Single particle electron microscopy is a powerful method that biophysicists employ to learn about the structure of biological macromolecules. In contrast to the more traditional crystallographic methods, this method images "unconstrained" particles, thus posing a variety of statistical problems. We formulate and study such a problem, one that is essentially of a random tomographic nature. Although unidentifiable (ill-posed), this problem can be seen to be amenable to a statistical solution, once strict parametric assumptions are imposed.It can also be seen to present challenges from a data analysis point of view (e.g. uncertainty estimation and presentation). In addition, motivated by the "physics" involved in the data-collection process, we define and explore a new type of diffusions in D.G. Kendall's shape space. These diffusions arise as the stochastic evolution of the orbits (under a group action) of a projected motion.