## Abstract

In our previous paper [J. Opt. Soc. Am. A **15**, 2849 (1998)], we investigated the behavior of the hybrid input–output algorithm (HIO) used for phase retrieval and showed that the HIO with an infinitesimally small feedback parameter has two abilities that are extremely important in finding a solution: the ability to locate an output-stagnation object and the ability to emerge from an output-stagnation object if it is not a solution. The behavior of the HIO is analyzed further, and an additional aspect is demonstrated that is also important in finding a solution. That is, outside the support of the input object of the HIO, the output-object values outside the support that have been produced so far accumulate so that the probability is conjectured to be extremely low that the HIO will fall into a periodic state. On the other hand, it is also demonstrated that this aspect of the HIO is not usually effective in the combination algorithm composed of the HIO and the error-reduction algorithm. To give credibility to our arguments, we show results that were obtained by reexamining the examples adopted in Seldin and Fienup’s numerical experiments [J. Opt. Soc. Am. A **7**, 412 (1990)].

© 1999 Optical Society of America

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### Equations (17)

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(1)
$$F(u,v)=|F(u,v)|exp[i\varphi (u,v)]=\sum _{x=0}^{M-1}\sum _{y=0}^{N-1}f(x,y)exp[-i2\pi (\mathit{ux}/M+\mathit{vy}/N)].$$
(3)
$$=\left\{\begin{array}{ll}{g}_{m}^{\prime}(x,y)& (x,y)\in D\\ {g}_{m}(x,y)-\beta {g}_{m}^{\prime}(x,y)& (x,y)\notin D\end{array},\right.$$
(4)
$${g}_{m+1}(x,y)=\left\{\begin{array}{ll}{g}_{m}^{\prime}(x,y)& (x,y)\in D\\ 0& (x,y)\notin D\end{array}.\right.$$
(5)
$${\u220a}_{\mathit{Fm}}^{2}=(\mathit{MN}{)}^{-1}\sum _{u}\sum _{v}[|{G}_{m}(u,v)|-|F(u,v)|{]}^{2}$$
(6)
$${\u220a}_{\mathit{om}}^{2}=\sum _{(x,y)\notin D}|{g}_{m}^{\prime}(x,y){|}^{2},$$
(7)
$${g}_{m+1}(x,y)=\left\{\begin{array}{ll}{g}_{m}^{\prime}(x,y)& (x,y)\in S\\ {g}_{m}(x,y)-\beta {g}_{m}^{\prime}(x,y)& (x,y)\notin S\end{array},\right.$$
(8)
$${g}_{m+1}(x,y)=\left\{\begin{array}{ll}{g}_{m}^{\prime}(x,y)& (x,y)\in S\\ 0& (x,y)\notin S\end{array},\right.$$
(9)
$${g}_{m+1}^{\mathrm{NS}}(x,y)={g}_{m}^{\mathrm{NS}}(x,y)-\beta {g}_{m}^{\prime \mathrm{NS}}(x,y),$$
(10)
$${g}_{m}^{\mathrm{NS}}(x,y)=\left\{\begin{array}{ll}0,& (x,y)\in S\\ {g}_{m}(x,y),& (x,y)\notin S\end{array}\right..$$
(11)
$${g}_{m+1}^{\mathrm{NS}}(x,y)={g}_{0}^{\mathrm{NS}}(x,y)-\beta \sum _{k=0}^{m}{g}_{k}^{\prime \mathrm{NS}}(x,y).$$
(12)
$${g}_{m}^{\mathrm{NS}}(x,y)={g}_{n}^{\mathrm{NS}}(x,y)$$
(13)
$$f=\left[\begin{array}{ccc}0.476& 3.244& 1.379\\ 1.659& 2.939& 1.102\end{array}\right],$$
(14)
$${g}_{s1}=\left[\begin{array}{ccc}0.353& 2.143& 3.172\\ 0.684& 2.470& 1.976\end{array}\right]$$
(15)
$${g}_{s2}=\left[\begin{array}{ccc}0.266& 1.876& 2.971\\ 0.746& 2.711& 2.222\end{array}\right].$$
(16)
$${E}_{\mathit{om}}={\left[\frac{{\u220a}_{\mathit{om}}^{2}}{(\mathit{MN}{)}^{-1}{\displaystyle \sum _{u}}{\displaystyle \sum _{v}}|F(u,v){|}^{2}}\right]}^{1/2},$$
(17)
$${W}_{m}={\left\{\sum _{x}\sum _{y}[{g}_{m}^{\mathrm{NS}}(x,y){]}^{2}\right\}}^{1/2},$$