## Abstract

In general, the problem of reconstructing an object from its
Fourier modulus has no solution when the Fourier modulus is
contaminated by noise. Therefore a quasi solution, which we call
the ideal estimate of the object to be reconstructed, is defined here
based on the concept of territories of the convergence objects of the
error-reduction algorithm, and a method that attempts to find that
solution is presented. Keeping in mind that the ideal estimate is
one of the output-stagnation objects of the hybrid input–output
algorithm, we modify the hybrid input–output algorithm so that the
output-stagnation objects can be located even when the value of the
feedback parameter is not infinitesimally small, and this modified
algorithm is combined with the hybrid input–output algorithm
itself. The results of computer simulations carried out to test the
performance of the proposed method are shown.

© 1999 Optical Society of America

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### Equations (21)

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(1)
$$F\left(u,v\right)=|F\left(u,v\right)|exp\left[i\mathrm{\varphi}\left(u,v\right)\right]=\sum _{x=-M/2+1}^{M/2}\sum _{y=-N/2+1}^{N/2}f\left(x,y\right)\times exp\left[-i2\mathrm{\pi}\left(\mathit{ux}/M+\mathit{vy}/N\right)\right],$$
(2)
$$f\left(x,y\right)={\left(\mathit{MN}\right)}^{-1}\sum _{u=-M/2+1}^{M/2}\sum _{v=-N/2+1}^{N/2}F\left(u,v\right)\times exp\left[i2\mathrm{\pi}\left(\mathit{ux}/M+\mathit{vy}/N\right)\right],$$
(3)
$${g}_{m+1}\left(x,y\right)=\left\{\begin{array}{ll}{g}_{m}^{\prime}\left(x,y\right),& \left(x,y\right)\in D\\ 0,& \left(x,y\right)\notin D\end{array}\right.,$$
(4)
$${g}_{m+1}\left(x,y\right)=\left\{\begin{array}{ll}{g}_{m}^{\prime}\left(x,y\right),& \left(x,y\right)\in D\\ {g}_{m}\left(x,y\right)-\mathrm{\beta}{g}_{m}^{\prime}\left(x,y\right),& \left(x,y\right)\notin D\end{array}\right.,$$
(5)
$$\mathrm{\u220a}_{\mathit{Fm}}{}^{2}={\left(\mathit{MN}\right)}^{-1}\sum _{u}\sum _{v}{\left[|{G}_{m}\left(u,v\right)|-|F\left(u,v\right)|\right]}^{2},$$
(6)
$$\mathrm{\u220a}_{\mathit{om}}{}^{2}=\sum _{\left(x,y\right)}\sum _{\notin D}|{g}_{m}^{\prime}\left(x,y\right){|}^{2},$$
(7)
$${N}_{R}={\left\{\frac{{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}{\left[|{F}_{n}\left(u,v\right)|-|F\left(u,v\right)|\right]}^{2}}{{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}|F\left(u,v\right){|}^{2}}\right\}}^{1/2}\times 100\%,$$
(8)
$${G}_{m+1}\left(u,v\right)={G}_{m}\left(u,v\right)+\mathrm{DFT}\left\{{\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}\right\}-i\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right){|}^{2}}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}\right]{G}_{m}^{\prime}\left(u,v\right)-\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right)|}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial |G\prime \left(u,v\right)|}\right|}_{G\prime ={G}_{m}^{\prime}}\right]{G}_{m}^{\prime}\left(u,v\right),$$
(9)
$${\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}=\left\{\begin{array}{ll}-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}},& \left(x,y\right)\in D\\ 0,& \left(x,y\right)\notin D\end{array}\right.$$
(10)
$${G}_{m+1}\left(u,v\right)={G}_{m}\left(u,v\right)+\mathrm{DFT}\left\{{\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}\right\}-i\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right){|}^{2}}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}\right]{G}_{m}^{\prime}\left(u,v\right),$$
(11)
$${G}_{m+1}\left(u,v\right)=G_{m}{}^{\mathit{ND}}\left(u,v\right)+{G}_{m}^{\prime D}\left(u,v\right)+i\mathrm{\beta}\frac{1}{|F\left(u,v\right){|}^{2}}\mathrm{Im}\left\{{G}_{m}^{\prime}\left(u,v\right)\times \left[{G}_{m}^{\prime \mathit{ND}}\left(u,v\right)\right]*\right\}{G}_{m}^{\prime}\left(u,v\right),$$
(12)
$$g_{m}{}^{\mathit{ND}}\left(x,y\right)=\left\{\begin{array}{ll}0,& \left(x,y\right)\in D\\ {g}_{m}\left(x,y\right),& \left(x,y\right)\notin D\end{array}\right.,$$
(13)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}=2\left[{g}_{m}\left(x,y\right)-{g}_{m}^{\prime}\left(x,y\right)\right]=0,\left(x,y\right)\in D,$$
(14)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}=-\frac{4}{\mathit{MN}}\mathrm{Im}\left\{{G}_{m}^{\prime}\left(u,v\right)\times \left[{G}_{m}^{\prime \mathit{ND}}\left(u,v\right)\right]*\right\}=0,$$
(15)
$${G}_{m+k}\left(u,v\right)={G}_{m}\left(u,v\right)-k\mathrm{\beta}{G}_{s}^{\prime \mathit{ND}}\left(u,v\right)$$
(16)
$$|{G}_{m}\left({u}_{0},{v}_{0}\right)|{k}_{0}\mathrm{\beta}|{G}_{s}^{\prime \mathit{ND}}\left({u}_{0},{v}_{0}\right)|,$$
(17)
$$r\left({u}_{0},{v}_{0}\right)=\frac{|{G}_{m}\left({u}_{0},{v}_{0}\right)|}{|{G}_{s}^{\prime \mathit{ND}}\left({u}_{0},{v}_{0}\right)|}.$$
(18)
$${G}_{m+1}\left(u,v\right)={G}_{m}\left(u,v\right)-\frac{{r}_{1}+{r}_{2}}{2}{G}_{s}^{\prime \mathit{ND}}\left(u,v\right).$$
(19)
$${T}_{m}={\left[\frac{{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}{\left(\mathrm{Im}\left\{{G}_{m}^{\prime}\left(u,v\right)\left[{G}_{m}^{\prime \mathit{ND}}\left(u,v\right)\right]*\right\}\right)}^{2}}{{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}|{G}_{m}^{\prime}\left(u,v\right){|}^{2}{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}|{G}_{m}^{\prime \mathit{ND}}\left(u,v\right){|}^{2}}\right]}^{1/2},$$
(20)
$${E}_{\mathit{om}}={\left[\frac{\mathrm{\u220a}_{\mathit{om}}{}^{2}}{{\left(\mathit{MN}\right)}^{-1}{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}|F\left(u,v\right){|}^{2}}\right]}^{1/2}$$
(21)
$${E}_{\mathit{tm}}={\left[\frac{{\displaystyle \sum _{x}}{\displaystyle \sum _{y}}|{\tilde{g}}_{m}^{\prime}\left(x,y\right)-f\left(x,y\right){|}^{2}}{{\displaystyle \sum _{x}}{\displaystyle \sum _{y}}|f\left(x,y\right){|}^{2}}\right]}^{1/2},$$