Phase retrieval from exactly oversampled diffraction intensity through deconvolution

We have shown that, when the linear oversampling ratio $⩾2$, exactly oversampled diffraction patterns can be directly obtained from measured data through deconvolution. By using computer simulations and experimental data, we have demonstrated that exact oversampling of diffraction patterns distinctively improves the quality of phase retrieval. Furthermore, phase retrieval based on the exact sampling scheme is independent of the oversampling ratio, which can significantly reduce the radiation dosage to the samples. We believe that the present work will contribute to high-quality image reconstruction of materials science samples and biological structures using x-ray diffraction microscopy.

Figure 1

Images reconstructed from intensity-integrated diffraction patterns with (a) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=2$ and (b) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=10$, and from exactly oversampled diffraction patterns with (c) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=2$ and (d) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=10$. (e) The discrepancy error $\left(R_d\right)$ of phase retrieval as a function of the linear oversampling ratio.

Figure 2

(Color online) Images reconstructed from measured diffraction patterns with (a) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=2$ and (b) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=18$, and from exactly oversampled diffraction patterns with (c) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=2$ and (e) ${\sigma }_{\mathrm{x}}={\sigma }_{\mathrm{y}}=18$. The red, yellow and blue colors represent the high, the medium and the low electron density.