  The number of noisy images required for molecular reconstruction in
single-particle cryo-electron microscopy (cryo-EM) is governed by the
autocorrelations of the observed, randomly-oriented, noisy projection images.
In this work, we consider the effect of imposing sparsity priors on the
molecule. We use techniques from signal processing, optimization, and applied
algebraic geometry to obtain new theoretical and computational contributions
for this challenging non-linear inverse problem with sparsity constraints. We
prove that molecular structures modeled as sums of Gaussians are uniquely
determined by the second-order autocorrelation of their projection images,
implying that the sample complexity is proportional to the square of the
variance of the noise. This theory improves upon the non-sparse case, where the
third-order autocorrelation is required for uniformly-oriented particle images
and the sample complexity scales with the cube of the noise variance.
Furthermore, we build a computational framework to reconstruct molecular
structures which are sparse in the wavelet basis. This method combines the
sparse representation for the molecule with projection-based techniques used
for phase retrieval in X-ray crystallography.
