Recovering the three-dimensional structure of proteins is an
important tool for understanding their properties and functionality.
In this talk we consider the mathematical aspects of reconstructing
the three-dimensional structure of a molecule from its CryoEM images
taken at random unknown orientations. We show that by constructing
the parametric graph Laplacian that corresponds to the given
samples, we can recover the orientation of each projection.
Mathematically, we consider the problem of constructing a
parametrization of a domain using only samples of a function $f$
defined on this domain. This leads to the parametric
graph-Laplacian, and we show conditions on $f$ for which such a
parametrization is possible. As an example, we demonstrate that this
allows to reconstruct an object, given only its Radon projections at
unsorted random unknown angles.
We then describe the extension of this approach to the
three-dimensional CryoEM reconstruction problem. We consider each
CryoEM image as the value of a high dimensional projection function,
evaluated at some point in $SO(3)$ (the unknown random orientation
of the molecule). By combining the Fourier slice theorem with the
parametric graph Laplacian, we are able to recover the orientation
of each projection, assuming only uniform sampling of $SO(3)$, thus
transforming the problem into a standard tomography problem.
This is a joint work with Ronald Coifman, Amit Singer, and Fred
Sigworth (Yale University).