## 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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### Equations (15)

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(1)
$$P(A\hspace{0.17em}\text{or}\hspace{0.17em}B)=P(A)+P(B),$$
(2)
$$P(A\hspace{0.17em}\text{and}\hspace{0.17em}B)=P(A)P(B).$$
(3)
$$\begin{array}{c}\mathrm{\Pi}(A\hspace{0.17em}\text{or}\hspace{0.17em}B)=\text{max}[\mathrm{\Pi}(A),\mathrm{\Pi}(B)],\\ \mathrm{\Pi}(A\hspace{0.17em}\text{and}\hspace{0.17em}B)=\text{min}[\mathrm{\Pi}(A),\mathrm{\Pi}(B)].\end{array}$$
(4)
$$P(A\hspace{0.17em}\text{or}\hspace{0.17em}B)\ge \text{max}[P(A),P(B)]$$
(5)
$$P(A\hspace{0.17em}\text{and}\hspace{0.17em}B)\le \text{min}[P(A),P(B)].$$
(6)
$$\underset{m=1,\dots ,N}{{\mathrm{\Pi}}_{m}}=\sum _{n=1}^{N}\text{min}({i}_{m},{i}_{n}).$$
(7)
$$\mathrm{\Pi}(A)=\text{max}({\mathrm{\Pi}}_{m}),$$
(8)
$$P(A)=\sum _{m}{i}_{m}.$$
(9)
$${i}_{n}=\sum _{m=n}^{N}{m}^{-1}({\mathrm{\Pi}}_{m}-{\mathrm{\Pi}}_{m+1}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{\mathrm{\Pi}}_{N+1}=0.$$
(10)
$${\mathrm{\Pi}}_{1}\ge {\mathrm{\Pi}}_{2}\ge \dots \ge {\mathrm{\Pi}}_{N},$$
(11)
$${i}_{1}\ge {i}_{2}\ge \dots \ge {i}_{N}.$$
(12)
$$\mathrm{\Pi}(y)=\int \text{d}x\hspace{0.17em}\text{min}[i(y),i(x)].$$
(13)
$$\mathrm{\Pi}(y)=(fy/\pi -1)\text{cos}\hspace{0.17em}fy-{\pi}^{-1}\hspace{0.17em}\text{sin}\hspace{0.17em}fy+1$$
(14)
$${i}_{m}=\sum _{n=-M}^{M}{o}_{n}{s}_{mn},$$
(15)
$${i}_{m}=\text{max}[\text{min}({o}_{-M},{s}_{m,-M}),\text{min}({o}_{-M+1},{s}_{m,-M+1}),\dots ,\text{min}({o}_{M},{s}_{mM})],$$