Turns out you can beat the Nyquist frequency.... by a lot.

This week-end I stumbled across a field of signal processing called Compressed Sensing, or sparse sampling.

These sampling and reconstruction techniques have been applied to real space and fourier space images, such as MRI, and CT scanning. It has led to people being able to reconstruct images when sampled with a single pixel. Turns out it's also starting to be applied to cryo-EM.

The only requirements are that your system have a single solution, and that it be "sparse", that is to say that most of its coefficients are close or equal to zero when represented by some transform. Furthermore, your samples have to be random (are you excited yet?)

As it turns out virtually all real data that is not pure noise is sparse.

This technique has already been applied to random projections, though in the field of medical tomography.

I attach a reference in which they reconstruct 2D real space images from a small number of 4-fold binned samples. I am sure you will see the parallels to the work that we do every day.

Also, if someone can get the following reference:

Cryo-electron microscopy single particle reconstruction of virus particles using compressed sensing theory

Proc. SPIE 6498, 64981G (2007); doi:10.1117/12.705008