Abstract

Given M sampled image values of an incoherent object, what can be deduced as the most likely object? Using a communication-theory model for the process of image formation, we find that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited. The derivation is an adaptation to optics of a formulation by Jaynes for unbiased estimates of positive probability functions. The restoring formula is tested, via computer simulation, upon noisy images of objects consisting of random impulses. These are found to be well restored, with resolution often exceeding the Rayleigh limit and with a complete absence of spurious detail. The proviso is that the noise in each image input must not exceed about 40% of the signal image. The restoring method is applied to experimental data consisting of line spectra. Results are consistent with those of the computer simulations.

© 1972 Optical Society of America

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References

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  1. Y. Biraud, Astron. Astrophys. 1, 124 (1969).
  2. P. A. Jansson, R. H. Hunt, and E. K. Plyler, J. Opt. Soc. Am. 60, 596 (1970).
    [Crossref]
  3. R. S. Hershel, J. Opt. Soc. Am. 60, 1546A (1970).
  4. D. P. MacAdam, J. Opt. Soc. Am. 60, 1617 (1970).
    [Crossref]
  5. E. T. Jaynes, Trans. IEEE SSC-4, 227 (1968).
  6. S. Goldman, Information Theory (Prentice–Hall, New York, 1955), Appendix II.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).
  8. B. L. Phillips, J. ACM,  9, 84 (1962).
    [Crossref]
  9. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964), p. 41.
  10. Reference 9, p. 57.
  11. See, e.g., L. A. Pipes, Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1958), p. 116.
  12. A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).
  13. A. van den Bos, Trans. IEEE IT-17, 494 (1971).
  14. See, e.g., R. Nathan, J. Opt. Soc. Am. 57, 578A (1967).
  15. The data in question cover the first three maxima of curve A in Fig. 2 of Ref. 2. Jansson’s corresponding restoration is Fig. 2(c).

1971 (1)

A. van den Bos, Trans. IEEE IT-17, 494 (1971).

1970 (4)

A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).

P. A. Jansson, R. H. Hunt, and E. K. Plyler, J. Opt. Soc. Am. 60, 596 (1970).
[Crossref]

R. S. Hershel, J. Opt. Soc. Am. 60, 1546A (1970).

D. P. MacAdam, J. Opt. Soc. Am. 60, 1617 (1970).
[Crossref]

1969 (1)

Y. Biraud, Astron. Astrophys. 1, 124 (1969).

1968 (1)

E. T. Jaynes, Trans. IEEE SSC-4, 227 (1968).

1967 (1)

See, e.g., R. Nathan, J. Opt. Soc. Am. 57, 578A (1967).

1962 (1)

B. L. Phillips, J. ACM,  9, 84 (1962).
[Crossref]

Biraud, Y.

Y. Biraud, Astron. Astrophys. 1, 124 (1969).

Dowson, D. C.

A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).

Goldman, S.

S. Goldman, Information Theory (Prentice–Hall, New York, 1955), Appendix II.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

Hershel, R. S.

R. S. Hershel, J. Opt. Soc. Am. 60, 1546A (1970).

Hunt, R. H.

Jansson, P. A.

Jaynes, E. T.

E. T. Jaynes, Trans. IEEE SSC-4, 227 (1968).

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964), p. 41.

MacAdam, D. P.

Nathan, R.

See, e.g., R. Nathan, J. Opt. Soc. Am. 57, 578A (1967).

Phillips, B. L.

B. L. Phillips, J. ACM,  9, 84 (1962).
[Crossref]

Pipes, L. A.

See, e.g., L. A. Pipes, Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1958), p. 116.

Plyler, E. K.

van den Bos, A.

A. van den Bos, Trans. IEEE IT-17, 494 (1971).

Wragg, A.

A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).

Astron. Astrophys. (1)

Y. Biraud, Astron. Astrophys. 1, 124 (1969).

J. ACM (1)

B. L. Phillips, J. ACM,  9, 84 (1962).
[Crossref]

J. Opt. Soc. Am. (4)

P. A. Jansson, R. H. Hunt, and E. K. Plyler, J. Opt. Soc. Am. 60, 596 (1970).
[Crossref]

R. S. Hershel, J. Opt. Soc. Am. 60, 1546A (1970).

D. P. MacAdam, J. Opt. Soc. Am. 60, 1617 (1970).
[Crossref]

See, e.g., R. Nathan, J. Opt. Soc. Am. 57, 578A (1967).

Trans. IEEE (3)

A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).

A. van den Bos, Trans. IEEE IT-17, 494 (1971).

E. T. Jaynes, Trans. IEEE SSC-4, 227 (1968).

Other (6)

S. Goldman, Information Theory (Prentice–Hall, New York, 1955), Appendix II.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964), p. 41.

Reference 9, p. 57.

See, e.g., L. A. Pipes, Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1958), p. 116.

The data in question cover the first three maxima of curve A in Fig. 2 of Ref. 2. Jansson’s corresponding restoration is Fig. 2(c).

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Figures (6)

Fig. 1
Fig. 1

Restoration by maximum entropy (starred curves) of random-impulse patterns (solid), compared with the perfect-band-limited version (boxed curves) of each object. A value ρ=1 was used. The input image points {Im} (not shown) have a maximum relative error of 5% due to uniformly random noise.

Fig. 2
Fig. 2

Maximum-entropy restorations of the shaded object, all based upon the same image data. Dotted curve, ρ=1; solid curve, ρ=20; dashed curve, ρ=100. Where the curves for ρ=20 and 100 coincide, only the dashed one is shown. Noise amplitude β=5%.

Fig. 3
Fig. 3

The effect of ρ upon the relative rms error α over the class of objects, for various levels β of image noise. The values of β label the curves. Level α=1 represents eqality of rms error with that of the perfect-band-limited versions of the true objects.

Fig. 4
Fig. 4

As in Fig. 2. Here, however, noise amplitude β=40%.

Fig. 5
Fig. 5

Restoration by maximum entropy (solid) of a portion of the Q branch of the ν3 band of the spectrum for CH4. Triangles indicate the experimental inputs {Im} used in the restoring scheme, some of which lie outside the plotted region and are not shown.

Fig. 6
Fig. 6

Comparison of the spectrum (dot-dashed curve) of a maximum-entropy restoration with that (solid curve) of the true object. Image noise amplitude β was 5%. Because the true object was symmetric, only the real part of each spectrum is shown.

Equations (31)

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O ( x ) 0 ,             all             x .
O j = o j Δ O ,
j = 1 J o j = O ,             where O 1 ,
P 0 = O Δ O .
p j = O j / O ,
W 0 ( o 1 , , o J ) = O ! o 1 ! o 2 ! o J ! .
ln W 0 ( O 1 , , O J ) = - Δ O - 1 j = 1 J O j ln O j
Δ x < y m - y m - 1 .
I m = j = 1 J O j S ( y m , x j ) + n m ,             m = 1 , , M
N m = n m + B ,             N m 0 ,             B 0.
B n neg .
N m 0 ,
ρ Δ O / Δ N .
N m = η m Δ N ,
ln W N ( N 1 , , N M ) = - Δ N - 1 m = 1 M N m ln N m
W 0 W N = maximum ,
- j = 1 J Ô j ln Ô j - ρ m = 1 M N ˆ m ln N ˆ m - m = 1 M λ m [ j = 1 J Ô j S ( y m , x j ) + N ˆ m - B - I m ] - μ ( j = 1 J Ô j - P 0 ) = maximum .
Ô j = exp [ - 1 - μ - m = 1 M λ m S ( y m , x j ) ] .
N ˆ m = exp [ - 1 - λ m / ρ ] .
I m = j = 1 J Ô j S ( y m , x j ) + N ˆ m - B ,             m = 1 , , M
P 0 = j = 1 J Ô j .
- scene d x Ô ( x ) ln Ô ( x ) - P 0 ln .
Ô ( x ) = exp [ - 1 - μ - m = 1 M λ m S ( y m , x ) ] .
const × [ 1 - m = 1 m λ m S ( y m , x ) ] ,
d ( n ) d x ( n ) Ô ( x ) Ô ( x ) ,             n = 1 , 2 , .
( n ) Ô ( x ) λ m 1 λ m 2 λ m n Ô ( x ) ,             n = 1 , 2 , , M .
e 2 = J - 1 j = 1 J ( O j - O j ) 2 , e { e m e for O j = Ô j e p b l for O j = O j p b l .
α = 9 - 1 n = 1 9 ( e m e / e p b l ) n
H = - j = 1 J O j ln O j .
j = 1 J O j = const ,
Δ H = - ( 1 + ln O m ) Δ O m - ( 1 + ln O n ) Δ O n = ln ( O m / O n ) .