Abstract

Inspired by the work of Lane and Bates on automatic multidimensional deconvolution [ J. Opt. Soc. Am. A 4, 180 ( 1987)], we have developed a systematic approach and an operational code for performing the deconvolution of multiply-convolved two-dimensional complex data sets in the absence of noise. We explain, in some detail, the major algorithmic steps, where noise or numerical errors can cause problems, our approach in dealing with numerical rounding errors, and where special noise-mitigating techniques can be used toward making blind de-convolution practical. Several examples of deconvolved imagery are presented, and future research directions are noted.

© 1993 Optical Society of America

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References

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  1. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [CrossRef]
  2. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
    [CrossRef]
  3. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [CrossRef]
  4. D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  5. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
    [CrossRef]
  6. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
  7. J. W. Goodman, International Trends in Optics (Academic, Boston, Mass., 1991).
  8. H. Stark, Image Recovery, Theory and Applications (Academic, Orlando, Fla., 1987).
  9. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

1990 (1)

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

1987 (3)

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Goodman, J. W.

J. W. Goodman, International Trends in Optics (Academic, Boston, Mass., 1991).

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Parker, C. R.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Quek, B. K.

Stark, H.

H. Stark, Image Recovery, Theory and Applications (Academic, Orlando, Fla., 1987).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

IEEE Trans. Acoust. Speech Signal Process. (2)

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Other (4)

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).

J. W. Goodman, International Trends in Optics (Academic, Boston, Mass., 1991).

H. Stark, Image Recovery, Theory and Applications (Academic, Orlando, Fla., 1987).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

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Figures (9)

Fig. 1
Fig. 1

Typical rendition of u space and v space for a complex quadratic polynomial.

Fig. 2
Fig. 2

Typical root distribution resulting from specific u.

Fig. 3
Fig. 3

Typical u path causing roots vi and vj to converge at some point v0.

Fig. 4
Fig. 4

Typical root grouping.

Fig. 5
Fig. 5

Geometrical rendition of variable u-path step.

Fig. 6
Fig. 6

Blind deconvolution example. (a) Magnitude of a small (16 × 16 pixel) multiply-convolved complex image. (b), (c) Magnitudes of deconvolved (8 × 8 pixel) complex-image components.

Fig. 7
Fig. 7

(a) Magnitude of a multiply-convolved complex image. (b)–(d) Magnitudes of 8 × 8 pixel deconvolved complex-image components.

Fig. 8
Fig. 8

(a) 64 × 64 pixel magnitude of a multiply-convolved complex image. (b), (c) Magnitudes of 32 × 32 pixel deconvolved complex-image components.

Fig. 9
Fig. 9

(a) Magnitude of a 32 × 32 complex image convolved with an 8 × 8 real-valued point-spread function. (b) Magnitude of a 32 × 32 real-valued image convolved with same 8 × 8 point-spread function. (c) Magnitude of 32 × 32 deconvolved complex-image component [real valued in (b)]. (d) Magnitude of a deconvolved 8 × 8 point-spread function.

Equations (58)

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g ( x ) = f ( x - τ ) h ( τ ) d τ ,
f 1 ( x , y ) f 2 ( x , y ) f k ( x , y ) F 1 ( ω x , ω y ) F 2 ( ω x , ω y ) F k ( ω x , ω y ) ,
F ( ω x , ω y ) = l = 1 k F l ( ω x , ω y ) .
F ( k , l ) = n = 0 N - 1 m = 0 M - 1 f ( m , n ) exp [ - j ( 2 π k m / M ) ] × exp [ - j ( 2 π l n / N ) ] .
F ( u , v ) = n = 0 N - 1 m = 0 M - 1 f ( m , n ) u m v n ,
F ( u , v ) = n = 0 N - 1 a n ( u ) v n ,
a n ( u ) = m = 0 M - 1 f ( m , n ) u m .
F ( u , v ) = F 1 ( u , v ) F 2 ( u , v ) .
v F ( u ˜ , v ˜ ) = 0.
u F ( u ˜ , v ˜ ) = F 1 ( u ˜ , v ˜ ) u F 2 ( u ˜ , v ˜ ) + F 2 ( u ˜ , v ˜ ) u F 1 ( u ˜ , v ˜ ) .
u F ( u ˜ , v ˜ ) = 0.
F 1 ( u , v ) = A ( u ) Q 1 ( u , v ) = A ( u ) i [ v - v i ( u ) ] ,             v i S 1 .
F 1 ( u , v ) = B ( v ) P 1 ( u , v ) = B ( v ) i [ u - u i ( v ) ] ,             u i T 1 .
Q 1 ( u 1 , v ) = B ( v ) P 1 ( u 1 , v ) .
F u [ u ( s ) , v i ( s ) ] u ˙ ( s ) + F v [ u ( s ) , v i ( s ) ] v ˙ i ( s ) = 0 ,
F u [ u ( s ) , v j ( s ) ] u ˙ ( s ) + F v [ u ( s ) , v j ( s ) ] v ˙ j ( s ) = 0 ,
F u = F u ,             F v = F v ,             u ˙ = d u d s ,             v ˙ = d v d s ,
v ˙ i = - F u ( u , v i ) u ˙ F v ( u , v i ) ,
v ˙ j = - F u ( u , v j ) u ˙ F v ( u , v j ) ,
D 2 = v i - v j 2 = ( v i - v j ) ( v i * - v j * ) ,
d d s D 2 = 2 Re [ ( v i * - v j * ) ( v ˙ i - v ˙ j ) ] .
( v ˙ i - v j ) = [ F u ( u , v j ) F v ( u , v j ) - F u ( u , v i ) F v ( u , v i ) ] u ˙ .
d d s D 2 = 2 Re { ( v i * - v j * ) [ F u ( u , v j ) F v ( u , v j ) - F u ( u , v i ) F v ( u , v i ) ] u ˙ } .
( v i * - v j * ) [ F u ( u , v j ) F v ( u , v j ) - F u ( u , v i ) F v ( u , v i ) ] = ρ 0 exp ( j φ 0 ) .
Δ u = u ˙ exp [ j ( π - φ 0 ) ] Δ s ,
φ 0 = phase { ( v i * - v j * ) [ F u ( u , v j ) F v ( u , v j ) - F u ( u , v i ) F v ( u , v i ) ] } ,
[ F u ( u , v j ) F v ( u , v j ) - F u ( u , v i ) F v ( u , v i ) ] 0.
v + Δ v v + d v d u Δ u + 1 2 d 2 v d u 2 ( Δ u ) 2 .
1 2 v u u ( Δ u max ) 2 v u Δ u max 0.1.
Δ u max 0.1 v u u 0.5 v u .
F u Δ u + F v Δ v + 1 2 [ F u u ( Δ u ) 2 + 2 F u v Δ u Δ v + F v v ( Δ v ) 2 ] = 0.
( F u + 1 2 F u u Δ u ) + ( F v + 2 F u v Δ u ) ( Δ v Δ u ) + F v v ( Δ v Δ u ) 2 = 0.
F ( u , v ) = 0 ,
F v ( u , v ) = 0.
J = [ F u u F u v F u v F v v ] .
S ( i , j ) = F u ( u 0 , v 0 ) λ max .
S = [ 1 0.12 0 0 0 0.34 0 0 0.12 1 0 0 0 0.25 0 0 0 0 1 0.22 0 0 0.18 0 0 0 0.22 1 0 0 0.16 0 0 0 0 0 1 0 0 0.42 0.34 0.25 0 0 0 1 0 0 0 0 0.18 0.16 0 0 1 0 0 0 0 0 0.42 0 0 1 ] .
S ¯ = [ 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 ] .
[ 1 1 0 0 0 1 0 0 ]             group 1 ,
[ 0 0 1 1 0 0 1 0 ]             group 2 ,
[ 0 0 0 0 1 0 0 1 ]             group 3.
A i ( u , v ) = m = 1 N R [ v n - v m ( u k ) ] ,
Δ F v 2 = 2 Re [ F v * ( F u v + F v v v u ) Δ u ] .
A i j = c j a i j ,
B i j = d j a i j ,
d i A i j = c j B i j ,
d i A i j - c j B i j = 0.
Γ x = 0 ,
x = [ d 1 d 2 d M c 1 c 2 c N ] T ,
Γ T Γ x = λ x ,             x T x = 1
C i j = 1 2 [ d i A i j + c j B i j ] .
A = [ 1 1 / 2 3 1 5 3 / 2 7 2 ] ,
B = [ 1 2 9 / 2 6 10 12 35 / 2 20 ] .
Γ = [ 1 0 0 0 - 1 0 0 3 0 0 - 9 / 2 0 0 0 5 0 - 10 0 0 0 0 7 - 35 / 2 0 1 / 2 0 0 0 0 - 2 0 1 0 0 0 - 6 0 0 3 / 2 0 0 - 12 0 0 0 2 0 - 20 ] .
x = [ 0.2620 0.3931 0.5241 0.6551 0.2620 0.0655 ] .
x = [ 1.0 1.5 2.0 2.5 1.0 0.25 ] = [ d 1 d 2 d 3 d 4 c 1 c 2 ] .
A e = A = [ 1 1 / 2 3 1 5 3 / 2 7 2 ] [ 1 / c 1 0 0 1 / c 2 ] = [ 1 2 3 4 5 6 7 8 ] ,
B e = [ 1 / d 1 0 0 0 0 1 / d 2 0 0 0 0 1 / d 3 0 0 0 0 1 / d 4 ] [ 1 2 9 / 2 6 10 12 35 / 2 20 ] = [ 1 2 3 4 5 6 7 8 ] .

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