  Single particle cryo-electron microscopy (EM) is an increasingly popular
method for determining the 3-D structure of macromolecules from noisy 2-D
images of single macromolecules whose orientations and positions are random and
unknown. One of the great opportunities in cryo-EM is to recover the structure
of macromolecules in heterogeneous samples, where multiple types or multiple
conformations are mixed together. Indeed, in recent years, many tools have been
introduced for the analysis of multiple discrete classes of molecules mixed
together in a cryo-EM experiment. However, many interesting structures have a
continuum of conformations which do not fit discrete models nicely; the
analysis of such continuously heterogeneous models has remained a more elusive
goal. In this manuscript, we propose to represent heterogeneous molecules and
similar structures as higher dimensional objects. We generalize the basic
operations used in many existing reconstruction algorithms, making our approach
generic in the sense that, in principle, existing algorithms can be adapted to
reconstruct those higher dimensional objects. As proof of concept, we present a
prototype of a new algorithm which we use to solve simulated reconstruction
problems.
