## Abstract

Given only a portion of an arbitrary finite-energy [*L*_{2}(−∞, ∞)] image and a portion of that image's spectrum, we present a closed-form method by which the entire image can be reconstructed. The algorithm is a basic augmentation of a recently proposed iterative restoration algorithm presented by Stark *et al.* [J. Opt. Soc. Am. 71, 635 (1981); Opt. Lett. 6, 259 (1981)]. Experimental results are presented.

© 1981 Optical Society of America

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### Equations (15)

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(1)
$${f}_{k}(x)=\sum _{r=0}^{k-1}{({\mathcal{\text{Q}}}_{b}\mathcal{\text{P}}a)}^{r}{m}_{1}(x),$$
(2)
$${m}_{1}(x)={\mathcal{\text{P}}}_{b}f(x)+{\mathcal{\text{Q}}}_{b}[{\mathcal{\text{Q}}}_{a}f(x)].$$
(3)
$${P}_{a}=\text{diag}\{0,0,\dots ,0,1,\dots ,1,1,1,\dots ,1,0,\dots ,0,0\},$$
(4)
$${\mathcal{\text{P}}}_{b}=1-{\mathcal{\text{Q}}}_{b}$$
(5)
$${P}_{b}={D}^{-1}BD,$$
(6)
$$B=\text{diag}\{0,0,\dots ,0,1,\dots ,1,1,1,\dots ,1,0,\dots ,0,0\}$$
(7)
$${Q}_{b}=I-{P}_{b},$$
(8)
$$\begin{array}{ll}{\mathbf{\text{f}}}_{k}\hfill & =\text{\u2211}_{r=0}^{k-1}{({Q}_{b}{P}_{a})}^{r}{\mathbf{\text{m}}}_{1}\\ & ={R}_{k}{\mathbf{\text{m}}}_{1,}\end{array}$$
(9)
$${R}_{k}=\sum _{r=0}^{k-1}{({Q}_{b}{P}_{a})}^{r}$$
(10)
$$\begin{array}{ll}R\phantom{\rule{0em}{0ex}}\hfill & \triangleq {R}_{\infty}\hfill \\ \hfill & =\text{\u2211}_{r=0}^{\infty}{({Q}_{b}{P}_{a})}^{r}\hfill \\ \hfill & ={(I-{Q}_{b}{P}_{a})}^{-1},\hfill \end{array}$$
(11)
$${\mathbf{\text{f}}}_{\infty}=R\phantom{\rule{0.1em}{0ex}}{\mathbf{\text{m}}}_{1}.$$
(12)
$${\mathbf{\text{m}}}_{1}={P}_{b}\mathbf{\text{f}}+{Q}_{b}{Q}_{a}\mathbf{\text{f}}.$$
(13)
$$R={(I-{Q}_{a}{Q}_{b})}^{-1}.$$
(14)
$$\mathbf{\text{r}}={P}_{a}\mathbf{\text{f}}+{Q}_{a}{P}_{b}\mathbf{\text{f}}$$
(15)
$${\mathbf{\text{f}}}_{\infty}=R\phantom{\rule{0.1em}{0ex}}\mathbf{\text{r}}.$$