My Para(bellum)llax Is With Me

In Defence of the Holy Inquisition

``Amazing! They do not allow a single thought that differs even slightly from their picture of the world!''

I mumbled this while reading the response to my notes from some highly ranked scientific journal.

James Bond, who happened to be nearby, replied mockingly:

``Do you imagine yourself as a new Galileo Galilei?''

``I do not,'' I said angrily. ``But the dogmatism of society has been horrible at all times. Today as well as at Galileo's time.''

``I am afraid you are following the vulgar version of the Galileo story. The members of the High Court of the Holly Inquisition who investigated his case were not necessarily ignorant fanatics. Some of them were among the most educated people of their time, and there were serious scientific arguments against the Copernican system [Scientific American, January 2014].''

``What? Galileo, one of the founders of modern experimental science, was wrong?''

``I did not say he was wrong. I mean that, in this particular case, the evidence available at the time was not as decisive as we like to imagine today. One of the weakest points of the Copernican system was the so-called stellar parallax. If the Earth revolves around the Sun, the distant stars should appear to shift slightly in the sky during the year. But no such shift could be detected.''

``Because the stars are too far away. The displacement was simply too small to measure with the instruments of that time.''

``Exactly. And the scholars considered that possibility. But there was a problem. Stars did not appear as point-like objects through the telescopes of the time. Astronomers could measure their apparent discs. They did not yet understand that these apparent sizes were artifacts of the passage of light waves through a circular aperture of a telescope rather than the true angular diameters of the stars. If the stars were distant enough to make their parallax undetectable, and if those apparent diameters were real, then the stars had to be absurdly enormous, something like several light-years in modern terms. Copernicus, whom you like to regard as an exemplary rationalist, appealed to the immeasurable power of God: if God wished to create unimaginably large and distant stars, why should He not? His opponents could answer that God was also rational and had no reason to fill the heavens with useless gigantic objects. ''

``I still remain on Galileo's side.''

``I am not on anybody's side,'' said James. ``I advocate rational and logical thinking. Galileo's conviction did great credit to his intuition. Whether it always did equal credit to his objectivity is another question.''

Parallax in 4DSTEM

Leaving aside James's extravagant opinions, parallax is an effect that must be taken into account in many areas of science.

Consider, for example, Scanning Transmission Electron Microscopy (STEM). A small probe is formed by focusing a convergent electron beam onto the specimen. Because the illumination is convergent, different parts of the probe-forming aperture correspond to slightly different illumination angles.

For a sufficiently thin specimen at the correct focus, this does not cause a troublesome displacement of the image and that makes atomic-resolution STEM possible. The situation becomes more interesting when the specimen has appreciable depth, or when the beam, for any reason, is defocused. Features lying at different depths can then appear at different lateral positions when viewed through different parts of the diffraction pattern. This is the STEM analogue of parallax.

In conventional STEM, the detector integrates intensity over a chosen angular range. The information associated with the individual scattering directions is therefore mixed together, and parallax shifts contribute to image blurring.

The situation changes in 4DSTEM. For each $(x,y)$ pixel, instead of recording only an integrated detector signal, we record the complete two-dimensional diffraction pattern $(k_x,k_y)$. We can therefore construct virtual images using different regions of the diffraction pattern. For a defocused object, these virtual images are displaced relative to one another. The displacement depends systematically on the scattering angle and increases as we move away from the centre of the diffraction pattern.

Once these shifts have been measured or calibrated, the process can be reversed. The virtual images corresponding to different scattering angles can be shifted back into registration and combined. In this way, information that would otherwise contribute to a blurred conventional image can be used to reconstruct a sharper image of the object.

I sketched a simple Python example to illustrate the principle. The scattering model is deliberately primitive, and parallax is simulated only along the $x$ direction. Nevertheless, it gives a useful visual impression of what is happening: different angular views are displaced by different amounts, and bringing them back into registration recovers the sharp object.

The Python codes can be found in the pdf version of this document: Full Text with Codes.

If you have any comments or suggestions, please email pavel@temdm.com ".

Posted Sept 10, 2026