## Abstract

Superresolution has been an active field during the last 50 years. Many approaches for achieving superresolution have been suggested, all based on having *a priori* information on the signal. We discuss the space–bandwidth product adaptation process, which is a generalization of all previous approaches. This new point of view is based on handling the Wigner chart of the input signal as well as the Wigner chart of the signal that can be accepted by the system. We also take into account the number of degrees of freedom of the signal and the system. Examples that demonstrate the suggested approach are illustrated in a companion paper [J. Opt. Soc. Am. A **14**, 563 (1997)].

© 1997 Optical Society of America

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### Equations (19)

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(1)
$$\delta {x}_{\mathrm{RES}}\approx \mathrm{\lambda}{f}_{\#}=\frac{\mathrm{\lambda}f}{B},$$
(2)
$$\mathrm{SW}=\mathrm{\Delta}x\mathrm{\Delta}\nu .$$
(3)
$$W(x,\nu )={\int}_{-\infty}^{\infty}f\left(x+\frac{{x}^{\prime}}{2}\right){f}^{*}\left(x-\frac{{x}^{\prime}}{2}\right)exp(-i2\pi \nu {x}^{\prime})\mathrm{d}{x}^{\prime}.$$
(4)
$${\mathrm{SW}}_{B}(x,\nu )=\left\{\begin{array}{ll}1& \u3008W(x,\nu )\u3009{W}_{\mathrm{thresh}}\\ 0& \mathrm{otherwise}\end{array}\right..$$
(5)
$$\mathrm{SW}(x,\nu )={S}_{T}{\mathrm{SW}}_{B}(x,\nu ),$$
(6)
$$\mathrm{Total}\mathrm{energy}=\iint {\mathrm{SW}}_{B}(x,\nu )W(x,\nu )\mathrm{d}x\mathrm{d}\nu =\iint \mathrm{SW}(x,\nu )\mathrm{d}x\mathrm{d}\nu .$$
(7)
$${S}_{T}=\frac{\iint {\mathrm{SW}}_{B}(x,\nu )W(x,\nu )\mathrm{d}x\mathrm{d}\nu}{\iint {\mathrm{SW}}_{B}(x,\nu )\mathrm{d}x\mathrm{d}\nu}.$$
(8)
$${N}_{\mathrm{OUT}}\u2a7d{N}_{\mathrm{IN}}.$$
(9)
$${N}_{\mathrm{OUT}}={N}_{\mathrm{IN}}.$$
(10)
$$\mathrm{SWI}(x,\nu )\subset \mathrm{SWY}(x,\nu ).$$
(11)
$$[\mathrm{area}(\mathrm{SWI})=]{N}_{\mathrm{signal}}\u2a7d{N}_{\mathrm{system}}[=\mathrm{area}(\mathrm{SWY})].$$
(12)
$$\mathrm{SWI}(x,\nu )\not\subset \mathrm{SWY}(x,\nu ).$$
(13)
$${N}_{\mathrm{signal}}\u2a7d{N}_{\mathrm{system}},\hspace{1em}\hspace{1em}\mathrm{SWI}(x,\nu )\not\subset \mathrm{SWY}(x,\nu ).$$
(14)
$$\mathrm{SW}(x,{\nu}_{x},y,{\nu}_{y},t,\mathrm{\lambda},\mathrm{POL},\dots ).$$
(15)
$$\varphi =\frac{\pi}{2}P.$$
(16)
$$f(x)=\frac{1}{{f}^{*}(0)}\int W({x}^{\prime}/2,\nu )exp(i2\pi \nu {x}^{\prime})\mathrm{d}\nu .$$
(17)
$$\iint W(x,\nu )\mathrm{d}x\mathrm{d}\nu =\iint \mathrm{SW}(x,\nu )\mathrm{d}x\mathrm{d}\nu .$$
(18)
$$\int W(x,\nu )\mathrm{d}\nu =|f(x){|}^{2},$$
(19)
$$\iint {\mathrm{SW}}_{\mathrm{IN}}(x,\nu )\mathrm{d}x\mathrm{d}\nu =\iint {\mathrm{SW}}_{\mathrm{OUT}}(x,\nu )\mathrm{d}x\mathrm{d}\nu .$$