Abstract

The theory of an unconstrained single deblurring filter is developed considering the effect of the higher-order terms. It is based on the interference pattern of the Fourier transform of the blurred point spread function (PSF) h and the doubly blurred PSF hD = hh, ⊛ being the convolution symbol. Deblurring results of transverse sinusoidal motion blurred images obtained by using the present theory and the filter are presented.

© 1983 Optical Society of America

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References

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  1. G. W. Stroke, R. G. Zech, Phys. Lett. A 25, 89 (1967).
    [CrossRef]
  2. A. W. Lohmann, H. W. Warlech, Phys. Lett. A 25, 570 (1967).
    [CrossRef]
  3. G. W. Stroke, Opt. Acta 16, 401 (1969).
    [CrossRef]
  4. R. M. Vasu, G. L. Rogers, Appl. Opt. 19, 469 (1980).
    [CrossRef] [PubMed]
  5. J. Tsujiuchi, Prog. Opt. 2, 133 (1963).
  6. G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
    [CrossRef]
  7. C. Zetzsche, Appl. Opt. 21, 1077 (1982).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. L. Levi, Appl. Opt. 10, 38 (1971).
    [CrossRef] [PubMed]
  10. A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
    [CrossRef]

1982 (1)

1980 (1)

1977 (1)

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

1972 (1)

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

1971 (1)

1969 (1)

G. W. Stroke, Opt. Acta 16, 401 (1969).
[CrossRef]

1967 (2)

G. W. Stroke, R. G. Zech, Phys. Lett. A 25, 89 (1967).
[CrossRef]

A. W. Lohmann, H. W. Warlech, Phys. Lett. A 25, 570 (1967).
[CrossRef]

1963 (1)

J. Tsujiuchi, Prog. Opt. 2, 133 (1963).

Goodman, J. W.

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

Halioua, M.

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

Levi, L.

Lohmann, A. W.

A. W. Lohmann, H. W. Warlech, Phys. Lett. A 25, 570 (1967).
[CrossRef]

Rogers, G. L.

Sawchuk, A. A.

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

Stroke, G. W.

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

G. W. Stroke, Opt. Acta 16, 401 (1969).
[CrossRef]

G. W. Stroke, R. G. Zech, Phys. Lett. A 25, 89 (1967).
[CrossRef]

Thon, F.

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

Tsujiuchi, J.

J. Tsujiuchi, Prog. Opt. 2, 133 (1963).

Vasu, R. M.

Warlech, H. W.

A. W. Lohmann, H. W. Warlech, Phys. Lett. A 25, 570 (1967).
[CrossRef]

Willasch, D. H.

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

Zech, R. G.

G. W. Stroke, R. G. Zech, Phys. Lett. A 25, 89 (1967).
[CrossRef]

Zetzsche, C.

Appl. Opt. (3)

Opt. Acta (1)

G. W. Stroke, Opt. Acta 16, 401 (1969).
[CrossRef]

Phys. Lett. A (2)

G. W. Stroke, R. G. Zech, Phys. Lett. A 25, 89 (1967).
[CrossRef]

A. W. Lohmann, H. W. Warlech, Phys. Lett. A 25, 570 (1967).
[CrossRef]

Proc. IEEE (2)

G. W. Stroke, M. Halioua, F. Thon, D. H. Willasch, Proc. IEEE 65, 39 (1977).
[CrossRef]

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

Prog. Opt. (1)

J. Tsujiuchi, Prog. Opt. 2, 133 (1963).

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

Holographic system used in preparation of an unconstrained single deblurring filter showing blurred PSF h, doubly blurred PSF hD, Fourier transform lens (focal length = f′) L, photographic film P, and amplitude attenuator Aa.

Fig. 2
Fig. 2

Blurred PSF h and OTF H ¯ for the transverse sinusoidal motion blur (amplitude = A).

Fig. 3
Fig. 3

Fourier spectra of h and hD (= hh).

Fig. 4
Fig. 4

Unconstrained single deblurring filter.

Fig. 5
Fig. 5

Blurred images and deblurred images: (a) rectangles; (b) a Korean word x 0 as an object.

Equations (25)

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i ( x , y ) = o ( x , y ) h ( x x , y y ) d x d y , = o ( x , y ) h ( x , y ) ,
I ( ω x , ω y ) = O ( ω x , ω y ) H ( ω x , ω y ) = c 1 O ( ω x , ω y ) H ¯ ( ω x , ω y ) ,
h D ( x , y ) = h ( x , y ) h ( x , y ) .
J ( ω x , ω y ) = [ c 2 H D ( ω x , ω y ) exp ( j ω x a ) + H ( ω x , ω y ) ] × [ c 2 H * D ( ω x , ω y ) exp ( j ω x a ) + H * ( ω x , ω y ) ] ,
J ( ω x , ω y ) = | H | 2 [ 1 + c 2 2 | H | 2 + c 2 H exp ( j ω x a ) + c 2 H * exp ( j ω x a ) ] .
J ( ω x , ω y ) = c 1 2 | H ¯ | 2 [ 1 + c 3 2 | H ¯ | 2 + c 3 H ¯ exp ( j ω x a ) + c 3 H ¯ * exp ( j ω x a ) ] ,
t ( ω x , ω y ) = J 1 ( ω x , ω y ) = c 1 2 | H ¯ | 2 [ 1 + c 3 2 | H ¯ | 2 + c 3 H ¯ exp ( j ω x a ) + c 3 H ¯ * exp ( j ω x a ) ] 1 .
t ( ω x , ω y ) = c 1 2 | H ¯ | 2 n = 0 ( 1 ) n [ c 3 2 | H ¯ | 2 + c 3 H ¯ exp ( j ω x a ) + c 3 H ¯ * exp ( j ω x a ) ] n .
t ( ω x , ω y ) = n = 0 c 3 2 n ( H H ¯ * ) n 1 { 1 + n = 1 ( c 3 ) n [ H ¯ n exp ( j n ω x a ) + H ¯ * n exp ( j n ω x a ) ] } ,
G n ( ω x , ω y ) = ( c 3 ) n [ H ¯ n exp ( j n ω x a ) + H ¯ * n exp ( j n ω x a ) ] , D n ( ω x , ω y ) = c 3 2 n ( H H ¯ * ) n 1 , ( n 0 ) .
t ( ω x , ω y ) = ( 1 + n = 1 G n ) n = 0 D n .
c 3 < 1 ;
= n = 0 D n = [ 1 / ( H H ¯ * ) ] n = 1 ( c 3 2 H H ¯ * ) n 1 , = [ 1 / ( H H ¯ * ) ] [ 1 / ( 1 c 3 2 H H ¯ * ) ] , 1 / ( H H ¯ * ) = D 0 , n = 1 G n = n = 1 [ c 3 H ¯ exp ( j ω x a ) ] n + n = 1 [ c 3 H ¯ * exp ( j ω x a ) ] n , = [ c 3 H ¯ exp ( j ω x a ) ] / [ 1 + c 3 H ¯ exp ( j ω x a ) ] + [ c 3 H ¯ * exp ( j ω x a ) ] / [ 1 + c 3 H ¯ * exp ( j ω x a ) ] , c 3 H ¯ exp ( j ω x a ) c 3 H ¯ * exp ( j ω x a ) = G 1 ,
t ( ω x , ω y ) ( 1 + G 1 ) D 0 , = | H ¯ | 2 c 3 ( 1 / H ¯ * ) exp ( j ω x a ) c 3 ( 1 / H ¯ ) exp ( j ω x a ) .
K = H H * / [ ( c 2 · H D ) ( c 2 · H * D ) ] = c 3 2 [ 1 / ( H H ¯ * ) ] c 3 2 > 1.
t ( ω x , ω y ) = ( 1 + G 1 ) D 0 + n = 1 D n + n = 1 D n G 1 + n = 0 m = 2 D n G m ,
t h ( ω x , ω y ) = n = 1 D n + n = 1 D n G 1 + n = 0 m = 2 D n G m .
G n ( ω x , ω y ) = ( c 3 ) n H ¯ n [ exp ( j n ω x a ) + exp ( j n ω x a ) ] , D n ( ω x , ω y ) = c 3 2 n H ¯ 2 ( n 1 ) , ( n 0 ) ,
E h ( ω x , ω y ) = t h ( ω x , ω y ) ( H ¯ O ) , = H ¯ O n = 1 D n + ( H ¯ O n = 1 D n ) G 1 + ( H ¯ O n = 0 D n ) m = 2 G m .
( H ¯ O n = 1 D n ) G 1 exp [ j ( ω x x + ω y y ) ] d ω x d ω y = n = 1 c 3 2 n + 1 O H ¯ 2 n [ exp ( j ω x a ) + exp ( j ω x a ) ] exp [ j ( ω x x + ω y y ) ] d ω x d ω y , = n = 1 c 3 2 n + 1 o ( x , y ) [ h ( x , y ) h ( x , y ) ] 2 n - fold convolution [ δ ( x a ) + δ ( x + a ) ] .
o h ( x , y ) = n = 1 c 3 2 n + 1 o ( x , y ) [ h ( x , y ) h ( x , y ) ] 2 n - fold convolution .
S = c 3 2 n + 1 c 3 = c 3 2 n ,
h ( x , y ) = ( 2 T π 1 A 2 x 2 ) δ ( y ) ( A x A ) = 0 ( otherwise ) , H ¯ ( ω x ) = J 0 ( A ω x ) ,
f c = a / ( λ f ) = 70 lines / mm .
t ( ω x , ω y ) = | H ¯ | 2 n = 0 ( 1 ) n [ c 3 2 | H ¯ | 2 + c 3 H ¯ exp ( j ω x a ) + c 3 H ¯ * exp ( j ω x a ) ] n , = [ 1 / ( H H ¯ * ) ] [ ( 1 + c 3 2 H H ¯ * + c 3 4 H ¯ 2 H ¯ * 2 + c 3 6 H ¯ 3 H ¯ * 3 + ) ( c 3 H ¯ + c 3 3 H ¯ 2 H ¯ * + c 3 5 H ¯ 3 H ¯ * 2 + c 7 3 H ¯ 4 H ¯ * 3 + ) × exp ( j ω x a ) ( c 3 H ¯ * + c 3 3 H H ¯ * 2 + c 3 5 H ¯ 2 H ¯ * 3 + c 3 7 H ¯ 3 H ¯ * 4 + ) × exp ( j ω x a ) + ( c 3 2 H ¯ 2 + c 3 4 H ¯ 3 H ¯ * + c 3 6 H ¯ 4 H ¯ * 2 + ) exp ( j 2 ω x a ) + ( c 3 2 H ¯ * 2 + c 3 4 H H ¯ * 3 + c 3 6 H ¯ 2 H ¯ * 4 + ) exp ( j 2 ω x a ) ( c 3 3 H ¯ 3 + c 3 5 H ¯ 4 H ¯ * + c 3 7 H ¯ 5 H ¯ * 2 + ) exp ( j 3 ω x a ) ( c 3 3 H ¯ * 3 + c 3 5 H H ¯ * 4 + c 3 7 H ¯ 2 H ¯ * 5 + ) × exp ( j 3 ω x a ) + ] , = [ 1 / ( H H ¯ * ) ] ( 1 + c 3 2 H H ¯ * + c 3 4 H ¯ 2 H ¯ * 2 + c 3 6 H ¯ 3 H ¯ * 3 + ) × { 1 c 3 [ H ¯ exp ( j ω x a ) + H ¯ * exp ( j ω x a ) ] + c 3 2 [ H ¯ 2 exp ( j 2 ω x a ) + H ¯ * 2 exp ( j 2 ω x a ) ] c 3 3 [ H ¯ 3 exp ( j 3 ω x a ) + H ¯ * 3 × exp ( j 3 ω x a ) ] + } , = n = 0 c 3 2 n ( H H ¯ * ) n 1 { 1 + n = 1 ( c 3 ) n [ H ¯ n × exp ( j n ω x a ) + H ¯ * n exp ( j n ω x a ) ] } .

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