Abstract

A method for rapid limited-angle tomography is suggested that allows the reconstruction of density distribution in hardwood samples with high accuracy and spatial resolution from only a few x-ray projections. The sparsity of the experimental x-ray data is compensated by incorporation of a priori knowledge about generic wood macrostructure, as well as some morphological information obtainable from optical images of the sample surface, in the mathematical framework of the tomographic analysis. Numerical tests of the proposed method confirm its potential as a practical technique for nondestructive testing of trees for scientific and industrial purposes. It can also be useful for the analysis of other types of object with an underlying layered structure.

© 1998 Optical Society of America

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References

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  1. R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994).
    [CrossRef]
  2. 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 J. 48, 134–138 (1995).
  3. R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).
  4. T. E. Gureyev, R. Evans, S.-A. Stuart, M. Cholewa, “Quasi-one-dimensional tomography,” J. Opt. Soc. Am. A 13, 735–742 (1996).
    [CrossRef]
  5. K. C. Tam, V. Perez-Mendez, “Tomographical imaging with limited-angle input,” J. Opt. Soc. Am. 71, 582–592 (1981).
    [CrossRef]
  6. H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
    [PubMed]
  7. H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification, 2nd ed. (Syracuse U. Press, Syracuse, N.Y., 1979).
  8. Consider the simplest example of uniform density ρc(x, y) ≡ C. Obviously, ρc(x, y) can be described by a 1D+ model with the uniform 1D density g(s) ≡ C and arbitrary structure angle f(s).
  9. T. E. Gureyev, R. Evans, “X-ray microdensitometry of wood. Numerical and experimental evaluation of quasi-one-dimensional tomographic methods,” Report DFFP521 (CSIRO Forestry and Forest Products, Melbourne, Australia, 1996).
  10. It is a frequent feature of hardwood species that the regions of rapid density variation near the boundaries of growth rings are free from vessel elements.

1997

R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).

1996

1995

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 J. 48, 134–138 (1995).

1994

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994).
[CrossRef]

1990

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

1981

Barrett, H. H.

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

Cholewa, M.

Core, H. A.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification, 2nd ed. (Syracuse U. Press, Syracuse, N.Y., 1979).

Cote, W. A.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification, 2nd ed. (Syracuse U. Press, Syracuse, N.Y., 1979).

Day, A. C.

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification, 2nd ed. (Syracuse U. Press, Syracuse, N.Y., 1979).

Downes, G. M.

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 J. 48, 134–138 (1995).

Evans, R.

R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).

T. E. Gureyev, R. Evans, S.-A. Stuart, M. Cholewa, “Quasi-one-dimensional tomography,” J. Opt. Soc. Am. A 13, 735–742 (1996).
[CrossRef]

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 J. 48, 134–138 (1995).

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994).
[CrossRef]

T. E. Gureyev, R. Evans, “X-ray microdensitometry of wood. Numerical and experimental evaluation of quasi-one-dimensional tomographic methods,” Report DFFP521 (CSIRO Forestry and Forest Products, Melbourne, Australia, 1996).

Gureyev, T. E.

T. E. Gureyev, R. Evans, S.-A. Stuart, M. Cholewa, “Quasi-one-dimensional tomography,” J. Opt. Soc. Am. A 13, 735–742 (1996).
[CrossRef]

T. E. Gureyev, R. Evans, “X-ray microdensitometry of wood. Numerical and experimental evaluation of quasi-one-dimensional tomographic methods,” Report DFFP521 (CSIRO Forestry and Forest Products, Melbourne, Australia, 1996).

Kibblewhite, R. P.

R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).

Menz, D. N. J.

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 J. 48, 134–138 (1995).

Perez-Mendez, V.

Stringer, S.

R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).

Stringer, S. L.

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 J. 48, 134–138 (1995).

Stuart, S.-A.

Tam, K. C.

Appita J.

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 J. 48, 134–138 (1995).

R. Evans, R. P. Kibblewhite, S. Stringer, “Kraft pulp fiber property prediction from wood properties in eleven radiata pine clones,” Appita J. 50, 25–33 (1997).

Holzforschung

R. Evans, “Rapid measurement of the transverse dimensions of tracheids in radial wood sections from Pinus radiata,” Holzforschung 48, 168–172 (1994).
[CrossRef]

J. Nucl. Med.

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

H. A. Core, W. A. Cote, A. C. Day, Wood Structure and Identification, 2nd ed. (Syracuse U. Press, Syracuse, N.Y., 1979).

Consider the simplest example of uniform density ρc(x, y) ≡ C. Obviously, ρc(x, y) can be described by a 1D+ model with the uniform 1D density g(s) ≡ C and arbitrary structure angle f(s).

T. E. Gureyev, R. Evans, “X-ray microdensitometry of wood. Numerical and experimental evaluation of quasi-one-dimensional tomographic methods,” Report DFFP521 (CSIRO Forestry and Forest Products, Melbourne, Australia, 1996).

It is a frequent feature of hardwood species that the regions of rapid density variation near the boundaries of growth rings are free from vessel elements.

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Figures (12)

Fig. 1
Fig. 1

Dimensions of a typical wood sample.

Fig. 2
Fig. 2

Quasi-one-dimensional structure.

Fig. 3
Fig. 3

Example of a transverse section of a hardwood sample: A, vessels; B, ring boundary; C, parenchyma tissue; D, fiber.

Fig. 4
Fig. 4

Example of a case in which average wood density, average fiber density, and average vessel-free density, over the trajectory of an x ray, are all significantly different. Let ρ lo = 500 kg/m3 and ρ hi = 1000 kg/m3, l(s, σ) = 4 mm, l χ(s, σ) = 2 mm. Then j(s, σ) = (0 kg/m3 × 0.002 m + 1000 kg/m3 × 0.002 m)/0.004 m = 500 kg/m3, j f (s, σ) = (500 kg/m3 × 0.002 m + 1000 kg/m3 × 0.002 m)/0.004 m = 750 kg/m3, j χ(s, σ) = (0 kg/m3 × 0.002 m + 1000 kg/m3 × 0.002 m)/0.002 m = 1000 kg/m3.

Fig. 5
Fig. 5

(a) One-dimensional density profile and (b) structure angle profile of the 1D+ fiber density distribution used in the hardwood models.

Fig. 6
Fig. 6

Numerically simulated hardwood samples (a) 0 and (b) 3, 20 ≤ x ≤ 60 mm.

Fig. 7
Fig. 7

Part of the vessel-free projection j 0(s, σ-) (solid line), the uncorrected projection j 2(s, σ-) (dashed line), and the corrected projection j f 2(s, σ-) (dotted line) of the second sample at σ- = tan(-20°).

Fig. 8
Fig. 8

Average relative error in the corrected projections j f m (s, σα) (squares) and uncorrected projections j m (s, σα) (circles), m = 1, 2, and 3, compared with the vessel-free projections j 0(s, σα) at σα = tan α(s).

Fig. 9
Fig. 9

Part of the vessel-free projection j 0(s, σ-) (solid line), the uncorrected projection j 2(s, σ-) (dashed line), and the corrected projection j f 2(s, σ-) (dotted line) of the second sample at σ- = tan(-20°) obtained by use of the ternary map of vessels and parenchyma containing 40% uniform random error.

Fig. 10
Fig. 10

Average relative error in projection j f 2(s, σ-) at σ- = tan(-20°) corrected by use of the map with various amounts of random error.

Fig. 11
Fig. 11

Average absolute error in the structure angle α(x) in sample 2 reconstructed by use of projections j 2(s, σ) with γ(≡tan-1 σ) = -20° and 0° and ternary maps of vessels and parenchyma containing various amounts of random error. The circle corresponds to the error in the structure angle obtained by use of vessel-free projections j 0(s, σ).

Fig. 12
Fig. 12

Average relative error in the 1D density G(x) in sample 2 reconstructed by use of projections j 2(s, σ), γ(≡ tan-1 σ) = -20°, and 0° and ternary maps of vessels and parenchyma containing various amounts of random error.

Tables (1)

Tables Icon

Table 1 Main Parameters of Simulated Samples

Equations (18)

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p s ,   σ = M s , σ   ρ x t ,   y t d t ,
j s ,   σ = p s ,   σ l s ,   σ = 1 d - d / 2 d / 2   ρ x + σ y ,   y d y ,   x = sd / 2 .
j s ,   σ = 1 2 σ - f s - σ + f s σ - f s   g s - t d t .
f s - σ σ σ 2 - ss 2 j s ,   σ - 2 σ j s ,   σ = 0 .
g s = j s ,   f s .
f s 0 = σ 0 + 2 σ j s 0 ,   σ 0 σ σ 2 - ss 2 j s 0 ,   σ 0 .
ρ ˆ d p ,   - σ p = j ˜ p ,   σ = g ˜ p sinc p σ - f ,
J p ,   σ 1 ,   σ 2 j ˜ p ,   σ 2 / j ˜ p ,   σ 1 = sinc p σ 2 - f / sinc p σ 1 - f .
f σ 1 + σ 2 2 + 3   J p ,   σ 1 ,   σ 2 - 1 p 2 σ 2 - σ 1 ,   | p |     | σ i - f | - 1 , i = 1 ,   2 .
ρ x ,   y = ρ f x ,   y χ x ,   y ,
ρ f x ,   y = G x - F ξ y
χ x ,   y = 1   in fiber , 0   in vessels , ρ p / ρ f x ,   y in parenchyma ,
j f s ,   σ = p f s ,   σ l s ,   σ = 1 d - d / 2 d / 2   ρ f x + σ y ,   y d y ,   x = sd / 2 ,
j χ s ,   σ = p s ,   σ l χ s ,   σ = - d / 2 d / 2   ρ f χ x + σ y ,   y d y × - d / 2 d / 2   χ x + σ y ,   y d y ,
l χ s ,   σ = - d / 2 d / 2   χ x + σ y ,   y 1 + σ 2 1 / 2 d y .
j χ s ,   σ = j s ,   σ l s ,   σ l χ s ,   σ .
j f s ,   σ j χ s ,   σ .
d 2 l 0 2 l / d   | j s ,   σ - j 0 s ,   σ | / j 0 s ,   σ d s

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