While the development of cryo-electron microscopy (cryo-EM) has already proven to revolutionize the field of structural biology by imaging biomolecules in solution, the vast majority of proteins cannot be reconstructed at a satisfying resolution. Among them, membrane proteins still remain an imaging challenge for biologists - despite being prominent targets for over half of prescription drugs on the pharmaceutical market. The proposed work develops a novel mathematical and statistical method that will enhance the resolution of cryo-EM, targeting the 3D imaging and reconstruction of membrane proteins. The main challenges that we solve, and the novelty of this approach, come from statistics and differential geometry. From a statistical perspective, the proposal revisits the paradigm of cryo-EM shape reconstruction, by replacing the traditional "expectation-maximization" learning procedure by its faster and scalable counterpart, called "variational inference". In order to apply "variational inference" in this context, our solution implements the differential geometry of 3D shape spaces within the recent and popular dimension reduction method of "variational autoencoders". By improving the efficiency of the image reconstruction algorithm, while benchmarking its accuracy, we leverage the extraordinary amount of raw data produced by cryoEM. In turn, processing more data improves the resolution of the reconstructed biomolecular shapes. The originality of the proposed project is to leverage statistics and mathematics that have not yet penetrated the communities of machine learning and biological imaging. Our proposal is also broadly applicable beyond cryo-EM and biological imaging. Reconstructing biomolecular shapes is the focus of several imaging modalities, such as coherent diffraction for single particle imaging, that produce images whose analysis is an on-going research of members of our team. The proposal translates to the representation of shapes for these technologies.