Abstract

The long-standing problem of the superresolving reconstruction (restoration) of an object of known finite spatial extent from a noisy linearly degraded image is considered. The resolution of two-point sources (objects) spaced less than one Rayleigh distance apart is an ill-posed problem. To determine a superresolving inverse of an ill-conditioned linear degradation operator with a known set of input/output training signals, a linear associative memory (LAM) technique is employed. By limiting the set of reconstructable signals, an exceptionally robust inverse filter has been obtained. This filter is based on a new constrained LAM matrix operator technique. Superresolving restoration of 1-D and 2-D two-point sources as well as some typical edge-type signals in the presence of considerable measurement noise is demonstrated.

© 1987 Optical Society of America

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  1. R. Barakat, G. Newsam, “Algorithms for Reconstruction of Partially Known, Band-Limited Fourier-Transform Pairs from Noisy Data,” J. Opt. Soc. Am. A 2, 2027 (1985).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

1985 (2)

G. S. Stiles, Dong-Lih Denq, “On the Effect of Noise on the Moore-Penrose Generalized Inverse Associative Memory,” IEEE Trans. Pattern. Anal. Machine Intell. PAMI-3, 358 (1985).
[CrossRef]

R. Barakat, G. Newsam, “Algorithms for Reconstruction of Partially Known, Band-Limited Fourier-Transform Pairs from Noisy Data,” J. Opt. Soc. Am. A 2, 2027 (1985).
[CrossRef]

1984 (1)

1982 (2)

1981 (5)

1980 (1)

1979 (1)

J. A. Cadzow, “An Extrapolation Procedure for Bandlimited Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 4 (1979).
[CrossRef]

1978 (2)

M. S. Sabri, W. Steenart, “An Approach to Bandlimited Extrapolation: The Extrapolation Matrix,” IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[CrossRef]

D. C. Youla, “Generalized Image Restoration by the Method of Alternating Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

1976 (1)

H. C. Andrews, C. L. Patterson, “Singular Value Decomposition and Digital Image Processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

1975 (1)

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1972 (1)

R. Gerschberg, W. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optic 35, 237 (1972).

1967 (1)

1966 (1)

B. Rust, W. R. Burrus, C. Schneeberger, “A Simple Algorithm for Computing the Generalized Inverse of a Matrix,” Commun. ACM 9, 381 (1966).
[CrossRef]

1964 (2)

L. D. Pyle, “Generalized Inverse Computation Using the Gradient Projection Method,” J. Assoc. Comput. Mach. 11, 422 (1964).
[CrossRef]

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

1961 (1)

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty I,” Bell Syst. Tech. J. 40, 43 (1961).

1960 (1)

T. N. E. Greville, “Some Applications of the Pseudoinverse of a Matrix,” SIAM Rev. 2, 15 (1960).
[CrossRef]

Albert, A.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

Andrews, H. C.

H. C. Andrews, C. L. Patterson, “Singular Value Decomposition and Digital Image Processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, Washington, DC, 1977).

Baker, C. T.

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

Barakat, R.

Burrus, W. R.

B. Rust, W. R. Burrus, C. Schneeberger, “A Simple Algorithm for Computing the Generalized Inverse of a Matrix,” Commun. ACM 9, 381 (1966).
[CrossRef]

Cadzow, J. A.

J. A. Cadzow, “An Extrapolation Procedure for Bandlimited Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 4 (1979).
[CrossRef]

Cahana, D.

Crawford, A. E.

Denq, Dong-Lih

G. S. Stiles, Dong-Lih Denq, “On the Effect of Noise on the Moore-Penrose Generalized Inverse Associative Memory,” IEEE Trans. Pattern. Anal. Machine Intell. PAMI-3, 358 (1985).
[CrossRef]

Eichmann, G.

Fox, L.

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Maximum-Information Data Processing: Application to Optical Signals,” J. Opt. Soc. Am. 71, 294 (1981).
[CrossRef]

B. R. Frieden, “Image Enhancement and Restoration,” in Picture Processing and Digital Filtering,” T. S. Huang, Ed. (Springer-Verlag, New York, 1975).

Frost, R. L.

Gerschberg, R.

R. Gerschberg, W. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optic 35, 237 (1972).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins Press, Baltimore, 1983).

Greville, T. N. E.

T. N. E. Greville, “Some Applications of the Pseudoinverse of a Matrix,” SIAM Rev. 2, 15 (1960).
[CrossRef]

Helstrom, C. W.

Howard, S. J.

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

Kohonen, T.

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).

Mammone, R. J.

Meada, J.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constraint Iterative Restoration Algorithms,” Proc. IEEE 69, 432 (1981).
[CrossRef]

Murata, K.

Muyers, U. F.

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

Newsam, G.

Papoulis, A.

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Patterson, C. L.

H. C. Andrews, C. L. Patterson, “Singular Value Decomposition and Digital Image Processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty I,” Bell Syst. Tech. J. 40, 43 (1961).

Pyle, L. D.

L. D. Pyle, “Generalized Inverse Computation Using the Gradient Projection Method,” J. Assoc. Comput. Mach. 11, 422 (1964).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constraint Iterative Restoration Algorithms,” Proc. IEEE 69, 432 (1981).
[CrossRef]

Rushforth, C. K.

Rust, B.

B. Rust, W. R. Burrus, C. Schneeberger, “A Simple Algorithm for Computing the Generalized Inverse of a Matrix,” Commun. ACM 9, 381 (1966).
[CrossRef]

Sabri, M. S.

M. S. Sabri, W. Steenart, “An Approach to Bandlimited Extrapolation: The Extrapolation Matrix,” IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[CrossRef]

Saxton, W.

R. Gerschberg, W. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optic 35, 237 (1972).

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constraint Iterative Restoration Algorithms,” Proc. IEEE 69, 432 (1981).
[CrossRef]

Schneeberger, C.

B. Rust, W. R. Burrus, C. Schneeberger, “A Simple Algorithm for Computing the Generalized Inverse of a Matrix,” Commun. ACM 9, 381 (1966).
[CrossRef]

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty I,” Bell Syst. Tech. J. 40, 43 (1961).

Stark, H.

Steenart, W.

M. S. Sabri, W. Steenart, “An Approach to Bandlimited Extrapolation: The Extrapolation Matrix,” IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[CrossRef]

Stiles, G. S.

G. S. Stiles, Dong-Lih Denq, “On the Effect of Noise on the Moore-Penrose Generalized Inverse Associative Memory,” IEEE Trans. Pattern. Anal. Machine Intell. PAMI-3, 358 (1985).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, Washington, DC, 1977).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins Press, Baltimore, 1983).

Webb, H.

Wright, K.

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

Youla, D. C.

D. C. Youla, “Generalized Image Restoration by the Method of Alternating Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

Zhou, Y.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty I,” Bell Syst. Tech. J. 40, 43 (1961).

Commun. ACM (1)

B. Rust, W. R. Burrus, C. Schneeberger, “A Simple Algorithm for Computing the Generalized Inverse of a Matrix,” Commun. ACM 9, 381 (1966).
[CrossRef]

Comput. J. (1)

C. T. Baker, L. Fox, U. F. Muyers, K. Wright, “Numerical Solution of Fredholm Integral Equations of the First Kind,” Comput. J. 7, 141 (1964).
[CrossRef]

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

H. C. Andrews, C. L. Patterson, “Singular Value Decomposition and Digital Image Processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

J. A. Cadzow, “An Extrapolation Procedure for Bandlimited Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 4 (1979).
[CrossRef]

IEEE Trans. Circuits Syst. (3)

M. S. Sabri, W. Steenart, “An Approach to Bandlimited Extrapolation: The Extrapolation Matrix,” IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[CrossRef]

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

D. C. Youla, “Generalized Image Restoration by the Method of Alternating Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

IEEE Trans. Pattern. Anal. Machine Intell. (1)

G. S. Stiles, Dong-Lih Denq, “On the Effect of Noise on the Moore-Penrose Generalized Inverse Associative Memory,” IEEE Trans. Pattern. Anal. Machine Intell. PAMI-3, 358 (1985).
[CrossRef]

J. Assoc. Comput. Mach. (1)

L. D. Pyle, “Generalized Inverse Computation Using the Gradient Projection Method,” J. Assoc. Comput. Mach. 11, 422 (1964).
[CrossRef]

J. Opt. Soc. Am. (7)

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

Optic (1)

R. Gerschberg, W. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optic 35, 237 (1972).

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constraint Iterative Restoration Algorithms,” Proc. IEEE 69, 432 (1981).
[CrossRef]

SIAM Rev. (1)

T. N. E. Greville, “Some Applications of the Pseudoinverse of a Matrix,” SIAM Rev. 2, 15 (1960).
[CrossRef]

Other (8)

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins Press, Baltimore, 1983).

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

B. R. Frieden, “Image Enhancement and Restoration,” in Picture Processing and Digital Filtering,” T. S. Huang, Ed. (Springer-Verlag, New York, 1975).

P. A. Jansson, Ed., Deconvolution with Applications in Spectroscopy (Academic, Orlando, FL, 1984).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, Washington, DC, 1977).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

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Figures (8)

Fig. 1
Fig. 1

(a) First twenty singular values of an unconstrained LAM matrix M; (b) as in (a) for the constrained M (P = 2).

Fig. 2
Fig. 2

Quadratic matrix norm of a superresolving matrix M as a function of the threshold parameter P.

Fig. 3
Fig. 3

Resolution of a degraded noisy two-point object with a SNR of 19 dB and SBP of 1 obtained with constrained M with singular values as in Fig. 1(b). This constrained matrix M is used in the next examples for two-point object superresolution.

Fig. 4
Fig. 4

(a) Superresolution of two-point objects with SBP = 0.15 and SNR of 19 dB; (b) with SNR of 13 dB; (c) with SNR of 7 dB and SBP of 0.3.

Fig. 5
Fig. 5

Sample of the noise-free degraded training patterns used in constructing the 2-D constrained M matrix.

Fig. 6
Fig. 6

Result of the restoration of a 2-D two-point object with an SNR of −6 dB and SBP of 0.5.

Fig. 7
Fig. 7

Sample of the input–output noise-free training patterns (input–output of ILPF) used to form a LAM matrix M; (a) and (b), a triangle; (c) and (d), a trapezoid; (e) and (f), a Gaussian pulse.

Fig. 8
Fig. 8

(a) Noisy degraded triangle from Fig. 7(a); SNR = −3 dB, SBP = 0.3; (b) the result of restoration of the signal in (a); (c) noisy degraded trapezoid from Fig. 7(c), SNR = 9 dB, SBP = 0.15; (d) the result of restoration of the signal in (c); (e) the noisy degraded Gaussian pulse from Fig. 7(e), SNR = 2 dB, SBP = 0.08; (f) the result of restoration of the signal in (e).

Equations (26)

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

g ( x ) = l 1 l 2 f ( t ) sin [ w c ( x - t ) ] u ¯ ( x - t ) d t .
f ( x ) = n = 0 λ n -1 a n q n ( x ) ,
a n = l 1 l 2 g ( y ) q n ( y ) d y .
g = Hf ,
f = Mg ,
g = H f + n ,
f = H + g ,
X = [ g 1 , g 2 , , g N ] , Y = [ f ^ 1 , f ^ 2 , , f ^ N ] .
Y = H + X .
Y = MX ,
M = YX + .
X r + = [ X r - 1 + ( I - g r c r T ) c r T ] ,
c r = { ( I - X r - 1 X r - 1 + ) g r ( I - X r - 1 X r - 1 + ) g r 2             if the numerator is 0 , ( X r - 1 + ) T X r - 1 + g r 1 + X r - 1 + g r 2             otherwise ,
M r = Y r - 1 + X r - 1 + + ( f r - Y r - 1 X r - 1 + g r ) c r T ,
M r = M r - 1 + ( f r - M r - 1 g r ) c r T .
M r = { M r - 1 + ( f r - M r - 1 g r ) g ˜ r T g ˜ r 2             for             g ˜ r 0 , M r - 1             otherwise ,
g ˜ r = g r - i = 1 r - 1 ( g r , g ˜ i ) g ˜ i 2 g ˜ i ,
R y x = ( 1 / N ) i = 1 N f i g i T ,
R x x = ( 1 / N ) i = 1 N g i g i T .
C g g = ( 1 / N ) i = 1 N g i g i T .
c ( M ) = δ 1 ( M ) δ N ( M ) ,
D = ( 1 / N ) j = 1 N ( g i j - g o j ) 2 ,
Y = [ f 1 , f 2 , , f 100 ] .
X = [ g 1 , g 2 , , g 100 ] .
Y = [ Y 1 , Y 2 , Y 3 , Y 4 ] .
X = [ X 1 , X 2 , X 3 , X 4 ] .

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