Abstract

We present a direct method based on the sampling theorem for computing eigenwavefronts associated with linear space-invariant imaging systems (including aberrated imaging systems). A potential application of the eigenwavefronts to inverse problems in imaging is discussed. A noise-dependent measure for the information-carrying capacity of an imaging system is also proposed.

© 2005 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [CrossRef]
  3. C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  4. G. Toraldo Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  5. F. Gori, “Integral equations for incoherent imagery,” J. Opt. Soc. Am. 64, 1237–1243 (1974).
    [CrossRef]
  6. M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom and eigenfunctions for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
    [CrossRef]
  7. M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: a singular value analysis,” Opt. Acta 29, 727–746 (1982).
    [CrossRef]
  8. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  9. K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave-functions,” J. Phys. A Math. Gen. 36, 10011–10021 (2003).
    [CrossRef]
  10. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  11. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).
  12. The linear independence follows from the fact that sinc(2Bx-m)and sinc(2Bx-k)are orthogonal over (-∞, ∞) for m≠k.
  13. K. Khare, “Mathematical topics in imaging: sampling theory and eigenfunction analysis of imaging systems,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 2004).

2003 (1)

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave-functions,” J. Phys. A Math. Gen. 36, 10011–10021 (2003).
[CrossRef]

1982 (1)

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: a singular value analysis,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

1974 (2)

1969 (1)

1968 (1)

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bendinelli, M.

Bertero, M.

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: a singular value analysis,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

Consortini, A.

Frieden, B. R.

George, N.

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave-functions,” J. Phys. A Math. Gen. 36, 10011–10021 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gori, F.

Harris, R. W.

Khare, K.

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave-functions,” J. Phys. A Math. Gen. 36, 10011–10021 (2003).
[CrossRef]

K. Khare, “Mathematical topics in imaging: sampling theory and eigenfunction analysis of imaging systems,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 2004).

Pike, E. R.

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: a singular value analysis,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Ronchi, L.

Rushforth, C. K.

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Toraldo Di Francia, G.

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. A Math. Gen. (1)

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave-functions,” J. Phys. A Math. Gen. 36, 10011–10021 (2003).
[CrossRef]

Opt. Acta (1)

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: a singular value analysis,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

Proc. IRE (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998).

The linear independence follows from the fact that sinc(2Bx-m)and sinc(2Bx-k)are orthogonal over (-∞, ∞) for m≠k.

K. Khare, “Mathematical topics in imaging: sampling theory and eigenfunction analysis of imaging systems,” Ph.D. thesis (University of Rochester, Rochester, N.Y., 2004).

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Figures (8)

Fig. 1
Fig. 1

Linear space-invariant imaging system.

Fig. 2
Fig. 2

Impulse response for a system with aperture function P 1 ( f ) in Eq. (13).

Fig. 3
Fig. 3

First six eigenwavefronts corresponding to the impulse response in Fig. 2, with L = 2 , B = 2 . Solid and dotted curves show real and imaginary parts, respectively.

Fig. 4
Fig. 4

Eigenvalues (absolute magnitude) corresponding to the impulse response in Fig. 2, with L = 2 , B = 2 .

Fig. 5
Fig. 5

Impulse response for a system with the aperture function P 2 ( f ) in Eq. (14).

Fig. 6
Fig. 6

First six eigenwavefronts corresponding to impulse response in Fig. 5, with L = 2 , B = 2 . Solid and dotted curves show real and imaginary parts, respectively.

Fig. 7
Fig. 7

Eigenvalues (absolute magnitude) corresponding to the impulse response in Fig. 5, with L = 2 , B = 2 .

Fig. 8
Fig. 8

Estimated inverse recovery for a point-source input. A system with the aperture function P 1 ( f ) is assumed, and correspondingly the eigenwavefronts in Fig. 3 are used for recovery.

Equations (18)

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g o ( x ) = - L L d x p ( x - x ) g i ( x ) ,
p ( x ) = m = - p m 2 B sinc ( 2 Bx - m ) .
μ n ψ n ( x ) = - L L d x p ( x - x ) ψ n ( x ) .
μ n ψ n ( x ) = - L L d x m = - p m 2 B - x × sinc ( 2 Bx - m ) ψ n ( x ) = μ n m = - ψ n m 2 B sinc ( 2 Bx - m ) .
μ n ψ n m 2 B = - L L d x p m 2 B - x k = - ψ n k 2 B × sinc ( 2 Bx - k ) = k = - A mk ψ n k 2 B ,
A mk = - L L d x p m 2 B - x sinc ( 2 Bx - k ) .
A w n = μ n w n ,
w n = ψ n m 2 B T ,
A mk = - L L d x   sinc ( 2 Bx - m ) sinc ( 2 Bx - k ) .
A mk = l = - p m - l 2 B - L L d x   × sinc ( 2 Bx - l ) sinc ( 2 Bx - k ) = l = - p m - l 2 B A lk .
A = h   A ,
h ml = p m - l 2 B
P 1 ( f ) = exp i   π 2   f 2 rect f 2 B .
P 2 ( f ) = exp i   π 2   f 2 + 1 2   f 3 + 1 2   f 4 + 1 4   f 5 rect f 2 B .
g o ( x ) = n b n ψ n ( x ) .
g i ( x ) = rect x 2 L n   b n μ n   ψ n ( x ) .
m = - n b n ψ n m 2 B - g m 2 B sinc ( 2 Bx - m ) = 0 .
n b n ψ n m 2 B = g o m 2 B

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