. Mod. 1, 25 (2004).

Inverse Probl. (3)

P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, Inverse Probl. 21, 565 (2005).
[CrossRef]

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, Inverse Probl. 16, 1119 (2000).
[CrossRef]

J. Biomed. Opt. (1)

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

E. T. Chung, T. F. Chan, and X.-C. Tai, J. Comput. Phys. 205, 357 (2005).
[CrossRef]

Med. Biol. Eng. Comput. (1)

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

M. Schweiger, S. R. Arridge, and I. Nissilä, Phys. Med. Biol. 50, 2365 (2005).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (USA) (1)

P. C. Hansen and D. P. O'Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

Other (2)

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

J. A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge U. Press, 1999).

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Figures (2)

Fig. 1
Fig. 1

Top row, absorption and, bottom row, diffusion parameter distribution of (left) the target image, (middle) pixel-based reconstruction, and (right) level-set reconstruction.

Fig. 2
Fig. 2

Left, level-set representation of evolving shape. The dashed line denotes the final cost of the pixel-based reconstruction. Right two panels, Recovered contrast of (left) μ a and (right) κ inclusions in the level-set reconstruction. Dashed lines indicate contrast targets.

Equations (15)

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[ κ ( x ) + μ a ( x ) + i ω c ] Φ ( x , ω ) = q ( x , ω ) ,
Φ ( y , ω ) + 2 κ ( y ) A Φ ( y , ω ) n = 0 .
A ( m 1 , m 2 ) = G ,
m l ( x ) = { m l , i in S l m l , e in Ω \ S l } .
m l ( x ) = { m l , i if ψ l ( x ) 0 m l , e if ψ l ( x ) > 0 } .
J ( ψ 1 , ψ 2 , m 1 , i , m 2 , i ) = 1 2 R ( ψ 1 , ψ 2 , m 1 , i , m 2 , i ) 2
d ψ l d t = f l ( x , t ) , d m l , i d t = g l ( t )
m l ( ψ l ) = m l , e H ( ψ l ) + m l , i ( 1 H ( ψ l ) ) ,
d J d t = l = 1 2 J m l m l ψ l d ψ l d t + l = 1 2 J m l m l m l , i d m l , i d t .
J m l δ m l = Re Ω R l ( m l ) * R δ m l d x ,
d J d t = l = 1 2 Re Ω R l ( m l ) * R ( m l , e m l , i ) δ ( ψ l ) f l d x + l = 1 2 Re { g l Ω [ 1 H ( ψ l ) ] R l ( m l ) * R d x } ,
f l , d ( x , t ) = Re ( m l , e m l , i ) R l ( m l ) * R ,
g l , d ( t ) = Re ( S l R l ( m l ) * R d x ) ,
ψ l ( n + 1 ) = ψ l ( n ) + τ l δ ψ l ( n ) , ψ l ( 0 ) = ψ l , 0 ,
δ ψ l ( n ) = ( α I β Δ ) 1 [ ( m l , i m l , e ) R l ( m l ) * R ] .

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