## Abstract

The hybrid input-output algorithm (HIO) used for phase retrieval is in many cases combined with the error-reduction algorithm (ER) to attempt to stabilize the HIO. However, in our previous paper [J. Opt. Soc. Am. A **16**, 2163 (1999)], it was demonstrated that this combination makes it more likely that the resultant algorithm will fall into a periodic state before reaching a solution because the values of the input object outside the support, which is imposed as the object-domain constraint, are set to be zero in the intervals in which the ER is implemented. This paper deals with this problem inherent in the combination algorithm. The converging part of the HIO (CPHIO), which is an algorithm we previously developed [J. Opt. Soc. Am. A **15**, 2849 (1998)], can be thought of as an extension of the ER for the case in which the input object can have nonzero values outside the support. Keeping this in mind, the algorithm is then constructed by combining the HIO with the CPHIO instead of with the ER. The computer simulation results that demonstrate the effectiveness of the proposed algorithm are given.

© 2002 Optical Society of America

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### Equations (34)

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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)
$$f\left(x,y\right)={\left(\mathit{MN}\right)}^{-1}\sum _{u=-M/2+1}^{M/2}\sum _{v=-N/2+1}^{N/2}\times F\left(u,v\right)exp\left[i2\mathrm{\pi}\left(\mathit{ux}/M+\mathit{vy}/N\right)\right],$$
(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)
$$\mathrm{\u220a}_{\mathit{Fm}}{}^{2}={\left(\mathit{MN}\right)}^{-1}\sum _{u}\sum _{v}{\left[|{G}_{m}\left(u,v\right)|-|F\left(u,v\right)|\right]}^{2},$$
(5)
$$\mathrm{\u220a}_{\mathit{om}}{}^{2}=\sum _{\left(x,y\right)}\sum _{\notin D}|{g}_{m}^{\prime}\left(x,y\right){|}^{2},$$
(6)
$$\mathrm{\u220a}_{\mathit{Fm}}{}^{2}=\sum _{x}\sum _{y}|{g}_{m}\left(x,y\right)-{g}_{m}^{\prime}\left(x,y\right){|}^{2}.$$
(7)
$$\mathrm{\u220a}_{F}{}^{2}=\sum _{x}\sum _{y}|g\left(x,y\right)-g\prime \left(x,y\right){|}^{2}.$$
(8)
$$\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}=2\left[g\left(x,y\right)-g\prime \left(x,y\right)\right].$$
(9)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}=2\left[{g}_{m}\left(x,y\right)-{g}_{m}^{\prime}\left(x,y\right)\right]$$
(10)
$${g}_{m+1}\left(x,y\right)=\left\{\begin{array}{ll}{g}_{m}\left(x,y\right)-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}& \left(x,y\right)\in D\\ 0& \left(x,y\right)\notin D\end{array}\right..$$
(11)
$${g}_{m+1}\left(x,y\right)={g}_{m}\left(x,y\right).$$
(12)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}=0\left(x,y\right)\in D$$
(13)
$${g}_{m}\left(x,y\right)={g}_{m}^{\prime}\left(x,y\right)\left(x,y\right)\in D$$
(14)
$${g}_{m}\left(x,y\right)={g}_{m}^{\prime}\left(x,y\right).$$
(15)
$$\mathrm{\u220a}_{\mathit{Fm}}{}^{2}\ge \mathrm{\u220a}_{\mathit{om}}{}^{2}\ge \mathrm{\u220a}_{F,m+1}{}^{2}\ge \mathrm{\u220a}_{o,m+1}{}^{2}.$$
(16)
$${g}_{m+1}\left(x,y\right)=\left\{\begin{array}{ll}{g}_{m}^{\prime}\left(x,y\right)& \left(x,y\right)\in D\\ {g}_{m}\left(x,y\right)-\mathrm{\beta}{g}_{m}^{\prime}\left(x,y\right)& \left(x,y\right)\notin D\end{array}\right.,$$
(17)
$${G}_{m+1}\left(u,v\right)={G}_{m}\left(u,v\right)+\mathrm{DFT}\left\{{\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}\right\}-i\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right){|}^{2}}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}\right]\times {G}_{m}^{\prime}\left(u,v\right)-\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right)|}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial |G\prime \left(u,v\right)|}\right|}_{G\prime ={G}_{m}^{\prime}}\right]\times {G}_{m}^{\prime}\left(u,v\right),$$
(18)
$${\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}=\left\{\begin{array}{cc}-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}& \left(x,y\right)\in D\\ 0& \left(x,y\right)\notin D\end{array}\right.$$
(19)
$${G}_{m+1}\left(u,v\right)={G}_{m}\left(u,v\right)+\mathrm{DFT}\left\{{\left[-\frac{1}{2}{\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}\right]}^{D}\right\}-i\mathrm{\beta}\frac{\mathit{MN}}{4|F\left(u,v\right){|}^{2}}\left[{\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}\right]{G}_{m}^{\prime}\left(u,v\right),$$
(20)
$${G}_{m+1}\left(u,v\right)={{G}_{m}}^{\mathrm{ND}}\left(u,v\right)+{{G}_{m}^{\prime}}^{\mathrm{D}}\left(u,v\right)+i\mathrm{\beta}\frac{1}{|F\left(u,v\right){|}^{2}}\mathrm{Im}\left\{{G}_{m}^{\prime}\left(u,v\right)\times \left[{{G}_{m}^{\prime}}^{\mathrm{ND}}\left(u,v\right)\right]*\right\}{G}_{m}^{\prime}\left(u,v\right),$$
(21)
$${{g}_{m}}^{\mathrm{ND}}\left(x,y\right)=\left\{\begin{array}{ll}0& \left(x,y\right)\in D\\ {g}_{m}\left(x,y\right)& \left(x,y\right)\notin D\end{array}\right.,$$
(22)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{F}{}^{2}\right)}{\partial g\left(x,y\right)}\right|}_{g={g}_{m}}=2\left[{g}_{m}\left(x,y\right)-{g}_{m}^{\prime}\left(x,y\right)\right]=0,\left(x,y\right)\in D,$$
(23)
$${\left.\frac{\partial \left(\mathrm{\u220a}_{o}{}^{2}\right)}{\partial \mathrm{\theta}\left(u,v\right)}\right|}_{G\prime ={G}_{m}^{\prime}}=-\frac{4}{\mathit{MN}}\mathrm{Im}\left\{{G}_{m}^{\prime}\left(u,v\right)\left[{{G}_{m}^{\prime}}^{\mathrm{ND}}\left(u,v\right)\right]*\right\}=0.$$
(24)
$${G}_{m+k}\left(u,v\right)={G}_{m}\left(u,v\right)-k\mathrm{\beta}{G}_{s}^{\prime \mathrm{ND}}\left(u,v\right)$$
(25)
$$|{G}_{m}\left({u}_{0},{v}_{0}\right)|{k}_{0}\mathrm{\beta}|{G}_{s}^{\prime \mathrm{ND}}\left({u}_{0},{v}_{0}\right)|$$
(26)
$$|{G}_{m}\left(-{u}_{0},-{v}_{0}\right)|{k}_{0}\mathrm{\beta}|{G}_{s}^{\prime \mathrm{ND}}\left(-{u}_{0},-{v}_{0}\right)|,$$
(27)
$${E}_{\mathit{om}}={\left[\frac{\mathrm{\u220a}_{\mathit{om}}{}^{2}}{{\left(\mathit{MN}\right)}^{-1}\sum _{u}\sum _{v}|F\left(u,v\right){|}^{2}}\right]}^{1/2}$$
(28)
$${E}_{\mathit{tm}}={\left[\frac{\sum _{\left(x,y\right)}\sum _{\in S}|{\tilde{g}}_{m}^{\prime}\left(x,y\right)-f\left(x,y\right){|}^{2}}{\sum _{x}\sum _{y}|f\left(x,y\right){|}^{2}}\right]}^{1/2},$$
(29)
$${N}_{R}={\left\{\frac{\sum _{u}\sum _{v}{\left[|{F}_{n}\left(u,v\right)|-|F\left(u,v\right)|\right]}^{2}}{\sum _{u}\sum _{v}|F\left(u,v\right){|}^{2}}\right\}}^{1/2}\times 100\%,$$
(30)
$$\left\{\left(x,y\right)|\left(x,y\right)\notin S\right\}\subseteq \left\{\left(x,y\right)|\left(x,y\right)\notin D\right\}$$
(31)
$$g_{m+1}{}^{\mathrm{NS}}\left(x,y\right)=g_{m}{}^{\mathrm{NS}}\left(x,y\right)-\mathrm{\beta}g\prime _{m}{}^{\mathrm{NS}}\left(x,y\right)$$
(32)
$$g_{m}{}^{\mathrm{NS}}\left(x,y\right)=\left\{\begin{array}{cc}0& \left(x,y\right)\in S\\ {g}_{m}\left(x,y\right)& \left(x,y\right)\notin S\end{array}\right.,$$
(33)
$$g_{m+1}{}^{\mathrm{NS}}\left(x,y\right)=g_{0}{}^{\mathrm{NS}}\left(x,y\right)-\mathrm{\beta}\sum _{k=0}^{m}g\prime _{k}{}^{\mathrm{NS}}\left(x,y\right).$$
(34)
$${{g}_{m}}^{\mathrm{NS}}\left(x,y\right)={{g}_{n}}^{\mathrm{NS}}\left(x,y\right)$$