Abstract

Fourier-transform division filters are discussed. Using integrated squared error as a fidelity criterion with the magnitude of the filter transfer function subject to a constraint, the optimum spatial filter is developed. By consideration of the problem of image transformation, various methods for improving image reconstruction by altering image phase are discussed. The input-image phase and the desired output-image phase may be chosen to improve the performance of the filtering system. The problem is to shape the spectra of the input and the desired output. An algorithm, previously used for computing kinoforms, effectively determines an image phase that significantly improves the image reconstructed from the spatial filters.

© 1974 Optical Society of America

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References

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  1. G. W. Stroke, IEEE Spectrum, p. 24 (Dec.1972).
    [Crossref]
  2. A. W. Lohmann, D. P. Paris, and H. W. Werlich, Appl. Opt. 6, 1139 (1967).
    [Crossref] [PubMed]
  3. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [Crossref] [PubMed]
  4. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [Crossref] [PubMed]
  5. R. A. Gabel, Thesis (Princeton Univ., July1969).
  6. R. A. Gabel and B. Liu, Appl. Opt. 9, 1180 (1970).
    [Crossref] [PubMed]
  7. A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill, New York, 1962), p. 62.
  8. N. C. Gallagher, Ph.D. thesis (Princeton Univ., 1974).
  9. N. C. Gallagher and B. Liu, Appl. Opt. 12, 2328 (1973).
    [Crossref] [PubMed]
  10. W. J. Dallas, Thesis (Univ. of California–San Diego, 1972).
  11. H. Akahori, Appl. Opt. 12, 2336 (1973).
    [Crossref] [PubMed]
  12. D. C. Chu, Thesis (Stanford Univ., 1974).

1973 (2)

1972 (1)

G. W. Stroke, IEEE Spectrum, p. 24 (Dec.1972).
[Crossref]

1970 (1)

1967 (2)

1966 (1)

Appl. Opt. (6)

IEEE Spectrum (1)

G. W. Stroke, IEEE Spectrum, p. 24 (Dec.1972).
[Crossref]

Other (5)

D. C. Chu, Thesis (Stanford Univ., 1974).

W. J. Dallas, Thesis (Univ. of California–San Diego, 1972).

A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill, New York, 1962), p. 62.

N. C. Gallagher, Ph.D. thesis (Princeton Univ., 1974).

R. A. Gabel, Thesis (Princeton Univ., July1969).

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Figures (17)

Fig. 1
Fig. 1

Standard spatial-filtering configuration.

Fig. 2
Fig. 2

Complex-plane representation of the quantities in Eq. (6).

Fig. 3
Fig. 3

Amplitude-quantization function.

Fig. 4
Fig. 4

Phase-quantization function.

Fig. 5
Fig. 5

Image of the letter L.

Fig. 6
Fig. 6

Magnitude of the Fourier transform of the L.

Fig. 7
Fig. 7

Image of the letter T.

Fig. 8
Fig. 8

Magnitude of the Fourier transform of the T.

Fig. 9
Fig. 9

{Φ} as a function of D.

Fig. 10
Fig. 10

Image reconstruction with zero phase on both the L and the T.

Fig. 11
Fig. 11

Image reconstruction with phase computed by method 3.

Fig. 12
Fig. 12

Image of the letter P.

Fig. 13
Fig. 13

Magnitude of the Fourier transform of the P.

Fig. 14
Fig. 14

Image of the phrase E.E. Dept.

Fig. 15
Fig. 15

Magnitude of the Fourier transform of E.E. Dept.

Fig. 16
Fig. 16

Reconstructed image with zero phase on both the P and E.E. Dept.

Fig. 17
Fig. 17

Reconstructed image with phase computed by use of method 3.

Equations (33)

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B ( u , v ) = J ( u , v ) H ( u , v ) .
H ( u , v ) = B ( u , v ) / J ( u , v ) .
H ( u , v ) 1.
lim u or v B ( u , v ) = 0             and             lim u or v J ( u , v ) = 0
Φ = 1 X 2 - X / 2 X / 2 b ( x , y ) exp [ i β ( x , y ) ] - h ( x , y ) * a ( x , y ) exp [ i α ( x , y ) ] 2 d x d y ,
Φ = 1 X 2 - B ( u , v ) exp [ i ϕ ( u , v ) ] - H ( u , v ) A ( u , v ) exp [ i ψ ( u , v ) ] 2 d u d v .
Δ 2 = B exp ( i ϕ ) - H A exp ( i ψ ) 2 ,
H ( u , v ) = { B ( u , v ) A ( u , v ) exp [ i ϕ ( u , v ) - i ψ ( u , v ) ] , B ( u , v ) A ( u , v ) 1 exp [ i ϕ ( u , v ) - i ψ ( u , v ) ] , B ( u , v ) A ( u , v ) > 1.
H ( u , v ) = { B ( u , v ) D A ( u , v ) exp [ i ϕ ( u , v ) - i ψ ( u , v ) ] , B ( u , v ) D A ( u , v ) 1 exp [ i ϕ ( u , v ) - i ψ ( u , v ) ] , B ( u , v ) D A ( u , v ) > 1.
P D = { ( u , v ) : B ( u , v ) D A ( u , v ) 1 } and R D = { ( u , v ) : B ( u , v ) D A ( u , v ) > 1 } .
c ( x , y ) exp [ i γ ( x , y ) ] = - H ( u , v ) D A ( u , v ) × exp [ i ψ ( u , v ) ] exp [ i 2 π ( x u + y v ) ] d u d v .
c ( x , y ) exp [ i γ ( x , y ) ] = 1 X 2 m , n = - H m n D A m n × exp ( i ψ m n ) exp [ i 2 π X ( m x + n y ) ] ,
T m n = { B m n / D A m n , ( m / X , n / X ) P D 1 , ( m / X , n / X ) R D
c ( x , y ) exp [ i γ ( x , y ) ] = 1 X 2 m , n I ( T m n + N m n ) exp [ i ( ρ m n + θ m n ) ] × D A m n exp ( i ψ m n ) exp [ i 2 π X ( m x + n y ) ] .
b ( x , y ) exp [ i β ( x , y ) ] = 1 X 2 m , n = - B m n exp ( i ϕ m n ) exp [ i 2 π X ( m x + n y ) ] .
{ Φ } = 1 X 2 - X / 2 X / 2 b 2 ( x , y ) d x d y + 1 X 4 { m , n I P D B m n / ( D A m n ) Q / 2 [ B m n 2 ( 1 - 2 sin Θ / 2 Θ / 2 ) + D 2 σ 2 A m n 2 ] + m , n I R D [ D 2 A m n 2 - 2 D A m n B m n sin Θ / 2 Θ / 2 ] } .
{ Φ } = 1 X 2 - X / 2 X / 2 b 2 ( x , y ) d x d y ,
lim D I P D = I , and lim D I R D
1 X 2 b 2 ( x , y ) d x d y = lim D { Φ } ,
e ( x , y ) = b ( x , y ) exp [ i β ( x , y ) - c ( x , y ) exp [ i γ ( x , y ) ] ,
e ( x , y ) = 1 X 2 m , n = - E m n exp [ i 2 π X ( m x + n y ) ] ,
E m n = { B m n exp ( i φ m n ) - D A m n ( B m n D A m n + N m n ) × exp [ i ( φ m n + θ m n ) ] ,             m , n I P D and B m n / ( D A m n ) Q / 2 B m n exp ( i φ m n ) - D A m n exp [ i ( φ m n + θ m n ) ] ,             m , n I R D B m n exp ( i φ m n ) ,             m , n I             or             B m n / ( D A m n ) < Q / 2.
Φ = 1 X 2 - X / 2 X / 2 e ( x , y ) 2 d x d y .
Φ = 1 X 4 m , n = E m n 2 ;
{ Φ } = 1 X 4 m , n = - { E m n 2 }
{ E m n 2 } = { 2 B m n 2 ( 1 - sin Θ / 2 Θ / 2 ) + D 2 A m n 2 σ 2 ,             m , n I P D and B m n / ( D A m n ) Q / 2 B m n 2 + D 2 A m n 2 - 2 D A m n B m n sin Θ / 2 Θ / 2 ,             m , n I R D B m n 2 ,             m , n I             or             B m n / ( D A m n ) < Q / 2
{ Φ } = 1 X 4 m , n = - B m n 2 + 1 X 4 { m , n I P D [ B m n 2 ( 1 - 2 sin Θ / 2 Θ / 2 ) + D 2 A m n 2 σ 2 ] + m , n I R D [ D 2 A m n 2 - 2 D A m n B m n sin Θ / 2 Θ / 2 ] } .
1 X 2 - X / 2 X / 2 b 2 ( x , y ) d x d y = 1 X 4 m , n = - B m n 2 ,
D A m n T m n = { B m n + N m n , m , n I P D             and             B m n Q / 2 D A m n + N m n , m , n I R D             and             D A m n Q / 2 0 , m , n I     or             D A m n T m n < Q / 2
E m n = B m n exp ( i ϕ m n ) - D A m n T m n exp [ i ( ϕ m n + θ m n ) ] ,
{ E m n 2 } = { 2 B m n 2 ( 1 - sin Θ / 2 Θ / 2 ) + σ 2 ,             m , n I P D             and             B m n Q / 2 B m n 2 + D 2 A m n 2 + σ 2 - 2 D A m n B m n sin Θ / 2 Θ / 2 ,             m , n I R D B m n 2 ,             m , n I             or             D A m n T m n < Q / 2
Φ ˆ = 1 X 4 m , n = - E m n 2 ;
{ Φ } = 1 X 2 - X / 2 X / 2 b 2 ( x , y ) d x d y + 1 X 4 { m , n I P D D A m n T m n Q / 2 [ B m n 2 ( 1 - 2 sin Θ / 2 Θ / 2 ) + σ 2 ] + m , n I R D [ D 2 A m n 2 + σ 2 - 2 D A m n B m n sin Θ / 2 Θ / 2 ] } .