Abstract

Theoretical and experimental investigations of a method for direct recording and reconstruction of 3-D x-ray images are described. When recording, an x-ray source is placed at discrete positions, and an x-ray grid is fixed on the x-ray plate. When reconstructing, a lenticular sheet matched with the grid is overlayed on the developed plate, and a parallax–panoramagram type 3-D image is seen with the naked eye. Using a small x-ray source designed for use in dental clinics and an x-ray grid matched with the lenticular sheet, 3-D x-ray images having a 0.001-rad resolvable angle and a 0.02-rad flipping angle are directly recognized with clear parallax.

© 1978 Optical Society of America

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References

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  1. L. P. Dudley, Stereoptics (McDonald & Company, London, 1951), pp. 96–109.
  2. G. N. Hounsfield, Br. J. Radiol. 46, 1016 (1973).
    [CrossRef] [PubMed]
  3. H. E. Johnes, Br. J. Radiol. 49, 745 (1976).
    [CrossRef]
  4. K. Sano, Ed., Atlas of Stereoscopic Neuroradiology (U. Tokyo Press, Tokyo, 1976), pp. 4–8.
  5. S. Ikeda, Y. Ono, K. Takakura, T. Soma, M. Sato, Y. Ueda, S. Kwak, Eds., X-ray 3-Dimensional Images (Eisai Company, Ltd., Japan, 1973).
  6. T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
    [CrossRef]
  7. G. Groh, M. Kock, Appl. Opt. 9, 775 (1970).
    [CrossRef] [PubMed]
  8. E. Klotz, H. Weiss, Opt. Commun. 11, 368 (1974).
    [CrossRef]
  9. E. Klotz, H. Weiss, Appl. Opt. 15, 1913 (1976).
    [CrossRef] [PubMed]
  10. C. H. MacGillavry, G. D. Rieck, K. Lonsdale, Eds., International Tables for X-ray Crystallography (Kynoch Press, Birmingham, U.K., 1968), pp. 157–192. Equation (20) is fitted to the data shown in Table 3.2.2C.
  11. T. Okoshi, Three-Dimensional Image Technology (Sangyo Tosho, Tokyo, 1972), pp. 56–58.
  12. J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

1977 (1)

J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

1976 (2)

1974 (1)

E. Klotz, H. Weiss, Opt. Commun. 11, 368 (1974).
[CrossRef]

1973 (1)

G. N. Hounsfield, Br. J. Radiol. 46, 1016 (1973).
[CrossRef] [PubMed]

1970 (1)

1969 (1)

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

Dudley, L. P.

L. P. Dudley, Stereoptics (McDonald & Company, London, 1951), pp. 96–109.

Groh, G.

Hamasaki, J.

J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

Hioki, R.

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

Hounsfield, G. N.

G. N. Hounsfield, Br. J. Radiol. 46, 1016 (1973).
[CrossRef] [PubMed]

Johnes, H. E.

H. E. Johnes, Br. J. Radiol. 49, 745 (1976).
[CrossRef]

Kasahara, T.

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

Kawabata, M.

J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

Kimura, Y.

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

Klotz, E.

Kock, M.

Okoshi, T.

T. Okoshi, Three-Dimensional Image Technology (Sangyo Tosho, Tokyo, 1972), pp. 56–58.

Tanaka, S.

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

Weiss, H.

Yokota, K.

J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

Appl. Opt. (2)

Br. J. Radiol. (2)

G. N. Hounsfield, Br. J. Radiol. 46, 1016 (1973).
[CrossRef] [PubMed]

H. E. Johnes, Br. J. Radiol. 49, 745 (1976).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Kasahara, Y. Kimura, R. Hioki, S. Tanaka, Jpn. J. Appl. Phys. 8, 124 (1969).
[CrossRef]

National Convention Rd. of IEE Japan (1)

J. Hamasaki, K. Yokota, M. Kawabata, National Convention Rd. of IEE Japan, 588, July (1977).

Opt. Commun. (1)

E. Klotz, H. Weiss, Opt. Commun. 11, 368 (1974).
[CrossRef]

Other (5)

L. P. Dudley, Stereoptics (McDonald & Company, London, 1951), pp. 96–109.

K. Sano, Ed., Atlas of Stereoscopic Neuroradiology (U. Tokyo Press, Tokyo, 1976), pp. 4–8.

S. Ikeda, Y. Ono, K. Takakura, T. Soma, M. Sato, Y. Ueda, S. Kwak, Eds., X-ray 3-Dimensional Images (Eisai Company, Ltd., Japan, 1973).

C. H. MacGillavry, G. D. Rieck, K. Lonsdale, Eds., International Tables for X-ray Crystallography (Kynoch Press, Birmingham, U.K., 1968), pp. 157–192. Equation (20) is fitted to the data shown in Table 3.2.2C.

T. Okoshi, Three-Dimensional Image Technology (Sangyo Tosho, Tokyo, 1972), pp. 56–58.

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Figures (10)

Fig. 1
Fig. 1

Formation of a 3-D x-ray plate.

Fig. 2
Fig. 2

Object space recorded on the plate from 2n0 + 1 directions. (a) Case 1 (m0a < n0c bounded by four lines); and (b) case 2 (m0a > n0c bounded by five lines).

Fig. 3
Fig. 3

Example of the mesh points constructed by 17 × 5 = 85 rays. If the n = ±1 positions are eliminated, sectional planes shown by broken lines appear. On these planes, sampled points by n = 0,±2 positions do not coincide with those by n = ±1 positions.

Fig. 4
Fig. 4

Reconstruction of a 3-D x-ray image.

Fig. 5
Fig. 5

Numbers of the sampled points P1, P2, and P3 in relation to m0, n0, lυ, and lw. P2 is shown by a broken line.

Fig. 6
Fig. 6

Deformation and flipping of the image space. When eyes move upward in the subzones of n = 0 and 1, the image space changes from the space by the chain line to the space by the broken line.

Fig. 7
Fig. 7

Cross section of a simple x-ray grid.

Fig. 8
Fig. 8

Pictures of a reconstructed 3-D image. A plastic object is taken from four directions at visible wavelengths.

Fig. 9
Fig. 9

Pictures of a reconstructed 3-D image. A seashell in a virtual image taken by x ray. Two out of four scenes are shown.

Fig. 10
Fig. 10

Pictures of a reconstructed 3-D image. A seashell in a real image taken by x ray. Two out of four scenes are shown.

Tables (1)

Tables Icon

Table I Relations for Taking 3-D x-Ray Images

Equations (28)

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x = n b + ( m a n b n c ) z z g 1 , = { m + n [ u ( z g z 1 1 ) υ ] } a z z g 1 ,
z l = z g u ( u + υ + l ) 1 , x l , k = k a l ,
z a = z g u ( u + υ + w ) 1 , z b = z g u ( u + υ ) 1 , z c = z g u ( u + υ w ) 1 i f w < u + υ , x b = m 0 a u ( u + υ ) 1 ,
l 0 l l w , m 0 + n 0 | l | k m 0 n 0 | l | ,
h m , n = [ m n ( u + υ ) ] a f z g 1 .
s = h m , n 1 h m , n = ( u + υ ) a f z g 1 ,
a + h m , n 0 h m 1 , n 0 = { 1 [ 2 n 0 ( u + υ ) 1 ] f z g 1 } a .
f = z g [ ( 2 n 0 + 1 ) ( u + υ ) 1 ] 1 .
x = [ m + n ( z g z 1 1 ) ( u + υ ) ] a z z g 1 .
z l = z g ( u + υ ) ( u + υ + l ) 1 , x l , k = k a l ,
M = ( u + υ ) u 1 ,
P = P 1 + P 2 + P 3 , P 1 = l w [ n 0 ( l w 1 ) + 1 ] , P 2 = 2 m 0 + 1 , P 3 = l 0 [ n 0 ( 2 l w l 0 1 ) + 1 ] .
F l = l a z g 1 ,
R l = [ a 2 + l 2 D 2 ( u + υ ) 2 ] 1 / 2 z g 1 ,
Q l = [ 2 q 0 ( l + υ ) u 1 a 1 ] + 1 ,
g = a 2 r sec θ , h θ = f tan θ .
g 0 = a ( 1 2 r a 1 ) if θ = 0 , g = 0.5 g 0 if θ ( g 0 a 1 ) 1 / 2 , g = 0 if θ ( 2 g 0 a 1 ) 1 / 2 .
0.25 a 3 f 2 g 0 a ( 2 n 0 + 1 ) 1 ,
n 0 max = [ 0.5 ( 4 f 2 a 2 1 ) ] ,
μ 2.65 × 10 4 λ 3 Z 3.3 ρ ( 1 0.75 × 10 4 λ Z 2 ) ,
x = m { a [ a f + ( a a ) z g ] z g f z g 1 } + { m z [ a f + ( a a ) z g ] + n ( z g z ) ( u + υ ) a f } f z g 1 .
L = 1 + ( 1 a a 1 ) z g f 1 ,
x = n a ( u + υ ) L 1 , n 0 n n 0 ,
z l = z g ( u + υ ) ( L l + u + υ ) 1 , x l , k = k a l ,
M 1 = a l a l 1 = M , M 2 = z l z l 1 = M z g z g 1 .
| 1 a a 1 | f z g 1
a l w
z l w

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