## Abstract

In coherent lensless imaging, the presence of image sidelobes, which arise as a natural consequence of the finite nature of the detector array, was early recognized as a convergence issue for phase retrieval algorithms that rely on an object support constraint. To mitigate the problem of truncated far-field measurement, a controlled analytic continuation by means of an iterative transform algorithm with weighted projections is proposed and tested. This approach avoids the use of sidelobe reduction windows and achieves full-resolution reconstructions.

© 2008 Optical Society of America

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### Equations (9)

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(1)
$$F\left(\mathbf{u}\right)=\mid F\left(\mathbf{u}\right)\mid \mathrm{exp}\left[i\varphi \left(\mathbf{u}\right)\right]=\int f\left(\mathbf{x}\right)\mathrm{exp}(-i2\pi \mathbf{u}\cdot \mathbf{x})\mathrm{d}\mathbf{x},$$
(2)
$${G}_{k}\left(\mathbf{u}\right)=\mid {G}_{k}\left(\mathbf{u}\right)\mid \mathrm{exp}\left[i{\theta}_{k}\left(\mathbf{u}\right)\right].$$
(3)
$${G}_{k}^{\prime}\left(\mathbf{u}\right)=\mid F\left(\mathbf{u}\right)\mid \mathrm{exp}\left[i{\theta}_{k}\left(\mathbf{u}\right)\right],$$
(4)
$${E}^{2}=\frac{\sum _{\mathbf{x}\notin S}{\mid {g}^{\prime}\left(\mathbf{x}\right)\mid}^{2}}{\sum _{\mathbf{x}}{\mid {g}^{\prime}\left(\mathbf{x}\right)\mid}^{2}},$$
(5)
$${g}_{k+1}\left(\mathbf{x}\right)=\{\begin{array}{cc}{g}_{k}^{\prime}\left(\mathbf{x}\right),\hfill & \phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{0.3em}{0ex}}\mathbf{x}\u220aS\hfill \\ 0\hfill & \phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{0.3em}{0ex}}\mathbf{x}\notin S\hfill \end{array},$$
(6)
$${g}_{k+1}\left(\mathbf{x}\right)=\{\begin{array}{cc}{g}_{k}^{\prime}\left(\mathbf{x}\right),\hfill & \phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{0.3em}{0ex}}\mathbf{x}\u220aS\hfill \\ {g}_{k}\left(\mathbf{x}\right)-\beta {g}_{k}^{\prime}\left(\mathbf{x}\right),\hfill & \phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{0.3em}{0ex}}\mathbf{x}\notin S\hfill \end{array},$$
(7)
$${G}_{k}^{\prime}\left(\mathbf{u}\right)=W\left(\mathbf{u}\right)\mid F\left(\mathbf{u}\right)\mid \mathrm{exp}\left[i{\theta}_{k}\left(\mathbf{u}\right)\right]+[1-W\left(\mathbf{u}\right)]{G}_{k}\left(\mathbf{u}\right),$$
(8)
$${\epsilon}_{k}^{2}=\underset{\alpha ,{\mathbf{x}}_{\mathbf{0}}}{\mathrm{min}}\left[\frac{\sum _{\mathbf{x}}{\mid f\left(\mathbf{x}\right)-\alpha {g}_{k}(\mathbf{x}-{\mathbf{x}}_{\mathbf{0}})\mid}^{2}}{\sum _{\mathbf{x}}{\mid f\left(\mathbf{x}\right)\mid}^{2}}\right],$$
(9)
$${W}_{\eta}\left(\mathbf{u}\right)=1-\eta +\eta \frac{\mid F\left(\mathbf{u}\right)\mid}{{\mathrm{max}}_{\mathbf{u}}\mid F\left(\mathbf{u}\right)\mid}$$