Abstract

The convergence of Van Cittert’s iterative method of deconvolution is studied from an algebraic point of view without any special prior condition with respect to the system matrix. The convergence criteria are expressed in terms of system eigenvalues. We choose bounds for the general relaxation coefficient μ of Van Cittert’s additional term so as to ensure convergence. We show that the bounds can be estimated from the system matrix. Many powerful nonlinear deconvolution techniques are derived from Van Cittert’s method, even though it appears outmoded. As an example, we demonstrate that Gold’s iterative algorithm is a special Van Cittert’s algorithm with a variable relaxation factor μ

© 1994 Optical Society of America

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References

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  1. A. N. Tickhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1974).
  2. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 199–231.
  3. C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 14–18.
  4. P. A. Jansson, Deconvolution with Application in Spectroscopy (Academic, Orlando, Fla., 1984).
  5. A. Bennia, S. M. Riad, “Filtering capabilities and convergence of Van Cittert deconvolution technique,”IEEE Trans. Instrum. Meas. 41, 246–250 (1992).
    [CrossRef]
  6. P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitatswerteilung in Spektrallinien II,” Z. Phys. 69, 298 (1931).
    [CrossRef]
  7. R. N. Bracewell, J. A. Roberts, “Aerial smoothing in radio astronomy,” Aust. J. Phys. 7, 615–640 (1954).
    [CrossRef]
  8. N. R. Hill, G. E. Ioup, “Convergence of the Van Cittert iterative method of deconvolution,”J. Opt. Soc. Am. 66, 487–489 (1976).
    [CrossRef]
  9. P. A. Jansson, “Method for determining the response function of a high resolution infrared spectrometer,”J. Opt. Soc. Am. 60, 184–191 (1970).
    [CrossRef]
  10. R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [CrossRef]
  11. B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).
  12. S. Kawata, Y. Ichioka, “Iterative image restoration for linearly degraded images. I. Basis,”J. Opt. Soc. Am. 70, 762–768 (1980).
    [CrossRef]
  13. S. Kawata, Y. Ichioka, “Iterative image restoration for linearly degraded images. II. Reblurring procedure,”J. Opt. Soc. Am. 70, 768–772 (1980).
    [CrossRef]
  14. S. Singh, S. Tandon, “An iterative restoration technique,” Signal Process. 11, 1–11 (1986).
    [CrossRef]
  15. P. B. Crilly, “Convergence issues for iterative deconvolution signal restoration algorithms,” Chemometrics Intell. Lab. Syst. 12, 291–298 (1991).
    [CrossRef]
  16. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).
  17. R. Gold, “An iterative unfolding method for matrices,” Mathematics and Computer Research and Development Rep. ANL-6984 (Argonne National Laboratory, Argonne, Ill., 1964).
  18. Ching-Bo Juang, “Design and application of a computer-controlled confocal scanning differential polarization microscope,” Rev. Sci. Instrum. 50, 2399–2408 (1988).
    [CrossRef]

1992 (1)

A. Bennia, S. M. Riad, “Filtering capabilities and convergence of Van Cittert deconvolution technique,”IEEE Trans. Instrum. Meas. 41, 246–250 (1992).
[CrossRef]

1991 (1)

P. B. Crilly, “Convergence issues for iterative deconvolution signal restoration algorithms,” Chemometrics Intell. Lab. Syst. 12, 291–298 (1991).
[CrossRef]

1988 (1)

Ching-Bo Juang, “Design and application of a computer-controlled confocal scanning differential polarization microscope,” Rev. Sci. Instrum. 50, 2399–2408 (1988).
[CrossRef]

1986 (1)

S. Singh, S. Tandon, “An iterative restoration technique,” Signal Process. 11, 1–11 (1986).
[CrossRef]

1981 (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

1980 (2)

1976 (1)

1970 (1)

1954 (1)

R. N. Bracewell, J. A. Roberts, “Aerial smoothing in radio astronomy,” Aust. J. Phys. 7, 615–640 (1954).
[CrossRef]

1931 (1)

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitatswerteilung in Spektrallinien II,” Z. Phys. 69, 298 (1931).
[CrossRef]

Arsenin, V. Y.

A. N. Tickhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1974).

Bellman, R.

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).

Bennia, A.

A. Bennia, S. M. Riad, “Filtering capabilities and convergence of Van Cittert deconvolution technique,”IEEE Trans. Instrum. Meas. 41, 246–250 (1992).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, J. A. Roberts, “Aerial smoothing in radio astronomy,” Aust. J. Phys. 7, 615–640 (1954).
[CrossRef]

Crilly, P. B.

P. B. Crilly, “Convergence issues for iterative deconvolution signal restoration algorithms,” Chemometrics Intell. Lab. Syst. 12, 291–298 (1991).
[CrossRef]

Gold, R.

R. Gold, “An iterative unfolding method for matrices,” Mathematics and Computer Research and Development Rep. ANL-6984 (Argonne National Laboratory, Argonne, Ill., 1964).

Hill, N. R.

Ichioka, Y.

Ioup, G. E.

Jansson, P. A.

Juang, Ching-Bo

Ching-Bo Juang, “Design and application of a computer-controlled confocal scanning differential polarization microscope,” Rev. Sci. Instrum. 50, 2399–2408 (1988).
[CrossRef]

Kawata, S.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 199–231.

Riad, S. M.

A. Bennia, S. M. Riad, “Filtering capabilities and convergence of Van Cittert deconvolution technique,”IEEE Trans. Instrum. Meas. 41, 246–250 (1992).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Roberts, J. A.

R. N. Bracewell, J. A. Roberts, “Aerial smoothing in radio astronomy,” Aust. J. Phys. 7, 615–640 (1954).
[CrossRef]

Rushforth, C. K.

C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 14–18.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Singh, S.

S. Singh, S. Tandon, “An iterative restoration technique,” Signal Process. 11, 1–11 (1986).
[CrossRef]

Tandon, S.

S. Singh, S. Tandon, “An iterative restoration technique,” Signal Process. 11, 1–11 (1986).
[CrossRef]

Tickhonov, A. N.

A. N. Tickhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1974).

Van Cittert, P. H.

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitatswerteilung in Spektrallinien II,” Z. Phys. 69, 298 (1931).
[CrossRef]

Wendroff, B.

B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).

Aust. J. Phys. (1)

R. N. Bracewell, J. A. Roberts, “Aerial smoothing in radio astronomy,” Aust. J. Phys. 7, 615–640 (1954).
[CrossRef]

Chemometrics Intell. Lab. Syst. (1)

P. B. Crilly, “Convergence issues for iterative deconvolution signal restoration algorithms,” Chemometrics Intell. Lab. Syst. 12, 291–298 (1991).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

A. Bennia, S. M. Riad, “Filtering capabilities and convergence of Van Cittert deconvolution technique,”IEEE Trans. Instrum. Meas. 41, 246–250 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Rev. Sci. Instrum. (1)

Ching-Bo Juang, “Design and application of a computer-controlled confocal scanning differential polarization microscope,” Rev. Sci. Instrum. 50, 2399–2408 (1988).
[CrossRef]

Signal Process. (1)

S. Singh, S. Tandon, “An iterative restoration technique,” Signal Process. 11, 1–11 (1986).
[CrossRef]

Z. Phys. (1)

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitatswerteilung in Spektrallinien II,” Z. Phys. 69, 298 (1931).
[CrossRef]

Other (7)

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).

R. Gold, “An iterative unfolding method for matrices,” Mathematics and Computer Research and Development Rep. ANL-6984 (Argonne National Laboratory, Argonne, Ill., 1964).

B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).

A. N. Tickhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1974).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 199–231.

C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 14–18.

P. A. Jansson, Deconvolution with Application in Spectroscopy (Academic, Orlando, Fla., 1984).

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Figures (1)

Fig. 1
Fig. 1

Resolution improvement of microscopic image by deconvolution. (a) Original image, (b) image after deconvolution, (c) profile of line 65 of the original image, (d) profile of line 65 of the image after deconvolution.

Tables (2)

Tables Icon

Table 1 Restoration of the Data Degraded by a Well-Conditioned Linear Systemaa

Tables Icon

Table 2 Restoration of the Data Degraded by a Near-Singular Linear Systemaa

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

g ( x , y ) = - - f ( u - x , v - y ) h ( u , v ) d u d v .
g ( i , j ) = n = - n = - h ( m , n ) f ( m - i , n - j ) .
Y = AX ,
f k + 1 ( x ) = f k ( x ) + [ g ( x ) = f x ( x ) h ( x ) ] ,
f k + 1 ( x ) = f k ( x ) + μ [ g ( x ) - f k ( x ) h ( x ) ] .
X k + 1 = X k + μ ( Y - A X k ) .
[ A ] i i > j i [ A ] i j .
D = I d - μ A
X k + 1 = μ Y + D X k .
X k = μ Y + μ DY + + μ D k - 1 Y + μ D k Y k = μ ( I d + D + + D k ) Y .
i { 1 n } ,             ( 1 - μ λ i ) k k 0 ,
D k k [ 0 ] ,
X k k A - 1 Y ,
1 - μ λ i < 1             for all i { 1 n } ;
( 1 - μ λ i ) ( 1 - μ λ i * ) < 1.
λ i = a i + j b i ,             λ i * = a i - j b i ,
μ [ μ ( a i 2 + b i 2 ) - 2 a i ] < 0.
μ = 0 ,             μ = 2 a i a i 2 + b i 2 .
0 < μ < 2 / λ i ,             i { 1 n } .
0 < μ < 2 / λ max .
( A T A A T ) Y = ( A T A A T A ) X .
X k + 1 = X k + μ [ ( A T A A T ) Y - ( A T A A T A ) X k ] .
X k + 1 = X k + μ ( Y - A X k ) .
A X = λ i X .
m = 1 n [ A ] j m [ X ] m = λ i [ X ] j , λ i = m = 1 n [ A ] j m [ X ] m [ X ] j , λ i m = 1 n [ A ] j m             for any i { 1 , , n } .
0 μ 2 / m = 1 n [ A ] j m .
μ i i k = [ X k ] [ A X k ] i
[ X k + 1 ] i = [ X k ] i + [ X k ] [ A X k ] i ( [ Y ] i - [ A X k ] i ) , [ X k + 1 ] i = [ Y ] i [ A X k ] i [ X k ] i .
X = [ 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , ] T ,
A 1 = [ 0.5 0.0 0.0 0.8 0.5 0.0 0.0 1.0 0.8 0.5 0.0 0.0 0.8 1.0 0.8 0.5 0.0 0.0 0.5 0.8 1.0 0.8 0.5 0.0 0.0 0.0 0.5 0.8 1.0 0.8 0.5 0.0 0.0 0.0 0.5 0.5 0.8 1.0 0.8 0.5 ] .
A 2 = [ 1.0 1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 ] .
h ( x , y ) = [ 0.000 0.004 0.008 0.010 0.008 0.004 0.000 0.004 0.014 0.021 0.028 0.021 0.014 0.004 0.008 0.021 0.046 0.063 0.046 0.021 0.008 0.010 0.028 0.063 0.076 0.063 0.028 0.010 0.008 0.021 0.046 0.063 0.046 0.021 0.008 0.004 0.014 0.021 0.028 0.021 0.014 0.004 0.000 0.004 0.008 0.010 0.008 0.004 0.000 ] .
f k + 1 = f k + μ [ h ( - x , - y ) g ( x , y ) - h ( - x , - y ) h ( x , y ) f k ] ,
X k + 1 = X k + μ ( A T Y - A T A X k ) ,

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