2.1.1. bcdi

Carnis et al. (2021a) tackled the pre-processing and post-processing of BCDI data (see Fig. 1).

Data pre-processing aims to improve the quality of the phase retrieval by optimizing the size and content of the 3D array used as input. Aside from the specific details of the experimental setup (diffractometer and setup geometry, detector type, file system), the majority of the pre-processing pipeline is actually beamline independent. On the basis of this observation, bcdi leverages inheritance and transforms the raw data to a common data format used for phase retrieval. This frees the user from having to learn or remember the technical details for each beamline and sets a common strategy across beamlines. New Beamline objects can be created for coherent beamlines which are unsupported as of yet.

The minimal processing consists of loading the raw data and stacking it together as an input for the phase retrieval. Intermediate optional steps can be added via a YAML (YAML Ain't Markup Language) configuration file. More complete pre-processing involves loading the data (with flatfield correction and masking of damaged detector pixels and detector gaps), normalizing each frame by a monitor (sensor at the beamline measuring the incoming X-ray flux), binning pixels, centering the Bragg peak and cropping the data set to a meaningful size, and interpolating missing intensities in damaged pixels. A mask is automatically created, starting from a 2D array representing the detector. It can be modified interactively to mask the parasitic scattering intensities coming from neighboring crystals via the interactive user interface. Hot pixels (saturated) or pixels with an abnormal statistical behavior are automatically masked during data loading. The result is a 3D array of the same shape as the intensity. Interpolating the data stack from the detector frame to an orthogonal frame (also called a geometric transformation) may be useful, for example, to compare Bragg peaks from different reflections. This geometric transformation can be realized using either the transformation matrix (Pfeifer, 2005) or the existing xrayutilities package (Kriegner et al., 2013) depending on which reference basis is needed.

Post-processing regroups methods applied to the complex output of the phase retrieval. The data can be interpolated in an orthogonal frame using the transformation matrix if still in the detector frame (geometric transformation not applied during pre-processing). After phase unwrapping, a refraction correction is optionally applied, and the phase ramp and phase offset are removed. At this point the displacement and the strain component are calculated from the phase. Note that at least three non-coplanar reflections are needed to derive the complete displacement field.

2.1.2. PyNX

PyNX2011 (Favre-Nicolin et al., 2011) is a toolkit with assorted Python modules and command-line scripts which can be used for the analysis of coherent X-ray imaging data, including phase retrieval (see Fig. 1). All calculations can be executed and distributed on multiple graphical processing units (GPUs) for accelerated computing using MPI.

Some facilities have started to make computational resources available to their users; those machines are an ideal environment for PyNX which is now available through SLURM at the ESRF and GRADES at the Optimized Light Source of Intermediate Energy of LURE (SOLEIL).

2.2.1. The Jupyter Notebook environment

The Jupyter Notebook environment (Perez & Granger, 2007; Kluyver et al., 2016) was chosen for its versatile, user-friendly and browser-based interface.

To simplify the data analysis pipelines in fourth-generation synchrotrons, it is of critical importance to offer the possibility for external users to analyze the collected data remotely with access to computational environments

Jupyter Notebook has proven to be an effective tool for the analysis of synchrotron data, both in terms of the GUI (Martini et al., 2020; Simonne et al., 2020) and in terms of supporting scientific communities looking for high-performance frameworks (Yin et al., 2017; Glick & Mache, 2018; Milligan, 2018; Stubbs et al., 2020; Parkinson et al., 2020). Moreover, computational environments and resources can be accessed from any computer via JupyterHub, a specific interface for computing clusters. JupyterHub offers the possibility to proceed to heavy computations without relying on specific hardware (e.g. GPUs) mandatory for accelerated phase retrieval with PyNX.

The Notebook can be used to take notes on the experiment or to create custom Python functions. They can be shared in .ipynb format or as .pdf documents. Gwaihir transforms each data set into a structured Python object that can be accessed and manipulated in the Notebook for e.g. personalized figures, parameter tests etc.

Moreover, researchers can create their own work spaces from shared resources, with direct access to tailored computational environments, without having to install multiple software products, keeping in mind that the use of GPUs for script optimization is far from accessible. Thus, system administrators can efficiently manage complex environments accessible to all users. Finally, remotely accessing the data avoids data storage issues, which can quickly become problematic with current CDI experiments.

2.2.2. Interactive data analysis/visualization

Interactive data analysis/visualization relies on the ipywidgets library.

Widgets are represented in the back-end by a single object linked to a single parameter. The front-end relies on JavaScript code; each time a widget is displayed, a new representation of that same object is created. The widget style, orientation and layout attributes can be edited to customize the final window; e.g. the layout attribute exposes a number of properties that impact how widgets are laid out, such as height and width.

During manipulation of the GUI, it is possible to quickly produce a final result owing to these intuitive and interactive widgets, whose values are passed as arguments to the data analysis functions (Fig. 4). For example, to designate the type of detector used, a value must be selected in a dropdown list that comprises the following options: Eiger2M, Maxipix, Eiger4M, Merlin, Timepix. These correspond to the different detectors currently supported in the bcdi package.

Figure 4 Widgets can be used to select folders and files and to tune the values of input parameters in functions. The selection of the data array, the color map and the contour of the resulting 3D object are performed through different widgets.

The final interface is created by setting the different widgets on a grid, with the output of different functions printed below.

Jupyter natively offers multiple options for interactive data plotting. Most of the figures displayed in the GUI are based on the matplotlib package (Hunter, 2007). For example, it is possible to select a list of 3D complex data arrays and visualize 2D slices in each dimension of their amplitude or phase, with different color maps (diverging, sequential, cyclic) and scale (linear, logarithmic).

Moreover, an interactive 3D visualization tool is provided which relies on the ipyvolume library (Breddeld, 2021), itself built on top of ipywidgets, and is specifically designed to quickly render large 3D data arrays. A concrete application of volume rendering is shown in Fig. 4, in which a specific contour of a 3D array is represented.

In the specific case of single volume rendering for the reconstructed object, the object surface is defined by a threshold of its maximum density (Fig. 4). Finally, the object surface can be color-mapped with the values of the displacement and strain retrieved during the data analysis (see Fig. 6).

To further interact with the figures (e.g. zoom, set the color bar range etc.), tools were implemented that rely on Bokeh (Bokeh Development Team, 2018), a Python library that transforms figures into interactive web pages (Fig. 5) that can also be displayed in Jupyter Notebook.

Figure 5 Detector images can be viewed interactively with Bokeh, to zoom in on the data, and visualize, for example, the intensity collected on each pixel.

In summary, the GUI regroups data reduction, analysis and visualization tools in the Jupyter Notebook interface via interactive methods.

3.1.1. Pre-processing

First, the correct beamline and instrumental parameters must be selected (e.g. sample–detector distance, probing energy, detector pixel size etc.), usually constant throughout the experiment. For example, the orthogonalization tab regroups the parameters needed to properly set up the transformation matrix between the sample frame and, for example, the laboratory frame. By choosing to transform the final object in a common, orthogonal frame, it becomes easier to compare the evolution of the object when probing different Bragg reflections (Lauraux et al., 2021).

Once the raw data have been collected, different pre-processing parameters can be modified to optimize the collected 3D diffraction intensity, e.g. the size and center of the array to analyze (either fixed manually as the center of the Bragg peak, if known, or determined as the center of mass of the 3D array during the analysis). The array can also be cropped to reduce its size, removing the points furthest from the Bragg peak where the signal-to-noise ratio is too low (Fig. 7).

It is important to create a detector mask prior to the experiment to correct the raw data for hypothetical hot pixels and background. Moreover, it is also possible to correct the raw data frame by frame for spurious data which taint the diffraction pattern. For example, it is possible, while recording the 3D diffracted intensity of a given reflection, to have signal coming either from the substrate or from neighboring objects that will be summed to the probed object intensity.

Finally, it is possible to normalize the raw data by an intensity monitor, or compute the scattering vector q of the measurement, from the instrumental geometry and parameters.

3.1.2. Phase retrieval

The diffracted intensity I(q) collected by the detector is proportional to the squared modulus of the structure factor F(q) [equations (1) and (2)]:

and

where f_j(q_hkl) is the atomic form factor of atom j, q_hkl is the scattering vector and is the phase lost during the measurement; hkl correspond to the Miller indices of the crystallographic planes probed by the incoming beam.

The phase of the Bragg electronic density (Fig. 7) Φ_hkl is proportional to the scalar component of the displacement u(r) that is parallel to q_hkl. A phase value of π is equivalent to a displacement from the equilibrium position of half the lattice spacing in the direction of q_hkl.

To retrieve the phase from the diffracted intensity, PyNX uses iterative algorithms (see Section 2.1.2). New methods such as convolution neural networks are under study and have started to show some encouraging results, but are not yet sufficiently robust, particularly in the case of strained particles (Cherukara et al., 2018b; Chan et al., 2021; Wu et al., 2021).

The parameters necessary for phase retrieval can be modified in the GUI (Fig. 8), such as the support threshold, the number of iterations for each algorithm, the object initialization procedure (square, sphere, auto-correlation) and so on. Each parameter is detailed in the GUI and also discussed in the literature (Fienup, 1982, 1978; Marchesini, 2007; Favre-Nicolin et al., 2020a). Initial guesses are given for each parameter but must be refined by the user to optimize the results.

Figure 8 Phase retrieval tab in Gwaihir. Parameters are separated into groups (files, support, point-spread function, algorithms etc.) and detailed in the README tab. The object can be reconstructed through a batch job, submitted to the computing cluster in the back-end, or with operators that will plot the evolution of the reconstruction object in the Notebook.

State-of-the-art GPUs available at large-scale facilities where data collection occurs enable almost real-time data reduction. This results in quick visualization of the amplitude and phase of the probed object, offering the possibility to synchrotron users to optimize their data acquisition during the experiment. Such live feedback is critical to the success of an experiment and provides broader opportunities in the type of experiments that can be carried out.

Recent updates of PyNX, including mathematical operators (Favre-Nicolin et al., 2020a) which represent most of the reconstruction operations traditionally used in phase retrieval (Gerchberg & Saxton, 1972; Fienup, 1978; Marchesini, 2007), laid the foundation for quick and interactive phase retrieval in Gwaihir.

It is possible to refine the input parameters directly in the GUI by visualizing their impact on the reconstructed object. This allows the refinement of the phase retrieval input parameters before submitting a batch job, which will spawn a sub-process on the computing cluster for phase retrieval. With a single click, several dozens of solutions can be computed in a matter of minutes on computing clusters. On a personal computer, one can use operators in the Notebook or spawn a job locally, yielding the same results with different execution times.

With well-tuned parameters and high-quality data sets, phase retrieval converges towards the same solution, but with minor differences between each reconstructed object, related to the phase retrieval process.

Gwaihir provides a wide range of selection criteria to find the best solution. Each reconstructed object has a list of final attributes that can be used as criteria for selection, such as the free log-likelihood (Favre-Nicolin et al., 2020b) or the standard deviation of the modulus of the reconstructed object. In the case of crystallographic defects, specific metrics (Chi, Sharp, Max Volume etc.) were derived that perform best depending on the type of object defect (Ulvestad et al., 2017).

Following the selection criteria, it is possible to quickly identify the solutions of poor quality that must be excluded to create a set of best solutions. PyNX then offers a method that merges this set into a single solution by computing eigen-vectors for the selected solutions (Favre-Nicolin et al., 2020b). An alternative approach also available in Gwaihir is to take the average of the best solutions (see e.g. Ulvestad et al., 2014).

3.1.3. Post-processing

Once the solution with the best figure of merit is selected, it is possible to use bcdi scripts to process the phase of the object. The phase origin can first have an impact when comparing the lattice displacement between different reconstructions. In the case of weak strain, it can be sufficient to consider the center of mass of the object as the origin of the phase. However, this can become quite complex in the case of defects or defaults in the object. Special methods are defined to target this issue in bcdi. For example, Guizar-Sicairos et al. (2011) and Hofmann et al. (2020) proposed a convenient method for the numerical calculation of phase gradients in the presence of phase jumps. More details and guidelines are given by Carnis et al. (2019, 2021a) as well as in the README tab of the GUI. Different methods can be tested and their results directly compared via the visualization tab.

3.1.4. Facet analysis

The surface in BCDI corresponds to the surface voxel layer defined by a threshold of its maximum density. Guidelines on how to select this threshold are given by Carnis et al. (2019), the objective being to correctly select the surface voxels of the object.

Once the threshold for the iso-surface is selected, it is possible to visualize a contour of the object directly in the GUI (Fig. 4) with specialized solutions such as Paraview (Ahrens et al., 2005) (Fig. 9). Crystallographic facets can be identified on the surface of the reconstructed object when studying faceted objects with a highly coherent beam (Richard et al., 2018), allowing in-depth studies of facet-dependent strain and displacement. Lattice strain and displacement are key factors in fields such as heterogeneous catalysis (Ulvestad et al., 2016; Kim et al., 2018; Fernández et al., 2019; Passos et al., 2020; Carnis et al., 2021b) or electrochemistry (Vicente et al., 2021).

Figure 9 In this screenshot taken from Paraview, the particle facets are identified by the white contours. The coloring represents the phase of the reconstructed object, proportional to the lattice displacement. This particle has a diameter of about 800 nm.

Retrieving the facets can be achieved by analyzing the probability distributions of the orientations of triangle normals on a mesh representation of the object (Grothausmann et al., 2012). This method is used in the Paraview plugin FacetAnalyser (Grothausmann & Beare, 2015), and yields a list of features detailed in Table 1. A similar method is used in bcdi.

In this example, we use the FacetAnalyser plugin (Fig. 9). Note that the edges and corners, sites of particular interest for heterogeneous catalysis (Taylor & Armstrong, 1925), are also retrieved together as voxels not belonging to any facets. The result depends on the algorithm input parameters, such as the minimum relative facet size, the angular acceptance for the facet normals etc.

The .vtk output file of FacetAnalyser can be opened and manipulated in the GUI. Each facet is assigned a crystallographic orientation that allows in fine facet-dependent structural analysis. The facets can be visualized individually in three dimensions in the GUI, together with the edges and corners, to help understand the particle shape.