Figure 1

Illustration of the effect of distortions on geodesics. The similarity metric of diffraction snapshots can be distorted, deviating from the round metric induced by $S^3$. (a) shows a 3D slice through the heavily distorted three-sphere $S^3$ and a geodesic line (red). Mapping back to $S^3$ (b) shows that even under heavy distortion the geodesic is approximately preserved, i.e., it is a great circle on $S^3$.Reuse & Permissions

Figure 2

Illustration of the geodesic and in-plane rotations algorithm (gipral). (a) Geometry of diffraction experiment. The triad (red) denotes the object and its orientation. In-plane rotations correspond to a rotation of both object and diffraction pattern around the x-ray axis. [(b) and (c)] Illustration of combined geodesic and in-plane rotations. For clarity, only the dark gray (red) object $O\left(P_1\right)$ corresponding to diffraction pattern $P_1$ is rotated in-plane around the x-ray beam [light gray (yellow line)]. The geodesic sequences connecting the orientations of each in-plane rotation of $O\left(P_1\right)$ [dark gray (red)] to $O\left(P_2\right)$ [dark gray (blue)] are shown in light gray (green). (b) In-plane rotations of $O\left(P_1\right)$ with arbitrary orientation (see www.gipral.org for an interactive illustration). (c) Maximum separation between the object orientations $O\left(P_1\right)$ and $O\left(P_2\right)$ leads to full coverage of SO(3). The orientations of the red and the blue arrows constitute the poles on $S^3$. Note that only its projection $S^2$ can be shown here.Reuse & Permissions

Figure 3

Relations between diffraction patterns and rotation operators $\mathit{R}$. The Solid vertical arrow describes geodesic out-of-plane operations (rotations about $\vec{\mathbf{a}}$), horizontal operators describe in-plane operations (rotations about $\vec{\mathbf{c}}$). Note that in the special case $\theta ={180}^{\circ }$ the dashed arrows are out-of-plane rotations, too.Reuse & Permissions

Figure 4

Application of the gipral method to experimental XFEL snapshot diffraction data of an ironoxide nanoparticle, dubbed nanorice (a) gipral output: orientation map of a subset of diffraction images collected at LCLS. Horizontal: in-plane; Vertical: geodesic out-of-plane rotations. (b) Slices through the 3D diffraction volume, assembled from the oriented 2D diffraction patterns. The color bar shows the the intensities in arbitrary digital units. (c) Reconstructed 3D electron density of a medium-sized 150-nm-long nanorice particle of the distribution shown in the TEM micrograph (inset). The bounding box depicts the oversampling volume. The magnified (red) object shows an isosurface representation.Reuse & Permissions

Figure 5

Fundamental cell in Rodrigues space for icosahedral symmetry (dodecahedron). The solid part can be reached by a combination out-of-plane geodesics and in-plane rotations of $P_1$ and $P_2$ in one go. Further iterations can then fill the whole fundamental cell. Due to the nonlinear deformation of Rodrigues space the gaps at the corner of the fundamental cell correspond to very small angular regions. In fact, the blue region corresponds to 92% of all possible orientations.Reuse & Permissions

Figure 6

Mirror symmetry together with a projection operation leads to symmetry in rotation such that rotations in positive and negative direction yield the same projection. The diffraction pattern based geodesic sequence depicted by red arrows on the right side is equivalent to the blue sequence and it therefore stops at the mirror plane. It does not continue to the blue arrows as it would without symmetry.Reuse & Permissions