Abstract

Can photographic images, e.g., corresponding to star clusters, be superresolved? Or alternatively, can already good (diffraction-limited) images be improved by restoring methods? To test this hypothesis, objects have been prepared that can be resolved only if the bandwidth in the restoration exceeds that of the image data. The objects are double and triple slits with separations equal to fractions (0.33, 0.50, or 0.65) of Rayleigh’s resolution distance. These are incoherently imaged in quasimonochromatic light by a slit-aperture optical system, and developed as photographs. The amount of diffraction blur is made comparable with the grain-limited resolution distance, since this is the usual situation for an efficient optics–film design. The photos are scanned across the slit directions, digitized, and computer restored by the method of maximum entropy. Results indicate that two object slits can be well resolved when separated by one-half of Rayleigh’s resolution distance, and that three object slits are resolvable subject to a definite hierarchy of confidence: Most accurately restored is the number of object impulses, next are their positions, and least accurate is information on their amplitudes.

© 1972 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), pp. 62, 96, 115.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

1972 (1)

1970 (2)

1969 (1)

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

1968 (1)

1967 (4)

1966 (1)

1964 (1)

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Figures (17)

F. 1
F. 1

Sketches, not to scale, of the double- and triple-slit test objects used in the experiments (top row), their radiance profiles (middle row), and perfect band-limited versions (bottom row) defined by Eq. (5).

F. 2
F. 2

Flow of the experiment, from left to right.

F. 3
F. 3

The four experimental images, on the scale seen by the scanning aperture. These correspond to objects (a)–(d) in Fig. 1. The white rectangle covers two successive aperture positions, and defines one datum input Im to the restoring algorithm.

F. 4
F. 4

Ideal image (– – –), its ME restoration (——), and perfect band-limited version (⋯) for the two-slit object with 0.33R separation. Nonphysical negative portions of the (⋯) curve are left out. Image (– – –) is not perfectly symmetric because it is not sampled symmetrically in y.

F. 5
F. 5

Restorations (——) of some representative lines of data. Experimental inputs {Im} are encircled, and (– – –) is the ideal image.

F. 6
F. 6

Composite photo of the first 15 restored lines, with the first one at the bottom. The top, 16th line, corresponds to the ideal ME restoration in Fig. 4. The exposure image data defined by each scanning aperture position within the image are also shown, on the same scale.

F. 7
F. 7

Ideal image (– – –), its ME restorations (——), and perfect band-limited version (⋯) for the two-slit object with 0.50R separation. Negative portions of the (⋯) curve are left out.

F. 8
F. 8

Restorations (——) of some representative lines of data. Experimental inputs {Im} are encircled, and (– – –) is the ideal image.

F. 9
F. 9

Composite photo of the first 30 restored lines. The top line of the restoration corresponds to the ideal ME restoration in Fig. 7. The exposure image data defined by each scanning aperture position within the image are also shown, on the same scale.

F. 10
F. 10

Ideal image (– – –), its ME restoration (——), and perfect band-limited version (⋯) for the three-slit object with 0.65R common separation. Negative portions of the (⋯) curve are left out.

F. 11
F. 11

Restorations (——) of some representative lines of data. Experimental inputs {Im} are encircled, and (– – –) is the ideal image.

F. 12
F. 12

Composite photo of the first 30 restored lines. The exposure image datum defined by each scanning-aperture position within the image is also shown, on the same scale.

F. 13
F. 13

Composite restorations of the object Fig. 1(c), based on (a) no averaging of the image data, (b) two-line averaging, and (c) four-line averaging.

F. 14
F. 14

Ideal image (– – –), its ME restoration (——), and perfect band-limited version (⋯) for the three-slit object with central-slit amplitude 0.50 of outer 2. Negative portions of the (⋯) are left out.

F. 15
F. 15

Restorations (——) of some representative lines of data. Experimental inputs {Im} are encircled, and (– – –) is the ideal image.

F. 16
F. 16

Composite photo of the first 30 restored lines. The exposure image data defined by each scanning-aperture position within the image are also shown, on the same scale.

F. 17
F. 17

Test use of a gaussian approximation to the true psf in the restoring algorithm. (a) Image; (b) restoration with gaussian psf; (c) restoration with sine2 psf.

Tables (2)

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Table I Experimental design parameters.

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Table II Summary of experiments.

Equations (6)

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I ( y ) = scene d x O ( x ) S ( y x ) + N ( y ) .
i ( ω ) = { o ( ω ) s ( ω ) + n ( ω ) for | ω | < Ω n ( ω ) for | ω | Ω .
2 π / Ω R ,
Ô ( x ) 0 .
O pbl ( x ) Ω Ω d ω o ( ω ) e j ω x , j = ( 1 ) 1 2
S ( y ) = sinc 2 ( Ω x / 2 π ) , sinc ( u ) ( π u ) 1 sin ( π u ) .