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## Avatar at the atomic scale: how to see in 3D with one eye

Here’s a great little contribution appearing in this week’s edition of our sister journal:

The ability to determine the structure of matter in three dimensions has profoundly advanced our understanding of nature… Here we present a 3D imaging modality, termed ankylography (derived from the Greek words ankylos meaning ‘curved’ and graphein meaning ‘writing’), which under certain circumstances enables complete 3D structure determination from a single exposure using a monochromatic incident beam.

This article is about lensless diffraction imaging, an evolution on crystallography and tomography that has developed slowly since 1980, and rapidly since the first experimental demonstration in 1999. Crystallography—of which the X-ray crystallography used to elucidate the structure of DNA is perhaps the most well-known example—exploits the duality between regular spatial arrangements of atoms and the periodic nature of electromagnetic waves. When a single-frequency beam illuminates a regular structure, the resulting pattern of interference between waves caused by the atoms is the Fourier transform of the structure.

Diffraction pattern of an icosahedral strucutre; from Ron Lifschitz's page on quasicrystals.

Fourier transformation arises in so many contexts in physics precisely because this set-up—a wave passing through a regular arrangement of objects—is so general. That it arises in this context is important, but not profound: indeed I would suggest taking the form of the diffraction pattern as a geometrical definition of the Fourier transform.

Taking a single diffraction snapshot is like looking at the world with one eye closed—everything is seen in a two-dimensional projection along your line-of-sight. Consequently, the diffraction pattern is the two-dimensional Fourier transformation of that image. What crystallographers do is take a sequence of snapshots from different angles to build up a gallery of the three-dimensional Fourier transform of the structure (in particular, of the electron density), and then invert it:

$f(\mathbf{r}) = \frac{1}{V}\int_{\mathcal{V}} F(\mathbf{q}) e^{+i\mathbf{q}\cdot\mathbf{r}} d^3\mathbf{q}.$

$F(\mathbf{q})$ is the Fourier (‘reciprocal’) space image determined directly from the snapshots; $f(\mathbf{r})$ is the real-space electron density that crystallographers want to find. Determining the image completely in reciprocal space is impossible, and the statistical problem of reconstructing f given a series of snapshots is an interesting one (whose description can be postponed for another day).

Tomographic reconstruction
from parallel slices

Let’s think about this problem from a slightly different angle. Another reconstruction method, that has found particularly wide-spread medical application, is tomography. Here, different locations, rather than orientations, are used when building up a representation of the object in the reciprocal space. Very commonly, a sequence of slices along a common axis are used, by firing radiation (the good kind) along a particular section and measuring the density from the drop in intensity on the other side. Armed with a series of such slices, the full three-dimensional distribution can be acquired, although the reconstruction transformation is a Radon, rather than Fourier, transform.

Despite these distinctions—and they are important distinctions—tomography and crystallographic diffraction imaging are the the same formal process of imaging and reconstruction. And, the same interesting statistical challenges emerge—indeed, these are sufficiently challenging and important that companies such as Blackford Analysis have been started-up to solve them.

As you can see from the picture just above, tomographic slicing works with non-periodic structures, but there is an important relationship between slicing and projection that is behind the new idea in this article. These lensless imagers argue that it is possible to reconstruct the atomic structure not just for objects that are less regular than crystals, but that this can be done with a single snapshot alone. Back to the authors:

We demonstrate that when the diffraction pattern of a finite object is sampled at a sufficiently fine scale on the Ewald sphere, the 3D structure of the object is in principle determined by the 2D spherical pattern… With further development, this approach of obtaining complete 3D structure information from a single view could find broad applications in the physical and life sciences.

How can this be? I’ve tried to give some hints throughout the article, and I’ll provide a description of the answer in the next post. (I know that my track record for follow-up posts is shocking, but as my co-bloggers can verify, the subsequent parts are already written.) For now, it’s over to you.

Update: I’ve now added a follow-up post with the mystery explained!