Abstract

Possibility theory offers an alternative to ordinary probability theory in describing uncertainty. Images are the visual manifestations of probability laws. Then can a given image be somehow transformed into a possibility image? We show one way of doing this and investigate the properties of such an image. The rules of possibility theory substitute logical operations (size comparisons) for arithmetic operations such as multiply and add. Thus a logical analog to the ordinary (arithmetic) convolution operation also exists. Properties of this logical convolution are investigated. These include superresolution and a kind of closure property that should aid in bandwidth compression.

© 1987 Optical Society of America

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References

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  1. B. R. Frieden, “Restoring with maximum likelihood,” (University of Arizona, Tucson, Ariz., 1971).
  2. W. H. Richardson, “Bayesian-based iterative method of image restoration,”J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  3. L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst. 1, 3–28 (1978).
    [CrossRef]
  4. D. Dubois, H. Prade, “Unfair coins and necessity measures: towards a possibilistic interpretation of histograms,” Fuzzy Sets Syst. 10, 15–20 (1983).
    [CrossRef]
  5. Analogous operations for the identification of spectral lines were recently reported by T. Blaffert, “Computer-assisted multicomponent spectral analysis with fuzzy data sets,” Anal. Chim. Acta 1, 135–148 (1984).
    [CrossRef]

1984 (1)

Analogous operations for the identification of spectral lines were recently reported by T. Blaffert, “Computer-assisted multicomponent spectral analysis with fuzzy data sets,” Anal. Chim. Acta 1, 135–148 (1984).
[CrossRef]

1983 (1)

D. Dubois, H. Prade, “Unfair coins and necessity measures: towards a possibilistic interpretation of histograms,” Fuzzy Sets Syst. 10, 15–20 (1983).
[CrossRef]

1978 (1)

L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst. 1, 3–28 (1978).
[CrossRef]

1972 (1)

Blaffert, T.

Analogous operations for the identification of spectral lines were recently reported by T. Blaffert, “Computer-assisted multicomponent spectral analysis with fuzzy data sets,” Anal. Chim. Acta 1, 135–148 (1984).
[CrossRef]

Dubois, D.

D. Dubois, H. Prade, “Unfair coins and necessity measures: towards a possibilistic interpretation of histograms,” Fuzzy Sets Syst. 10, 15–20 (1983).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Restoring with maximum likelihood,” (University of Arizona, Tucson, Ariz., 1971).

Prade, H.

D. Dubois, H. Prade, “Unfair coins and necessity measures: towards a possibilistic interpretation of histograms,” Fuzzy Sets Syst. 10, 15–20 (1983).
[CrossRef]

Richardson, W. H.

Zadeh, L. A.

L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst. 1, 3–28 (1978).
[CrossRef]

Anal. Chim. Acta (1)

Analogous operations for the identification of spectral lines were recently reported by T. Blaffert, “Computer-assisted multicomponent spectral analysis with fuzzy data sets,” Anal. Chim. Acta 1, 135–148 (1984).
[CrossRef]

Fuzzy Sets Syst. (2)

L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst. 1, 3–28 (1978).
[CrossRef]

D. Dubois, H. Prade, “Unfair coins and necessity measures: towards a possibilistic interpretation of histograms,” Fuzzy Sets Syst. 10, 15–20 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

B. R. Frieden, “Restoring with maximum likelihood,” (University of Arizona, Tucson, Ariz., 1971).

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Figures (5)

Fig. 1
Fig. 1

Π(y) is the shaded area defined by reference level i(y).

Fig. 2
Fig. 2

Sinusoidal input (bottom, solid) and possibility output (top, solid). Noisy input (bottom, dashed) and possibility output (top, dashed). Some noise suppression is apparent.

Fig. 3
Fig. 3

(Left) The jot in galaxy M87. (Middle) Its possibility transform. (Right) A logarithmic display of the image. All images are displayed with the same intensity mapping function. The possibility transform emphasizes midrange intensities, while the logarithmic image emphasizes low intensities. (Photos courtesy of D. Tody, Kitt Peak National Observatory.)

Fig. 4
Fig. 4

Impulsive object o(x), spread function s(x), arithmetic convolution of the two (dashed), and logical convolution (solid). The logical convolution shows superior resolution.

Fig. 5
Fig. 5

(Left) Object. (Middle) Blurred by arithmetic convolution. (Right) Blurred by logical convolution. Notice the exaggerated point- and line-source outputs in the logical convolution, a kind of superresolution. Note: Because of the halftone dots, proper viewing distance is 76 cm or more.

Equations (15)

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P ( A or B ) = P ( A ) + P ( B ) ,
P ( A and B ) = P ( A ) P ( B ) .
Π ( A or B ) = max [ Π ( A ) , Π ( B ) ] , Π ( A and B ) = min [ Π ( A ) , Π ( B ) ] .
P ( A or B ) max [ P ( A ) , P ( B ) ]
P ( A and B ) min [ P ( A ) , P ( B ) ] .
Π m m = 1 , , N = n = 1 N min ( i m , i n ) .
Π ( A ) = max ( Π m ) ,
P ( A ) = m i m .
i n = m = n N m - 1 ( Π m - Π m + 1 ) ,             Π N + 1 = 0.
Π 1 Π 2 Π N ,
i 1 i 2 i N .
Π ( y ) = d x min [ i ( y ) , i ( x ) ] .
Π ( y ) = ( f y / π - 1 ) cos f y - π - 1 sin f y + 1
i m = n = - M M o n s m n ,
i m = max [ min ( o - M , s m , - M ) , min ( o - M + 1 , s m , - M + 1 ) , , min ( o M , s m M ) ] ,

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