  Single-particle electron microscopy is a modern technique that biophysicists
employ to learn the structure of proteins. It yields data that consist of noisy
random projections of the protein structure in random directions, with the
added complication that the projection angles cannot be observed. In order to
reconstruct a three-dimensional model, the projection directions need to be
estimated by use of an ad-hoc starting estimate of the unknown particle. In
this paper we propose a methodology that does not rely on knowledge of the
projection angles, to construct an objective data-dependent low-resolution
approximation of the unknown structure that can serve as such a starting
estimate. The approach assumes that the protein admits a suitable sparse
representation, and employs discrete $L^1$-regularization (LASSO) as well as
notions from shape theory to tackle the peculiar challenges involved in the
associated inverse problem. We illustrate the approach by application to the
reconstruction of an E. coli protein component called the Klenow fragment.
