Abstract

A piecewise isoplanatic approximation (PIA) applicable to the classification of linear systems and the representation of space-variant linear systems is described. A new measure of the degree of invariance is developed and used as a tool to classify linear systems and determine optimal PIA patch sizes and critical points. Also, the relationship of the variation bandwidth to the new measure of invariance is utilized to support the validity of the new measure.

© 1987 Optical Society of America

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References

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  1. R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,”J. Opt. Soc. Am. 66, 918–921 (1976).
    [CrossRef]
  2. R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
    [CrossRef]
  3. R. J. Marks, “Holographic recording of optical space-variant systems,” M.S. thesis (Rose-Hulman Institute of Technology, Terre Haute, Ind., August1973).
  4. R. J. Marks, T. F. Krile, “Holographic representation of space-variant systems: system theory,” Appl. Opt. 15, 2241–2245 (1976).
    [CrossRef]
  5. A. W. Lohmann, D. P. Paris, “Space-variant image formation,”J. Opt. Soc. Am. 55, 1007–1013 (1965).
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).
  7. J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 238.
  8. J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, Fundamentals, S. Lee, ed. (Springer-Verlag, New York, 1980).

1978 (1)

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
[CrossRef]

1976 (2)

1965 (1)

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).

J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, Fundamentals, S. Lee, ed. (Springer-Verlag, New York, 1980).

Hagler, M. O.

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,”J. Opt. Soc. Am. 66, 918–921 (1976).
[CrossRef]

Jacobs, I. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 238.

Krile, T. F.

Lohmann, A. W.

Marks, R. J.

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
[CrossRef]

R. J. Marks, T. F. Krile, “Holographic representation of space-variant systems: system theory,” Appl. Opt. 15, 2241–2245 (1976).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,”J. Opt. Soc. Am. 66, 918–921 (1976).
[CrossRef]

R. J. Marks, “Holographic recording of optical space-variant systems,” M.S. thesis (Rose-Hulman Institute of Technology, Terre Haute, Ind., August1973).

Paris, D. P.

Walkup, J. F.

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,”J. Opt. Soc. Am. 66, 918–921 (1976).
[CrossRef]

Wozencraft, J. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 238.

Appl. Opt. (1)

IEEE Trans. Circuits Syst. (1)

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,”IEEE Trans. Circuits Syst. CAS-25, 228–233 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), p. 238.

J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, Fundamentals, S. Lee, ed. (Springer-Verlag, New York, 1980).

R. J. Marks, “Holographic recording of optical space-variant systems,” M.S. thesis (Rose-Hulman Institute of Technology, Terre Haute, Ind., August1973).

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Figures (8)

Fig. 1
Fig. 1

True and PIA outputs for magnifier with exponential input.

Fig. 2
Fig. 2

Mean-square difference between true and PIA outputs versus the patch density for the FT.

Fig. 3
Fig. 3

Mean-square difference between true and PIA outputs versus the patch density for the ET.

Fig. 4
Fig. 4

The curve of the degree of invariance for the FT.

Fig. 5
Fig. 5

The curve of the degree of invariance for the ET.

Fig. 6
Fig. 6

Line-spread functions of a certain patch for the magnifier.

Fig. 7
Fig. 7

Line-spread functions of a patch for the FT.

Fig. 8
Fig. 8

Line-spread functions of a patch for the ET.

Tables (1)

Tables Icon

Table 1 System Classification by the Degree of Invariance

Equations (55)

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g ( x ) = - f ( ξ ) h ( x - ξ ; ξ ) d ξ .
g ˜ ( x ) = n = 1 m l n t n f ( ξ ) h ( x - ξ ; x n ) d ξ ,
l n x n t n ,             l n + 1 = t n ,
g ( x ) = n = 1 m l n t n f ( ξ ) h ( x - ξ ; ξ ) d ξ .
lim n g ˜ ( x ) g ( x ) .
g ( x ) = ( 1 / M ) f ( x / M ) ,
h ( x - ξ ; ξ ) = δ ( x - M ξ ) ,
g ˜ ( x ) = n f [ x - ( M - 1 ) x n ] u [ x - l n - ( M - 1 ) x n ] × u [ - x + t n + ( M - 1 ) x n ] ,
E = - g ( x ) 2 d x = ( 1 / M ) - f ( x ) 2 d x ,
E ˜ = - g ( x ) 2 d x = - f ( x ) 2 d x = M E .
g ( x ) = - f ( ξ ) exp ( - j 2 π λ f l x ξ ) d ξ             ( FT )
g ( x ) = - f ( ξ ) exp ( - x ξ ) d ξ             ( ET ) ,
h ( x - ξ ; ξ ) = exp ( - j 2 π f x ξ )             ( FT )
h ( x - ξ ; ξ ) = exp ( - x ξ ) u ( ξ )             ( ET ) ,
f ( ξ ) = u ( ξ + a ) u ( a - ξ )             ( FT )
f ( ξ ) = u ( ξ ) u ( 2 a - ξ )             ( ET ) ,
g ˜ ( x ) = w [ 1 + 2 n = 1 k sinc ( n w 2 / λ f l ) cos ( 2 n π w f x ) ]             ( FT ) ,
g ˜ ( x ) = n = 1 2 k + 1 exp ( - x x n + x n 2 ) [ exp ( t n x n ) - exp ( l n x n ) ] / x n             ( ET ) ,
D ( x 0 , ξ 1 , ξ 2 ) = | h ( x + x 0 2 - ξ 1 ; ξ 1 ) - h ( x - x 0 2 - ξ 2 ; ξ 2 ) | 2 d x
C ( x 0 , ξ 1 , ξ 2 ) = h ( x + x 0 2 - ξ 1 ; ξ 1 ) h * ( x - x 0 2 - ξ 2 ; ξ 2 ) d x ,
σ ( ξ 1 , ξ 2 ) [ D ( ξ 1 - ξ 2 , ξ 1 , ξ 2 ) ] / [ C ( 0 , ξ 1 , ξ 1 ) + C ( 0 , ξ 2 , ξ 2 ) ] = - h [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 1 ] - h [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 2 ] 2 d x - h ( x - ξ 1 ; ξ 1 ) 2 d x + - h ( x - ξ 2 ; ξ 2 ) 2 d x .
σ ( ξ 1 , ξ 2 ) = 1 - Z = 1 - - [ P ( x ; ξ 1 , ξ 2 ) + P * ( x ; ξ 1 , ξ 2 ) ] d x - h ( x - ξ 1 ; ξ 1 ) 2 d x + - h ( x - ξ 2 ; ξ 2 ) 2 d x ,
P ( x ; ξ 1 , ξ 2 ) = h ( x - ξ 1 + ξ 2 2 ; ξ 1 ) h * ( x - ξ 1 + ξ 2 2 ; ξ 2 )
σ ( ξ 1 , ξ 2 ) = { 1 if ξ 1 ξ 2 0 if ξ 1 = ξ 2 .
σ ( ξ 1 , ξ 2 ) = ( 1 - Q / 4 L )             ( FT )
σ ( ξ 1 , ξ 2 ) = [ 1 - Q / ( Q 1 + Q 2 ) ]             ( ET ) ,
Q = ( 1 / λ f l ) sin [ ( 2 π / λ f l ) ( ξ 1 L - ξ 2 L + ξ 1 2 / 2 - ξ 2 2 / 2 ) ] - sin [ ( 2 π / λ f l ) ( - ξ 1 L + ξ 2 L + ξ 1 2 / 2 - ξ 2 2 / 2 ) ] / ( ξ 1 - ξ 2 ) π ,
Q 1 = exp [ - ( ξ 1 - ξ 2 ) ξ 1 ] sinh ( 2 L ξ 1 ) / ξ 1 ,
Q 2 = exp [ ( ξ 1 - ξ 2 ) ξ 2 ] sinh ( 2 L ξ 2 ) / ξ 2 ,
Q = 4 exp [ - ( ξ 1 - ξ 2 ) 2 / 2 ] sinh ( L ξ 1 + L ξ 2 ) / ( ξ 1 + ξ 2 ) .
σ LP ( ξ 1 , ξ 2 ) = { - h [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 1 ] × h * [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 2 ] d x } / [ - h ( x - ξ 1 ; ξ 1 ) 2 d x - h ( x - ξ 2 ; ξ 2 ) 2 d x ] 1 / 2
σ NM ( ξ 1 , ξ 2 ) = - h [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 1 ] - h [ x - ( ξ 1 + ξ 2 ) / 2 ; ξ 2 ] 2 d x / [ - h ( x - ξ 1 ; ξ 1 ) 2 d x + - h ( x - ξ 2 ; ξ 2 ) 2 d x ]
0 σ LP 1
0 σ NM σ max ,
σ LP = sinc [ 2 w ( M - 1 ) ( ξ 1 - ξ 2 ) ]
σ NM = 1 ,
σ ( ξ 1 , ξ 2 ) < s .
ρ x ( x n , ξ l n , ξ t n ) = | h ( x ; x n ) - ξ l n ξ t n h ( x ; ξ ) d ξ / ( ξ t n - ξ l n ) | 2 ,
ρ x ( x , ξ l n , ξ t n ) < c             for all x ,
H ξ ( x ; v ) = F ξ [ h ( x ; ξ ) ] ,
g ( x ) = - x f ( ξ ) d ξ ,
h ( x ; ξ ) = u ( x ) ;
H ξ ( x ; v ) = u ( x ) δ ( v ) .
h ( x ; ξ ) = δ [ x - ( M - 1 ) ξ ] ;
H ξ ( x ; v ) = exp [ - j 2 π v x / ( M - 1 ) ] / ( M - 1 ) .
h ( x ; ξ ) = exp [ ( x + ξ ) ξ ] u ( ξ ) ;
H ξ ( x ; v ) = [ 1 / ( x + j 2 π v ) ] * π 2 exp ( - π 2 v 2 ) ,
g ( ω ) = - f ( t ) exp ( - j ω t ) d t ,
h ( ω - t ; t ) = exp ( - j ω t ) ,
- h ( ω ) 2 d ω = - H ( u ) 2 d u .
- h ( ω - t ; t ) 2 d ω = - h ( ω ; t ) 2 d ω
- h ( ω , - t 1 + t 2 2 ; t 1 ) h * ( ω - t 1 + t 2 2 ; t 2 ) d ω = - H ( u ; t 1 ) H * ( u ; t 2 ) d u ,
σ ( t 1 , t 2 ) = 1 - - [ H ( u ; t 1 ) H * ( u ; t 2 ) + H * ( u ; t 1 ) H ( u ; t 2 ) ] d u - H ( u ; t 1 ) 2 d u + - H ( u ; t 2 ) 2 d u .
H ( u - t ; t ) = δ ( u ) exp ( j t 2 ) .
σ ( t 1 , t 2 ) = 1.

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