Optimized algorithm improves coherent X-ray diffraction imaging at ID10
Scientists have optimized the numerical algorithms used in coherent diffraction imaging at beamline ID10, enabling visualization of microscopic specimens at nanometre resolution. The improved algorithm significantly reduces image artefacts caused by incomplete datasets, providing a practical tool for enhanced imaging.
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High-resolution X-ray imaging is possible without using objective lenses by exploiting coherent beam illumination combined with numerical algorithms for image reconstruction [1]. In coherent diffraction imaging (CDI), the diffraction pattern encodes the electron density of a specimen. An iterative phase retrieval algorithm is applied to oversampled diffraction patterns to solve the phase problem and reconstruct a real-space image [2,3]. Although effective for simulated and high-quality data, this method often faces challenges with incomplete datasets.
The main sources of missing information in measured datasets are the use of a beamstop to block an intense direct beam, blind gaps in a detector, and a missing wedge in tomographic measurements due to restricted sample rotation during data acquisition, for example (Figure 1). When images are reconstructed from such data, they often contain artefacts that hinder accurate analysis and interpretation. To address this, a practical and effective approach has been proposed to mitigate reconstruction artefacts.
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Fig. 1: Slices through the measured 3D coherent diffraction volume. White regions indicate areas of incomplete data caused by detector gaps, the beamstop, or due to a missing wedge.
The enhanced algorithm improves the quality of the reconstructed images compared to the standard approach [2]. When the standard algorithm is applied to datasets with missing information, the resulting images frequently contain artefacts (Figure 2a). This occurs because no data exist in reciprocal space that could constrain the Fourier amplitudes and avoid image degradation.
The new approach incorporates an optimization step into the standard algorithm to converge towards a solution with minimum total variation (TV). The improvements are notable: images appear clearer, and fine details are more easily discernible (Figure 2b). Additionally, unconstrained Fourier amplitudes in reciprocal space align more effectively with the measured data (Figure 2b). These findings demonstrate that the algorithm is particularly effective in addressing the missing wedge problem encountered in tomography.
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Fig. 2: Reconstructed images obtained using the standard ERHIO (Error Reduction Hybrid Input-Output) algorithm (a) and with the TV minimization step (b). Top row: orthogonal slices through the 3D electron density. Middle row: 2D projected views of the sample. Bottom row: slices in Fourier space. Red shaded regions indicate areas of missing data.
The algorithm seeks a solution that both matches the measured data and minimizes total variation. This makes it more practical than methods based on neural networks [4], which require data for training, or TV regularization that relies on well-tuned regularization parameters [5]. Consequently, it has broad applicability for iterative phase retrieval from incomplete datasets.
Thanks to its simplicity, the new approach can be seamlessly integrated into the PyNX toolkit [6] for use in Bragg CDI and similar techniques. With the ESRF’s Extremely Brilliant Source now delivering an intense X-ray beam with an unprecedented degree of coherence, the TV minimization approach provides a powerful boost for high-resolution coherent diffraction imaging.
Principal publication and authors
Reduction of artifacts associated with missing data in coherent diffractive imaging, E. Malm (a) & Y. Chushkin (b), J. Synchrotron Radiat. 32, 210-216 (2025); https://doi.org/10.1107/S1600577524010956
(a) MAX IV Laboratory, Lund University, Lund (Sweden)
(b) ESRF
References
[1] H.N. Chapman et al., Nat. Photon. 4, 833-839 (2010).
[2] J.R. Fienup, Appl. Opt. 21, 2758-2769 (1982).
[3] J. Miao et al., Nature 400, 342-344 (1999).
[4] A. Bellisario et al., J. Appl. Cryst. 55, 122-132 (2022).
[5] Y. Yokoyama et al., J. Phys. Soc. Jpn. 91, 034701 (2022).
[6] V. Favre-Nicolin et al., J. Appl. Cryst. 53, 1404-1413 (2020).