Abstract

The spatial distribution of a radioactive fluid can be measured indirectly by observing the emerging gamma rays. A method is proposed and analyzed for gamma-ray imaging by stochastic time modulation and cross-correlation. Theoretical comparison is made to collimation and coded aperture techniques in gamma-ray image formation. Computed results are presented that illustrate the mean response and statistical error characteristics of this technique. Monte Carlo simulations are performed as a further verification. Because it relies upon a point-by-point reconstruction, rather than upon the integral properties of any particular aperture, the time modulation approach is seen to provide a theoretical basis for obtaining a smooth three-dimensional point response.

© 1974 Optical Society of America

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References

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  1. A. Gottschalk, R. N. Beck, Fundamental Problems of Scanning (Charles C Thomas, Springfield, 1968).
  2. N. F. Moody, Proc. IEEE 56, 218 (1970).
  3. H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
    [PubMed]
  4. H. H. Barrett, F. A. Horrigan, Raytheon Technical Memorandum T-926 (1972).
  5. W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
    [PubMed]
  6. W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
    [CrossRef]
  7. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  8. H. H. Barrett, G. D. DeMeester, Raytheon Technical Memorandum T-972 (1973).
  9. C. J. Oliver, E. R. Pike, Appl. Opt. 13, 159 (1974).
    [CrossRef]
  10. G. Goodrich, Bendix Electro-Optics Division; unpublished (1973).
  11. L. Mertz, Transformations in Optics (Wiley, New York, 1965), pp. 78–52.
  12. R. S. May, G. F. Knoll, A. Z. Akcasu, submitted to J. Nucl. Med.
  13. R. S. May, Ph.D. Thesis, University of Michigan (1974).
  14. G. F. Knoll, Report B/RG/3192 to Science Research Council, London (1973).
  15. F. Hossfeld, R. Amadori, Kernforschungsanlage Jülich Technical Report, Jül—684-FF (1970).
  16. R. A. Rydin, R. J. Hooper, Nucl. Sci. Eng. 38, 216 (1969).
  17. M. R. Buckner, T. W. Kerlin, Nucl. Sci. Eng. 49, 255 (1972).
  18. W. L. Rogers, Division of Nuclear Medicine, University of Michigan; personal communication (1973).
  19. D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).
  20. A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
    [CrossRef]

1974 (2)

C. J. Oliver, E. R. Pike, Appl. Opt. 13, 159 (1974).
[CrossRef]

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

1973 (1)

W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
[CrossRef]

1972 (3)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

M. R. Buckner, T. W. Kerlin, Nucl. Sci. Eng. 49, 255 (1972).

1970 (1)

N. F. Moody, Proc. IEEE 56, 218 (1970).

1969 (1)

R. A. Rydin, R. J. Hooper, Nucl. Sci. Eng. 38, 216 (1969).

1968 (1)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Akcasu, A. Z.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

R. S. May, G. F. Knoll, A. Z. Akcasu, submitted to J. Nucl. Med.

Amadori, R.

F. Hossfeld, R. Amadori, Kernforschungsanlage Jülich Technical Report, Jül—684-FF (1970).

Barrett, H. H.

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

H. H. Barrett, F. A. Horrigan, Raytheon Technical Memorandum T-926 (1972).

H. H. Barrett, G. D. DeMeester, Raytheon Technical Memorandum T-972 (1973).

D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).

Beck, R. N.

A. Gottschalk, R. N. Beck, Fundamental Problems of Scanning (Charles C Thomas, Springfield, 1968).

Beierwaltes, W. H.

W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
[CrossRef]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Buckner, M. R.

M. R. Buckner, T. W. Kerlin, Nucl. Sci. Eng. 49, 255 (1972).

DeMeester, G. D.

H. H. Barrett, G. D. DeMeester, Raytheon Technical Memorandum T-972 (1973).

D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).

Dicke, R. H.

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Farmelant, W. H.

D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).

Goodrich, G.

G. Goodrich, Bendix Electro-Optics Division; unpublished (1973).

Gottschalk, A.

A. Gottschalk, R. N. Beck, Fundamental Problems of Scanning (Charles C Thomas, Springfield, 1968).

Han, K. S.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Hooper, R. J.

R. A. Rydin, R. J. Hooper, Nucl. Sci. Eng. 38, 216 (1969).

Horrigan, F. A.

H. H. Barrett, F. A. Horrigan, Raytheon Technical Memorandum T-926 (1972).

Hossfeld, F.

F. Hossfeld, R. Amadori, Kernforschungsanlage Jülich Technical Report, Jül—684-FF (1970).

Jones, L. W.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
[CrossRef]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Kerlin, T. W.

M. R. Buckner, T. W. Kerlin, Nucl. Sci. Eng. 49, 255 (1972).

Knoll, G. F.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

G. F. Knoll, Report B/RG/3192 to Science Research Council, London (1973).

R. S. May, G. F. Knoll, A. Z. Akcasu, submitted to J. Nucl. Med.

Koral, K. F.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

May, R. S.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

R. S. May, G. F. Knoll, A. Z. Akcasu, submitted to J. Nucl. Med.

R. S. May, Ph.D. Thesis, University of Michigan (1974).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), pp. 78–52.

Moody, N. F.

N. F. Moody, Proc. IEEE 56, 218 (1970).

Oliver, C. J.

C. J. Oliver, E. R. Pike, Appl. Opt. 13, 159 (1974).
[CrossRef]

Pike, E. R.

C. J. Oliver, E. R. Pike, Appl. Opt. 13, 159 (1974).
[CrossRef]

Rogers, W. L.

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
[CrossRef]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

W. L. Rogers, Division of Nuclear Medicine, University of Michigan; personal communication (1973).

Rydin, R. A.

R. A. Rydin, R. J. Hooper, Nucl. Sci. Eng. 38, 216 (1969).

Wilson, D. T.

D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).

Appl. Opt. (1)

C. J. Oliver, E. R. Pike, Appl. Opt. 13, 159 (1974).
[CrossRef]

Astrophys. J. (1)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

J. Nucl. Med. (2)

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

Nucl. Sci. Eng. (2)

R. A. Rydin, R. J. Hooper, Nucl. Sci. Eng. 38, 216 (1969).

M. R. Buckner, T. W. Kerlin, Nucl. Sci. Eng. 49, 255 (1972).

Opt. Eng. (2)

W. L. Rogers, L. W. Jones, W. H. Beierwaltes, Opt. Eng. 12, 13 (1973).
[CrossRef]

A. Z. Akcasu, R. S. May, G. F. Knoll, W. L. Rogers, K. F. Koral, L. W. Jones, Opt. Eng. 13, 117 (1974).
[CrossRef]

Proc. IEEE (1)

N. F. Moody, Proc. IEEE 56, 218 (1970).

Other (11)

A. Gottschalk, R. N. Beck, Fundamental Problems of Scanning (Charles C Thomas, Springfield, 1968).

H. H. Barrett, F. A. Horrigan, Raytheon Technical Memorandum T-926 (1972).

H. H. Barrett, G. D. DeMeester, Raytheon Technical Memorandum T-972 (1973).

W. L. Rogers, Division of Nuclear Medicine, University of Michigan; personal communication (1973).

D. T. Wilson, H. H. Barrett, G. D. DeMeester, W. H. Farmelant, Raytheon Technical Memorandum T-945 (1973).

G. Goodrich, Bendix Electro-Optics Division; unpublished (1973).

L. Mertz, Transformations in Optics (Wiley, New York, 1965), pp. 78–52.

R. S. May, G. F. Knoll, A. Z. Akcasu, submitted to J. Nucl. Med.

R. S. May, Ph.D. Thesis, University of Michigan (1974).

G. F. Knoll, Report B/RG/3192 to Science Research Council, London (1973).

F. Hossfeld, R. Amadori, Kernforschungsanlage Jülich Technical Report, Jül—684-FF (1970).

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Figures (11)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Depth information.

Fig. 3
Fig. 3

(a) Plan for discrete derivations. (b) Subtended interval used to define conditional transmission probability.

Fig. 4
Fig. 4

Mean responses to a point source by some collections of randomly generated aperture realizations.

Fig. 5
Fig. 5

Mean response to a point source using a pseudostatistical string; M = Nf = 23.

Fig. 6
Fig. 6

(a) For various values of m, plots of the error kernal vs xx′; normalized with respect to peak value of PSRF. (b) Noise relative error vs m, evaluated at two values of xx′.

Fig. 7
Fig. 7

For three values of m, mean response curves with error bars for randomly generated realizations (M = 50).

Fig. 8
Fig. 8

Plot of relative error vs m for the case of nearly continous detection.

Fig. 9
Fig. 9

Out-of-focus response (M = 500).

Fig. 10
Fig. 10

(a) In-focus response to a simulated point source. (b) Peak weight vs focal depth for the simulated point source.

Fig. 11
Fig. 11

Simulated point source response for a random collection with mean transmission m = 0.50.

Equations (63)

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y = α z + β x ,
θ = tan 1 [ ( z x ) / l ] ,
C ν ( z ) = τ M d x f ( x ) T ν ( z , x ) λ ( z , x ) .
f ˆ ( x ) = Dectector d z ν = 1 M C ν ( z ) H ν ( z , x ) ,
= τ Dectector d z ν = 1 M H ν ( z , x ) 1 M × d x f ( x ) T ν ( z , x ) λ ( z , x ) ,
f ˆ ( x ) = τ d x f ( x ) Q ( x , x ) ,
Q ( x , x ) = Detector d z λ ( z , x ) 1 M ν = 1 M H ν ( z , x ) T ν ( z , x )
T ( y , θ ) = θ 0 Δ θ θ 0 + Δ θ d θ δ ( θ θ ) .
T ( z , x ) = 2 Δ θ l δ ( z x ) ,
Q ( x , x ) = 2 Δ θ l δ ( x x ) .
T ( y ) H ( y + z ) d y = δ ( z ) ,
d y T ( y ) H ( y + z ) = R ( z ) ,
T ( y ) = exp ( i γ y 2 ) ,
H ( y ) = ( γ / π ) exp ( i γ y 2 )
γ / π exp ( i γ y 2 ) exp [ i γ ( y + z ) 2 ] d y = δ ( z ) .
T ( y ) = ( 1 + cos γ y 2 ) / 2 .
lim L 1 L L / 2 L / 2 T ( y ) d y = 1 / 2 ,
H ( y ) = ( 4 γ / π ) ( cos γ y 2 sin γ y 2 ) ,
d y T ( y ) H ( y + z ) = δ ( z ) ( γ / π ) sin ( γ z 2 / 2 ) .
L / 2 L / 2 d y T ( y ) H ( y + z ) = R L ( z ) + g L ( z ) .
R L ( z ) δ ( z ) ,
g L ( z ) ( γ / π ) sin ( γ z 2 / 2 ) .
R L ( z ) = ( 1 / π ) ( sin γ z L / z ) ,
Q ( x , x ) = Detector d z H ( α z + β x ) T ( α z + β x ) .
y = α z + β x ,
Q ( x , x ) = 1 α L / 2 L / 2 d y T ( y ) H [ y + β ( x x ) ] .
Q ( x , x ) = ( 1 / α ) { R L [ β ( x x ) ] + g L [ β ( x x ) ] } .
H ( y ) = T ( y ) m .
Φ ( z ) = lim L 1 L L / 2 L / 2 d y T ( y ) [ T ( y + z ) m ] .
Φ ( z ) = σ 2 δ ( z ) .
t ν ( y ) T ν ( y ) m ( y ) ,
m ( y ) 1 M ν = 1 M T ν ( y ) ,
t ν ( y ) C ν ( z ) = 0 , t ν ( y ) C ν ( z ) f ( x ) ,
Φ ( y 1 y 2 ) = t ( y 1 ) t ( y 2 ) = lim M 1 M ν = 1 M t ν ( y 1 ) t ν ( y 2 ) ,
Q ( x , x ) = Detector d z λ ( z x ) 1 M ν = 1 M t ν ( α z + β x ) T ν ( α z + β x ) M Φ [ β ( x x ) ] · Detector d z λ ( z x ) .
Φ [ β ( x x ) ] = η δ [ β ( x x ) ] ,
Q ( x , x ) M η δ [ β ( x x ) ] Detector d z λ ( z x ) .
Φ M ( y 1 y 2 ) = 1 M ν = 1 M t ν ( y 1 ) t ν ( y 2 ) .
T ν = ( T 1 ν , , T N f ν ) .
t ν = ( T 1 ν m 1 , , T N f m N f ) ,
m j = 1 M ν = 1 M T j ν .
1 M ν = 1 M t ν ( y ) t ν ( y ) = { σ 2 if y , y in same interval , 0 otherwise .
( x , j ) [ α ( z j δ ) + β x , α ( z j + δ ) + β x ] .
J ( x , j ) = ν = 1 M C j ν t ¯ ν ( x , j ) ,
t ¯ ν ( x , j ) = 1 2 α δ ( x , j ) d y t ν ( y ) = 1 2 α δ α ( z j δ ) + β x α ( z j + δ ) + β x d y t ν ( y )
T ¯ ν ( x , j ) = 1 2 α δ ( x , j ) d y T ν ( y )
f ˆ ( x ) = j = 1 N J ( x , j ) = j = 1 N ν = 1 M C j ν t ¯ ν ( x , j ) .
f ˆ ( x ) = τ d x f ( x ) Q ( x , x )
Var [ f ˆ ( x ) ] = [ f ˆ ( x ) f ˆ ( x ) ] 2 = τ d x f ( x ) E ( x , x ) ,
Q ( x , x ) = 1 2 α δ j = 1 N ( x , j ) d y λ ( y x α ) × ( x , j ) d y 1 M ν = 1 M t ν ( y ) T ν ( y ) ,
E ( x , x ) = 1 ( 2 α δ ) 2 j = 1 N ( x , j ) d y λ ( y x α ) · ( x , j ) d y ( x , j ) d y 1 M ν = 1 M t ν ( y ) t ν ( y ) T ν ( y ) .
Var [ f ˆ ( x 0 ) ] = τ d x f ( x ) E ( x 0 , x ) ,
Var [ f ˆ ( x ) ] = f ˆ ( x ) 2 f ˆ ( x ) 2 ,
f ˆ ( x ) 2 = [ j = 1 N ν = 1 M C j ν ( x , j ) d y t ν ( y ) ] 2 = j , j ν , ν ( x , j ) d y ( x , j ) d y C j ν C j ν t ν ( y ) t ν ( y ) .
C j ν C j ν = δ ν ν δ j j C j ν 2 + ( 1 δ ν ν δ j j ) C j ν C j ν .
C j ν = k = 1 K l = 1 L C k , jl ν .
D k , j l ν = C k , j l ν .
C j ν 2 = l = 1 L l = 1 L k = 1 K k = 1 K C k , j l C k , j l .
C k , j l C k , j l = C k , j l = 0 C k , j l = 0 P ( C k , j l , C k , j l ) C k , j l C k , j l ,
P ( C k , j l , C k , j l ) = g k , g k N k , j l N k , j l C k , j l C k , j l P ( C k , j l , C k , j l , N k , j l , N k , j l , g k , g k ) = g k , g k P ( g k , g k ) N k , j l N k , j l P ( N k , j l , N k , j l | g k , g k ) · C k , j l C k , j l P ( C k , j l , C k , j l | N k , j l , N k , j l , g k , g k ) C k , j l C k j l .
P ( C k , j l , C k , j l | N k , j l , N k , j l , g k , g k ) = { P ( C k , j l | N k , j l ) δ C k , j l , C k , j l if k = k and l = l , P ( C k , j l | N k , j l ) P ( C k , j l | N k , j l ) otherwise .
C k , j l C k , j l P ( C k , j l , C k , j l | N k , j l , N k , j l , g k , g k ) C k , j l C k , j l = { δ k , k δ l l [ T k , j l 2 N k , j l 2 + T k , j l ( 1 T k , j l ) N k , j l ] + ( 1 δ k k δ l l ) ( T k , j l N k , j l ) ( T k , j l N k , j l ) } .
Var [ f ˆ ( x ) ] = k = 1 K f x τ [ j = 1 N l = 1 L λ k , j l ( x , j ) × d y ( x , j ) d y 1 M ν = 1 M T ν t ν ( y ) t ν ( y ) ] .

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