Abstract

A method of to mographic reconstruction is proposed that uses a priori known anisotropy of layered samples to reduce the amount of necessary projection data significantly. We show that an anisotropic scanning geometry, in accordance with the known anisotropy of a sample, allows the accurate reconstruction of the sample from the data of only a few angular projections with sufficiently fine translational step. A generic model for a particular class of samples with layered structure is presented, and the corresponding reconstruction algorithm is developed and tested on simulated and experimental data.

© 1996 Optical Society of America

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References

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  1. K. C. Tam, V. Perez-Mendez, “Tomographical imaging with limited-angle input,” J. Opt. Soc. Am. 71, 582–592 (1981).
    [CrossRef]
  2. R. Rangayyan, A. P. Dhawan, R. Gordon, “Algorithms for limited-view computed tomography: an annotated bibliography and a challenge,” Appl. Opt. 24, 4000–4012 (1985).
    [CrossRef] [PubMed]
  3. J. Davis, P. Wells, “Computed tomography measurements on wood,” Ind. Metrol. 2, 195–218 (1992).
    [CrossRef]
  4. R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994); R. Evans, G. M. Downes, D. N. J. Menz, S. L. Stringer, “Rapid measurement of variation in tracheid transverse dimensions in a radiata pine tree,” Appita 48, 134–138 (1995).
    [CrossRef]
  5. H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification (Syracuse U. Press, 1979).
  6. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  7. J. Howard, “Tomography and reliable information,” J. Opt. Soc. Am. A 5, 999–1014 (1988).
    [CrossRef]
  8. L. M. Cheng, A. S. Ho, R. E. Burge, “Use of a prioriknowledge in image reconstruction,” J. Opt. Soc. Am. A 1, 386–391 (1984).
    [CrossRef]
  9. M. I. Sezan, H. Stark, “Incorporation of a priorimoment information into signal recovery and synthesis problems,” J. Math. Anal. Appl. 122, 172–186 (1987).
    [CrossRef]
  10. A. Macovski, “Physical problems of computerized tomography,” Proc. IEEE 71, 373–378 (1983).
    [CrossRef]
  11. P. C. Sabatier, ed. Inverse Methods in Action (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  12. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  13. M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
    [CrossRef]
  14. A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
    [CrossRef]

1994

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994); R. Evans, G. M. Downes, D. N. J. Menz, S. L. Stringer, “Rapid measurement of variation in tracheid transverse dimensions in a radiata pine tree,” Appita 48, 134–138 (1995).
[CrossRef]

1992

J. Davis, P. Wells, “Computed tomography measurements on wood,” Ind. Metrol. 2, 195–218 (1992).
[CrossRef]

1988

1987

M. I. Sezan, H. Stark, “Incorporation of a priorimoment information into signal recovery and synthesis problems,” J. Math. Anal. Appl. 122, 172–186 (1987).
[CrossRef]

1986

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

1985

1984

1983

A. Macovski, “Physical problems of computerized tomography,” Proc. IEEE 71, 373–378 (1983).
[CrossRef]

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

1981

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Burge, R. E.

Cheng, L. M.

Core, H. A.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification (Syracuse U. Press, 1979).

Cote, W. A.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification (Syracuse U. Press, 1979).

Davis, J.

J. Davis, P. Wells, “Computed tomography measurements on wood,” Ind. Metrol. 2, 195–218 (1992).
[CrossRef]

Davison, M. E.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Day, A. C.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification (Syracuse U. Press, 1979).

Dhawan, A. P.

Evans, R.

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994); R. Evans, G. M. Downes, D. N. J. Menz, S. L. Stringer, “Rapid measurement of variation in tracheid transverse dimensions in a radiata pine tree,” Appita 48, 134–138 (1995).
[CrossRef]

Gordon, R.

Ho, A. S.

Howard, J.

Louis, A. K.

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

Macovski, A.

A. Macovski, “Physical problems of computerized tomography,” Proc. IEEE 71, 373–378 (1983).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Perez-Mendez, V.

Rangayyan, R.

Sezan, M. I.

M. I. Sezan, H. Stark, “Incorporation of a priorimoment information into signal recovery and synthesis problems,” J. Math. Anal. Appl. 122, 172–186 (1987).
[CrossRef]

Stark, H.

M. I. Sezan, H. Stark, “Incorporation of a priorimoment information into signal recovery and synthesis problems,” J. Math. Anal. Appl. 122, 172–186 (1987).
[CrossRef]

Tam, K. C.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Wells, P.

J. Davis, P. Wells, “Computed tomography measurements on wood,” Ind. Metrol. 2, 195–218 (1992).
[CrossRef]

Appl. Opt.

Holzforschung

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994); R. Evans, G. M. Downes, D. N. J. Menz, S. L. Stringer, “Rapid measurement of variation in tracheid transverse dimensions in a radiata pine tree,” Appita 48, 134–138 (1995).
[CrossRef]

Ind. Metrol.

J. Davis, P. Wells, “Computed tomography measurements on wood,” Ind. Metrol. 2, 195–218 (1992).
[CrossRef]

J. Math. Anal. Appl.

M. I. Sezan, H. Stark, “Incorporation of a priorimoment information into signal recovery and synthesis problems,” J. Math. Anal. Appl. 122, 172–186 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Numer. Math.

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

Proc. IEEE

A. Macovski, “Physical problems of computerized tomography,” Proc. IEEE 71, 373–378 (1983).
[CrossRef]

SIAM J. Appl. Math.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Other

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification (Syracuse U. Press, 1979).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

P. C. Sabatier, ed. Inverse Methods in Action (Springer-Verlag, Berlin, 1990).
[CrossRef]

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

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Figures (8)

Fig. 1
Fig. 1

Samples are taken from a tree from bark to pith.

Fig. 2
Fig. 2

Shape and dimensions of a typical sample.

Fig. 3
Fig. 3

1D+ structure of an XY slice of a sample.

Fig. 4
Fig. 4

Geometry of the projections.

Fig. 5
Fig. 5

Simulated 1D+ density distribution: (a) 2D density distribution, (b) structure angle f(2x/d), (c) 1D density g(2x/d).

Fig. 6
Fig. 6

Relative errors in the reconstructed 1D density g(2x/d) for the simulated sample from Fig. 5v, g(s) obtained from three conventional projections at angles γ = −10, 0, +10 deg, and one variable angle projection j[s, f(s)]: (a) noise-free projections, (b) projections with 3% noise, (c) 1D density obtained under the assumption of constant f(s).

Fig. 7
Fig. 7

Photograph of the sample used in the test of the 1D+ algorithm.

Fig. 8
Fig. 8

(a) Reconstructed 1D density grec(2x/d), (b) structure angle fmes(2x/d) obtained by processing of the optical image of the top face of the sample, (c) parts of the reconstructed 1D density profile grec(2x/d) (solid curve) and the experimental projection j[2x/d, fmes(2x/d)] at variable angle (dotted curve).

Equations (26)

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L ( ξ ) = { ( x , y ) x - F ( ξ ) ( y - d / 2 ) = ξ } ,
ρ ( x , y ) = G ( ξ ) = G [ x - F ( ξ ) ( y - d / 2 ) ] ,
F ( x ) = F [ ξ + F ( ξ ) η ] = F ( ξ ) + F ( ξ ˜ ) F ( ξ ) η = F ( ξ ) { 1 + F ( ξ ˜ ) η } F ( ξ ) ,
ln ( I 0 / I 1 ) = M ρ [ x ( t ) , y ( t ) ] d t ,
j ( s , σ ) = 1 d 0 d ρ [ s d / 2 + σ ( y - d / 2 ) , y ] d y ,
j ( s , σ ) = ln ( I 0 / I 1 ) / ( d 1 + σ 2 )
j ( s , σ ) = 1 d 0 d G { s d / 2 + [ σ - F ( ξ ) ] ( y - d / 2 ) } d y , ξ = ξ [ s d / 2 + σ ( y - d / 2 ) , y ] .
j ( s , σ ) = 1 2 [ σ - f ( s ) ] s - [ σ - f ( s ) ] s + [ σ - f ( s ) ] g ( t ) d t ,
ρ ( x , y ) = G ( ξ ) g ( 2 ξ / d ) ,             g ( t ) = a t + b , ξ = x - f ( y - d / 2 ) ,
j ( s , σ ) = a s + b = g ( s ) .
C 2 - C 1 = d ( B 1 - B 2 ) / 2.
Δ ρ ( x , y ) = ρ 2 ( x , y ) - ρ 1 ( x , y ) = C 2 - C 1 + y ( B 2 - B 1 ) = ( B 2 - B 1 ) ( y - d / 2 ) ,
s j ( s , σ ) g { s + [ σ - f ( s ) ] } - g { s - [ σ - f ( s ) ] } 2 [ σ - f ( s ) ] , σ j ( s , σ ) = g { s + [ σ - f ( s ) ] } + g { s - [ σ - f ( s ) ] } - 2 j ( s , σ ) 2 [ σ - f ( s ) ] .
g ( s + h ) - g ( s - h ) 2 h s j ( s , σ ) , g ( s + h ) + g ( s - h ) = 2 j ( s , σ ) + 2 h σ j ( s , σ ) ,
g ( s + h ) - g ( s - h ) 2 h s s 2 j ( s , σ ) s s 2 w ( s , σ ) , g ( s + h ) - g ( s - h ) = 4 σ j ( s , σ ) + 2 h σ σ 2 j ( s , σ ) = σ σ 2 w ( s , σ ) .
w ( s , σ ) = 0 ,
[ f ( s ) - σ ] j ( s , σ ) = 2 σ j ( s , σ ) .
f ( s 0 ) = σ ˜ + 2 σ j ( s 0 , σ ˜ ) j ( s 0 , σ ˜ ) .
g ( s 0 ) = j [ s 0 , f ( s 0 ) ] ,
g ( s 0 ) = j ( s 0 , σ 0 ) .
g ( s 0 + h ) + g ( s 0 - h ) = 2 j ( s 0 , σ 0 )
g ( s ) n = 0 g n ( s - s 0 ) n ,             g n = n g ( s 0 ) / n !
g 0 + k = 1 g 2 k ( σ - f ) 2 k = j ( s 0 , σ 0 ) .
j ( s 0 , σ k ) - j ( s 0 , σ k ˜ ) < 2 δ j ( s 0 )
Δ ( s , σ ) = 2 σ j ( s , σ ) j ( s , σ ) [ j ( s , σ ) ] 2 + χ 2 ,
g ( s ) = j [ s , σ + Δ ( s , σ ) ] .

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