Abstract

In dealing with optical imaging systems, it is more feasible experimentally to measure line spread functions than point spread functions. When the intensity distribution is known to possess rotational symmetry, the point spread function can be obtained mathematically from the corresponding line spread function by solving an integral equation. A direct solution of this equation is given which represents a procedure that is simpler for practical use than the usual one involving Fourier transforms.

© 1964 Optical Society of America

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References

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  1. R. C. Jones, J. Acoust. Soc. Am. 16, 147–171 (1945);J. Opt. Soc. Am. 48, 934–937 (1958).
    [Crossref]
  2. Thanks are due to F. Kottler, R. N. Wolfe, and H. J. Zweig, at present or formerly at these Laboratories, for helpful advice and discussion, and to S. A. Tuccio for working out the graphical example.
  3. M. Bôcher, An Introduction to the Study of Integral Equations, (Cambridge University Press, Cambridge, England, 1909), Vol. 10.
  4. E. L. O’Neill, The Modulation Function in Optics (Technical Note 110) (Boston University Optical Research Laboratory, Boston, Massachusetts, 1954).
  5. Tables of Integral Transforms, edited by A. Erdélyi (McGraw-Hill Book Company, Inc., New York, 1954), Vol. II.
  6. J. Shumaker and C. Yokley, Appl. Opt. 3, 83 (1964).
    [Crossref]
  7. M. Freeman and S. Katz, J. Opt. Soc. Am. 50, 826 (1960).
    [Crossref]
  8. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [Crossref]
  9. W. Barr, J. Opt. Soc. Am. 52, 885 (1962).
    [Crossref]
  10. R. Shack, J. Res. Natl. Bur. Std. 56, 245 (1956).
    [Crossref]
  11. A. Maréchal, Rev. Opt. 26, 257 (1947).

1964 (1)

1962 (1)

1961 (1)

1960 (1)

1956 (1)

R. Shack, J. Res. Natl. Bur. Std. 56, 245 (1956).
[Crossref]

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

1945 (1)

R. C. Jones, J. Acoust. Soc. Am. 16, 147–171 (1945);J. Opt. Soc. Am. 48, 934–937 (1958).
[Crossref]

Barr, W.

Bôcher, M.

M. Bôcher, An Introduction to the Study of Integral Equations, (Cambridge University Press, Cambridge, England, 1909), Vol. 10.

Bockasten, K.

Freeman, M.

Jones, R. C.

R. C. Jones, J. Acoust. Soc. Am. 16, 147–171 (1945);J. Opt. Soc. Am. 48, 934–937 (1958).
[Crossref]

Katz, S.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

O’Neill, E. L.

E. L. O’Neill, The Modulation Function in Optics (Technical Note 110) (Boston University Optical Research Laboratory, Boston, Massachusetts, 1954).

Shack, R.

R. Shack, J. Res. Natl. Bur. Std. 56, 245 (1956).
[Crossref]

Shumaker, J.

Yokley, C.

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

R. C. Jones, J. Acoust. Soc. Am. 16, 147–171 (1945);J. Opt. Soc. Am. 48, 934–937 (1958).
[Crossref]

J. Opt. Soc. Am. (3)

J. Res. Natl. Bur. Std. (1)

R. Shack, J. Res. Natl. Bur. Std. 56, 245 (1956).
[Crossref]

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Other (4)

Thanks are due to F. Kottler, R. N. Wolfe, and H. J. Zweig, at present or formerly at these Laboratories, for helpful advice and discussion, and to S. A. Tuccio for working out the graphical example.

M. Bôcher, An Introduction to the Study of Integral Equations, (Cambridge University Press, Cambridge, England, 1909), Vol. 10.

E. L. O’Neill, The Modulation Function in Optics (Technical Note 110) (Boston University Optical Research Laboratory, Boston, Massachusetts, 1954).

Tables of Integral Transforms, edited by A. Erdélyi (McGraw-Hill Book Company, Inc., New York, 1954), Vol. II.

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Figures (3)

F. 1
F. 1

Plots of spread functions. A, original line spread function; B, point spread function calculated from it.

F. 2
F. 2

Plots of A, function A(t) from Eq. (7), and B, function 0 A ( υ 2 + z ) d υ , from Fig. 3. Since the curves are bilaterally symmetrical, only one side of each is shown.

F. 3
F. 3

Plots of A(t) = A (υ2+z) vs v for values of z shown on curves.

Equations (44)

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i = i ( x , y ) ,
E = i ( x , y ) dydx .
Δ E = Δ x i ( x 0 , y ) d y .
I ( x ) = i ( x , y ) d y ,
i ( x , y ) = f ( x 2 + y 2 ) ,
I ( x ) = f ( x 2 + y 2 ) d y ,
I ( x ) = A ( x 2 ) = A ( t ) = 2 0 f ( t + y 2 ) d y , t = x 2 0 .
A ( t ) = t f ( u ) ( u t ) 1 2 d u .
z A ( t ) ( t z ) 1 2 d t = z t f ( u ) ( t z ) 1 2 ( u t ) 1 2 dudt .
z A ( t ) ( t z ) 1 2 d t = z [ z u ( t z ) 1 2 ( u t ) 1 2 d t ] f ( u ) d u .
s = ( t z ) / ( u z ) , d s = d t / ( u z ) ,
0 1 s 1 2 ( 1 s ) 1 2 d s ,
B ( m , n ) = 0 1 z m 1 ( 1 z ) n 1 d s .
B ( m , n ) = Γ ( m ) Γ ( n ) / Γ ( m + n ) .
Γ ( 1 2 ) = π 1 2 , Γ ( 1 ) = 1 ,
B ( 1 2 , 1 2 ) = Γ ( 1 2 ) [ Γ ( 1 2 ) / Γ ( 1 ) ] = π .
z A ( t ) ( t z ) 1 2 d t = π z f ( u ) d u ,
f ( z ) = 1 π d d z z A ( t ) ( t z ) 1 2 d t .
υ = ( t z ) 1 2 , 2 υ d υ = d t ,
f ( z ) = 2 π d d z 0 A ( υ 2 + z ) d υ .
f ( z ) = ( 2 π ) 0 A ( υ 2 + z ) d υ ,
i ( x , y ) = f ( x 2 + y 2 ) = f ( z ) .
υ 2 = t z ,
A ( υ 2 + z ) = A ( t ) ,
0 A ( υ 2 + z ) d υ .
A ( t ) = 2 0 f ( t + y 2 ) d y ,
A ( υ 2 + z ) = 2 0 f ( υ 2 + z + y 2 ) d y .
0 A ( υ 2 + z ) d υ = 2 0 0 f ( υ 2 + z + y 2 ) d υ d y ,
0 A ( υ 2 + z ) d υ = 2 0 π / 2 0 f ( r 2 + z ) rdrd θ = 0 π / 2 [ f ( r 2 + z ) ] 0 d θ .
lim r f ( r 2 + z ) = 0 ,
f ( z ) = 2 π 0 A ( υ 2 + z ) d υ ,
I # ( ν ) = I ( x ) e ι ν x d x = 2 0 f ( x 2 + y 2 ) e ι ν x dydx .
I # ( ν ) = 2 π / 2 π / 2 0 f ( r 2 ) e ι ν x cos θ rdrd θ = 2 π 0 f ( r 2 ) J 0 ( ν r ) rdr ,
f ( r 2 ) = 1 2 π 0 I # ( ν ) J 0 ( ν r ) ν d ν .
f ( r 2 ) = 1 π d d r r I ( x ) rdx x ( x 2 r 2 ) 1 2 ,
υ = ( x 2 r 2 ) 1 2 ,
f ( r 2 ) = 1 π d d r 0 A ( υ 2 + r 2 ) r d υ υ 2 + r 2 ,
I ( x ) = ( 1 + x 2 ) 1 = A ( x 2 ) .
f ( z ) = 2 π d d z 0 d υ 1 + υ 2 + z , = 2 π d d z [ 1 ( 1 + z ) 1 2 tan 1 υ ( 1 + z ) 1 2 ] 0 , = 2 π d d z [ π / 2 ( 1 + z ) 1 2 ] = 1 2 ( 1 + z ) 3 2 .
f ( r 2 ) = 1 2 ( 1 + r 2 ) 3 2 .
I ( x ) = exp ( x 2 ) = A ( x 2 ) .
f ( z ) = 2 π d d z 0 e ( υ 2 + z ) d υ = 2 π e z 0 e υ 2 d υ e z π 1 2 .
f ( r 2 ) = e r 2 / π 1 2 ,
0 A ( υ 2 + z ) d υ ,