The symmetries of image formation by scattering I Theoretical framework

Abstract

We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, and the remotest corners of the universe. Here, we deduce for the first time the fundamental symmetries underlying image formation. Intriguingly, these are similar to those of the anisotropic "Taub universe" of general relativity, with eigenfunctions closely related to spinning tops in quantum mechanics. This opens the possibility to apply the powerful arsenal of tools developed in two major branches of physics to new problems. We augment these tools with graph-theoretic means to recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. Our theoretical framework offers a potential link to recent observations on face perception in higher primates. In a later paper, we demonstrate the recovery of structure and dynamics from ultralow-signal random sightings of systems with no orientational or timing information.

Trajectories of the ribosome as a Brownian nanomachine

“…A collection of snapshots produces one or more data clouds, with discrete conformations producing separate clouds. We have previously shown that this approach is able to determine 3D structure from a collection of ultra-low signal 2D snapshots of unknown orientation (15)(16)(17)(18), and, in the presence of orientational heterogeneity and defocus variations, distinguish between discrete conformations with best-in-class performance (17).…”

“…Such a description can be achieved by one of many well-established machine-learning techniques, which also reveal the intrinsic dimensionality of the manifold, and hence the number of degrees of freedom exercised by the system under observation. Unfortunately, it is not possible to determine the conformational changes from such a description, because the local rates of change in multidimensional manifolds obtained from machinelearning techniques are, in general, unknown (21,22) and cannot be easily related to the underlying changes in the system under observation (15,23). To overcome this well-known difficulty, we introduce an additional step, in which the cloud of points is mapped to another coordinate system, where the local rates of change can be determined exactly, and related to the underlying conformational changes.…”

Reconstruction from limited single-particle diffraction data via simultaneous determination of state, orientation, intensity, and phase

“…Classification work includes manifold mapping (6), spectral clustering (7), principal component analysis, and support vector machines (8). Orientation methods include common curve approaches (9)(10)(11)(12), expectation maximization (13)(14)(15), and manifold embedding (16)(17)(18)(19). Once images are classified, oriented, and assembled into a 3D intensity function, iterative phasing (20) is often used to determine molecular structure.…”

Expansion-maximization-compression algorithm with spherical harmonics for single particle imaging with x-ray lasers

“…We note that this approach has been developed further recently in [12]. More recently, diffusion map techniques have been developed to compute low-dimensional manifolds from XFEL diffraction data [13]. In practice, these techniques can generate more than three significant dimensions revealing other experimental variables such as changing beam conditions or sample heterogeneity.…”