Abstract

The distortion of the image of an object, produced by diffraction, can be reduced by filtering. A class of filters is developed for minimizing the first term in the moment expansion of the error. The resulting spread functions have high attenuation rate, negligible side lobes, and are optimum for imaging smooth objects. One- and two-dimensional systems are considered; the analysis includes coherent and incoherent illumination.

© 1972 Optical Society of America

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References

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  1. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).
  2. A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill, New York, 1962).
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  4. J. E. Wilkins, J. Opt. Soc. Am. 53, 420 (1963).
    [Crossref]
  5. G. Lansraux and G. Boivin, Can. J. Phys. 76, 1696 (1958).
    [Crossref]
  6. R. Barakat, J. Opt. Soc. Am. 52, 264 (1962).
    [Crossref]
  7. R. Barakat and E. Levin, J. Opt. Soc. Am. 53, 274 (1963).
    [Crossref]
  8. A. Papoulis, J. Opt. Soc. Am. 62, 77 (1972).
    [Crossref]
  9. A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).
  10. N. I. Akhiezer, The Calculus of Variations (Blaisdell, New York, 1962).
  11. N. I. Akhiezer, Theory of Approximation, (Ungar, New York, 1956).
  12. E. Marchand, J. Opt. Soc. 54, 915 (1964).
    [Crossref]
  13. A. Papoulis, IEEE Trans. IT-14, 164 (1968).

1972 (1)

1968 (1)

A. Papoulis, IEEE Trans. IT-14, 164 (1968).

1964 (1)

E. Marchand, J. Opt. Soc. 54, 915 (1964).
[Crossref]

1963 (2)

1962 (1)

1958 (1)

G. Lansraux and G. Boivin, Can. J. Phys. 76, 1696 (1958).
[Crossref]

Akhiezer, N. I.

N. I. Akhiezer, The Calculus of Variations (Blaisdell, New York, 1962).

N. I. Akhiezer, Theory of Approximation, (Ungar, New York, 1956).

Barakat, R.

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 76, 1696 (1958).
[Crossref]

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 76, 1696 (1958).
[Crossref]

Levin, E.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Marchand, E.

E. Marchand, J. Opt. Soc. 54, 915 (1964).
[Crossref]

Papoulis, A.

A. Papoulis, J. Opt. Soc. Am. 62, 77 (1972).
[Crossref]

A. Papoulis, IEEE Trans. IT-14, 164 (1968).

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).

A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill, New York, 1962).

Wilkins, J. E.

Can. J. Phys. (1)

G. Lansraux and G. Boivin, Can. J. Phys. 76, 1696 (1958).
[Crossref]

IEEE Trans. (1)

A. Papoulis, IEEE Trans. IT-14, 164 (1968).

J. Opt. Soc. (1)

E. Marchand, J. Opt. Soc. 54, 915 (1964).
[Crossref]

J. Opt. Soc. Am. (4)

Other (6)

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).

N. I. Akhiezer, The Calculus of Variations (Blaisdell, New York, 1962).

N. I. Akhiezer, Theory of Approximation, (Ungar, New York, 1956).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).

A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill, New York, 1962).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

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Figures (6)

F. 1
F. 1

w ( x ) = B π 2 b [ sin b ( x + α ) π ( x + α ) + sin b ( x α ) π ( x α ) ] 2 , α = π 2 b .(i) w(x) = optimum spread function h ¯ 0 ( x ) of one-dimensional system for incoherent illumination; B = b/2π, pupil size a = b. (ii) w(x) = optimum amplitude spread function h0(x) of one-dimensional system for coherent illumination; B = 1, pupil size a = 2b. (iii) w(x) = optimum amplitude line spread function s0(x) of two-dimensional system for coherent illumination; B = 1, pupil size a = 2b.

F. 2
F. 2

Optimum transfer function H 0 ( u ) = cos π x / 2 a , | u | < aof one-dimensional system for incoherent illumination; pupil size a.

F. 3
F. 3

Optimum amplitude spread function h 0 ( x ) = sin a ( x + α ) 2 π ( x + α ) + sin a ( x α ) 2 π ( x α ) , α = π 2 aof one-dimensional system for incoherent illumination; pupil size a.

F. 4
F. 4

W ( u ) = B [ 1 π | sin α u | + ( 1 | u | 2 b ) cos α u ] , | u | 2 b .Fourier transform of the function w(x) in Fig. 1.

F. 5
F. 5

Optimum transfer function H 0 ( w ) = J 0 ( a w ) , | w | < aof two-dimensional system for incoherent illumination; pupil size a.

F. 6
F. 6

Optimum amplitude point spread function h 0 ( r ) = A J 0 ( a r ) α 2 r 2 , α = z 1 aof two-dimensional system for incoherent illumination; pupil size a.

Equations (82)

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p 1 ( x ) = p ( x ξ ) h ( ξ ) d ξ ,
q 1 ( x ) = q ( x ξ ) h ¯ ( ξ ) d ξ ,
P 1 ( u ) = P ( u ) H ( u ) , Q 1 ( u ) = Q ( u ) H ¯ ( u ) ,
H ¯ ( u ) = 1 2 π H ( u + ξ ) H * ( ξ ) d ξ
H ( u ) = 0 for | u | > a ,
q ( x ξ ) q ( x ) ξ q ( x ) + ξ 2 2 q ( x ) , | ξ | .
q 1 ( x ) m ¯ 0 q ( x ) + m ¯ 2 2 q ( x ) ,
m ¯ k = x k h ¯ ( x ) d x
m ¯ 2 | x | b x 2 h ¯ ( x ) d x b 2 | x | > b h ¯ ( x ) d x .
b b h ¯ ( x ) d x m ¯ 0 ( 1 m ¯ 2 b 2 m ¯ 0 ) .
p 1 ( x ) m 0 p ( x ) + m 2 2 p ( x ) ,
m k = x k h ( x ) d x
p 1 ( x , y ) = p ( x ξ , y η ) h ( ξ , η ) d ξ d η .
p ( x ξ , y η ) p ξ p x η p y + ξ 2 2 2 p x 2 + ξ η 2 p x y + η 2 2 2 p y 2 ,
p 1 ( x , y ) m 00 p + m 20 2 2 p x 2 + m 11 2 p x y + m 02 2 2 p y 2 ,
m k r = x k y r h ( x , y ) dxdy
h ( x , y ) = h ( r ) , r 2 = x 2 + y 2 .
M k = 0 r k h ( r ) d r ,
m 00 = 2 π M 1 , m 02 = m 20 = π M 3 , m 11 = 0 ,
p 1 ( x , y ) 2 π M 1 p + π M 3 2 ( 2 p x 2 + 2 p y 2 ) .
H ( u , υ ) = H ( w ) = 2 π 0 r h ( r ) J 0 ( w r ) d r , w 2 = u 2 + υ 2 .
H ( w ) = 0 for w > a
| h ( x ) | 2 d x = m ¯ 0 = constant .
x 2 | h ( x ) | 2 d x = m ¯ 2
a a | H ( u ) | 2 d u = 2 π m ¯ 0 ,
a a | H ( u ) | 2 d u = 2 π m ¯ 2 .
H ( a ) = H ( a ) = 0 .
a a F ( H , H ) d u ,
d d u F H = F H .
H ( u ) + λ H ( u ) = 0 .
λ = ( n π u / 2 a ) 2 .
H 0 ( u ) = A cos π u 2 a .
H 0 ( u ) = cos α u , α = π / 2 a
h 0 ( x ) = sin a ( x + α ) 2 π ( x + α ) + sin a ( x α ) 2 π ( x α )
H ¯ 0 ( u ) a 2 π 2 | sin α u | + a 2 π [ 1 | u | 2 a ] cos α u , | u | 2 a
h 0 2 ( x ) d x = a 2 π , x 2 h 0 2 ( x ) d x = π 8 a .
0 r 3 h 2 ( r ) d r = M ¯ 3
0 r h 2 ( r ) d r = M ¯ 1 = constant .
( u 2 + υ 2 ) h 2 ( r ) d u d υ = 1 4 π 2 [ ( H u ) 2 + ( H υ ) 2 ] d u d υ .
( H u ) 2 + ( H υ ) 2 = ( H w ) 2 ,
0 a w ( H w ) 2 d w = 4 π 2 M ¯ 3 .
0 a w H 2 ( w ) d w = 4 π 2 M ¯ 1 .
H ( a ) = 0 .
0 a [ w ( H ) 2 λ w H 2 ] d w .
w H ( w ) + H ( w ) + λ w H ( w ) = 0 .
H 0 ( w ) = J 0 ( α w ) , α = z 1 / a .
h 0 ( r ) = 1 2 π 0 a w J 0 ( α w ) J 0 ( r w ) d w = A J 0 ( a r ) α 2 r 2 ,
A = z 1 J 1 ( z 1 ) 2 π = 2.405 × 0.519 2 π
M 1 = 0 r h 0 2 ( r ) d r = 1 4 π 2 0 a w J 0 2 ( α w ) d w = a 2 8 π 2 J 1 2 ( z 1 ) ,
M 3 = 0 r 3 h 0 2 ( r ) d r = α 2 4 π 2 0 a w J 1 2 ( α w ) d w = α 2 4 π 2 0 a w J 0 2 ( α w ) d w = α 2 M ¯ 1 .
h ( x ) 0
x 2 h ( x ) d x = m 2
h ( x ) d x = H ( 0 ) = 1 .
h ( x ) 0 and H ( u ) = 0 for | u | > a ,
h ( x ) = | g ( x ) | 2 and G ( u ) = 0 for | u | > a / 2 .
x 2 | g ( x ) | 2 d x
| g ( x ) | 2 d x = 1 .
h 0 ( x ) = π a [ sin ( 1 2 a ) ( x + α ) π ( x + α ) + sin ( 1 2 a ) ( x α ) π ( x α ) ] 2 , α = π a
H 0 ( u ) = 1 π | sin α u | + ( 1 | u | a ) cos α u , | u | a
m 2 = x 2 h 0 ( x ) d x = | H 0 ( 0 ) | = π 2 a 2 , m 0 = H 0 ( 0 ) = 1 .
h ( r ) 0
0 r 3 h ( r ) d r = M 3
2 π 0 r h ( r ) d r = H ( 0 ) = 1 .
s ( x ) = h ( r ) d y , r 2 = x 2 + y 2
S ( u ) = s ( x ) e jux d x = h ( r ) e jux dxdy = H ( u , 0 ) .
S ( u ) = H ( u ) ,
μ n = x n s ( x ) d x
S ( u ) = μ 0 μ 2 2 μ 2 + μ 4 4 μ 4 + .
J 0 ( w r ) = 1 w 2 r 2 4 + w 4 r 4 64 + ,
H ( w ) = 2 π M 1 π M 3 2 w 2 + π M 5 32 w 4 + .
μ 0 = 2 π M 1 = 1 , μ 2 = 2 π M 3 .
s ( x ) 0 and S ( u ) = 0 for | u | > a ,
x 2 | h ( x ) | 2 d x π 2 4 a 2 | h ( x ) | 2 d x .
x 2 h ( x ) d x π 2 a 2 h ( x ) d x .
0 r 3 | h ( r ) | 2 d r z 1 2 a 2 0 r | h ( r ) | 2 d r ,
0 r 3 h ( r ) d r 2 π 2 a 2 0 r h ( r ) d r .
w ( x ) = B π 2 b [ sin b ( x + α ) π ( x + α ) + sin b ( x α ) π ( x α ) ] 2 , α = π 2 b .
H 0 ( u ) = cos π x / 2 a , | u | < a
h 0 ( x ) = sin a ( x + α ) 2 π ( x + α ) + sin a ( x α ) 2 π ( x α ) , α = π 2 a
W ( u ) = B [ 1 π | sin α u | + ( 1 | u | 2 b ) cos α u ] , | u | 2 b .
H 0 ( w ) = J 0 ( a w ) , | w | < a
h 0 ( r ) = A J 0 ( a r ) α 2 r 2 , α = z 1 a