Abstract

We present a new projection-based algorithm for solving the classical blind-deconvolution problem. In our approach all known a priori information about both the unknown source and the blurring functions is expressed through constraint sets. In computer simulations the algorithm performed well even when the prior information was not accurate. To see how well our algorithm compares against others, we compared it with another recently published deconvolution method [ J. Opt. Soc. Am. A 9, 932 ( 1992)]. The advantages of each method are discussed.

© 1994 Optical Society of America

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References

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  1. G. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  2. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
    [CrossRef]
  3. B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
    [CrossRef]
  4. A. K. Katsaggelos, R. L. Lagendijk, J. Biemond, “Constrained iterative identification and restoration of images,” in Proceedings of the European Signal Processing Conference, EURASIP 1988 (Elsevier, Amsterdam, 1988), pp. 1585–1588.
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  6. A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  7. H. Peng, H. Stark, “Signal recovery with similarity constraints,” J. Opt. Soc. Am. A 6,(1989).
    [CrossRef]
  8. S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).
  9. R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
    [CrossRef]
  10. B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
    [CrossRef]
  11. A. K. Katsaggelos, S. N. Efstratiadis, “A class of iterative signal restoration algorithms,”IEEE Trans. Acoustics Speech Signal Process. 38, 778–786 (1990).
    [CrossRef]

1992 (1)

1990 (2)

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

A. K. Katsaggelos, S. N. Efstratiadis, “A class of iterative signal restoration algorithms,”IEEE Trans. Acoustics Speech Signal Process. 38, 778–786 (1990).
[CrossRef]

1989 (1)

H. Peng, H. Stark, “Signal recovery with similarity constraints,” J. Opt. Soc. Am. A 6,(1989).
[CrossRef]

1988 (2)

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
[CrossRef]

G. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef] [PubMed]

1984 (1)

1982 (1)

1973 (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[CrossRef]

Ayers, G.

Biemond, J.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
[CrossRef]

A. K. Katsaggelos, R. L. Lagendijk, J. Biemond, “Constrained iterative identification and restoration of images,” in Proceedings of the European Signal Processing Conference, EURASIP 1988 (Elsevier, Amsterdam, 1988), pp. 1585–1588.

Boekee, D. E.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
[CrossRef]

Dainty, J. C.

Efstratiadis, S. N.

A. K. Katsaggelos, S. N. Efstratiadis, “A class of iterative signal restoration algorithms,”IEEE Trans. Acoustics Speech Signal Process. 38, 778–786 (1990).
[CrossRef]

Fienup, J. R.

Hunt, B. R.

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[CrossRef]

Jacoby, S. L. S.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Katsaggelos, A. K.

A. K. Katsaggelos, S. N. Efstratiadis, “A class of iterative signal restoration algorithms,”IEEE Trans. Acoustics Speech Signal Process. 38, 778–786 (1990).
[CrossRef]

A. K. Katsaggelos, R. L. Lagendijk, J. Biemond, “Constrained iterative identification and restoration of images,” in Proceedings of the European Signal Processing Conference, EURASIP 1988 (Elsevier, Amsterdam, 1988), pp. 1585–1588.

Kowalik, J. S.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
[CrossRef]

A. K. Katsaggelos, R. L. Lagendijk, J. Biemond, “Constrained iterative identification and restoration of images,” in Proceedings of the European Signal Processing Conference, EURASIP 1988 (Elsevier, Amsterdam, 1988), pp. 1585–1588.

Lane, R. G.

Levi, A.

McCallum, B. C.

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

Peng, H.

H. Peng, H. Stark, “Signal recovery with similarity constraints,” J. Opt. Soc. Am. A 6,(1989).
[CrossRef]

Pizzo, J. T.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Stark, H.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Regularized iterative restoration with ringing reduction,”IEEE Trans. Acoust. Speech Signal Process. 36, 1874–1888 (1988).
[CrossRef]

IEEE Trans. Acoustics Speech Signal Process. (1)

A. K. Katsaggelos, S. N. Efstratiadis, “A class of iterative signal restoration algorithms,”IEEE Trans. Acoustics Speech Signal Process. 38, 778–786 (1990).
[CrossRef]

IEEE Trans. Comput. (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

Opt. Lett. (1)

Other (2)

A. K. Katsaggelos, R. L. Lagendijk, J. Biemond, “Constrained iterative identification and restoration of images,” in Proceedings of the European Signal Processing Conference, EURASIP 1988 (Elsevier, Amsterdam, 1988), pp. 1585–1588.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

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Figures (10)

Fig. 1
Fig. 1

Block diagram of the proposed algorithm.

Fig. 2
Fig. 2

(a) Original image, (b) blurring function, (c) blurred image.

Fig. 3
Fig. 3

(a) Support window for f: Wf, (b) support window for h: Wh.

Fig. 4
Fig. 4

Images and blurring functions recovered by the proposed algorithm with different support windows: (a) and (b) case 1, (c) and (d) case 2, (e) and (f) case 3.

Fig. 5
Fig. 5

(a) Image, (b) blurring function recovered by the algorithm in of Ref. 2 with the same support window as in case 1.

Fig. 6
Fig. 6

Images and blurring functions recovered by the algorithm of Ref. 2 with the scaled initial conditions: (a) and (b) α = 0.1, (c) and (d) α = 0.025, (e) and (f) α = 0.01.

Fig. 7
Fig. 7

Images and blurring functions recovered by the algorithm of Ref. 2 for (a) and (b) case 2, (c) and (d) case 3.

Fig. 8
Fig. 8

Photon-limited data: (a) 105 counts, (b) 106 counts, (c) 5 × 106 counts.

Fig. 9
Fig. 9

Images and blurring functions recovered by the proposed algorithm from three levels of photon-limited data: (a) and (b) 105 counts, (c) and (d) 106 counts, (e) and (f) 5 × 106 counts.

Fig. 10
Fig. 10

Images and blurring functions recovered by the algorithm of Ref. 2 from three levels of photon-limited data: (a) and (b) 105 counts, (c) and (d) 106 counts, (e) and (f) 5 × 106 counts.

Tables (6)

Tables Icon

Table 1 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 4

Tables Icon

Table 2 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 5

Tables Icon

Table 3 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 6

Tables Icon

Table 4 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 7

Tables Icon

Table 5 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 9

Tables Icon

Table 6 Similarity Metric of the Recovered Source and Blurring Functions of Fig. 10

Equations (31)

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g ( x ) = f ( x - y ) h ( y ) d y + n ( x ) ,
g ( x ) = f ( x - y ) h ( y ) d y .
( u , v ) [ u ( x ) 2 d x + v ( x ) 2 d x ] 1 / 2 .
C g { ( u , v ) : u v = g }
g ( x ) = u ( x - y ) v ( y ) d y .
C f { ( u , v ) : u satisfying prescribed prior knowledge about the source f } , C h { ( u , v ) : v satisfying prescribed prior knowledge about the blurring funtion h } .
C f { ( u , v ) : u ( x ) = 0 , x Ω f } , C h { ( u , v ) : u ( x ) = 0 , x Ω h } ,
C f { ( u , v ) : u ( x ) ( f min , f max ) } ,
C h { ( u , v ) : v V ( ω ) = 0 , ω Ω H } ,
C h { ( u , v ) : v - h ˜ E } ,
( f , h ) ( k + 1 ) = P h P f P g ( f , h ) k ,             k = 0 , 1 , 2 , ,
F ˜ 4 - F ˜ 2 F ˜ F + H G * F ˜ = G 2 ,
H ˜ ( ω ) = G ( ω ) / F ˜ ( ω ) .
J ( u , v ) g - u v 2 = [ g ( x ) - ( u v ) ( x ) ] 2 d x 0 .
h k = arg { min h C h J ( f k - 1 , h ) } .
f k = arg { min f C f J ( f , h k ) } .
J ( f k , h k ) J ( f k - 1 , h k ) J ( f k - 1 , h k - 1 )
h l + 1 = P h [ h l - α k J ( f k - 1 , h l ) ] ,
J = 2 F k t F k h - 2 F k t g ,
J ( x , y ) = x 2 + α y 2 ,             α > 0 ,
C = { ( x , y ) : x 0 , y 0 , x + y = 1 } .
f l + 1 = P h [ f l - β k J ( f l , h k ) ] ,
E c = E i + E f ,
E f [ G ( ω ) - F ( ω ) H ( ω ) ] × [ G ( ω ) - F ( ω ) H ( ω ) ] * d ω ,
E i Ω ¯ f f ( x ) 2 d x + Ω ¯ h h ( x ) 2 d x ,
S f ( f ^ ) = max x 0 f ( x ) f ^ ( x - x 0 ) d x f · f ^ ,
h ( x 1 , x 2 ) = { 1 / 2 π σ 2 exp [ - ( x 1 2 + x 2 2 ) / 2 σ 2 ] for x 1 2 + x 2 2 4 σ 2 0 otherwise ,
C f = { ( u , v ) : u ( x ) is nonnegative for x W f ,             0 otherwise } ,
C h = { ( u , v ) : v ( x ) = 0 , x W h } ,
f 0 ( x ) = { g ( x ) if x W f 0 otherwise ,
f 0 ( x ) = { α g ( x ) if x W f 0 otherwise ,

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