  In this paper we propose an algorithm for aligning three-dimensional objects
when represented as density maps, motivated by applications in cryogenic
electron microscopy. The algorithm is based on minimizing the 1-Wasserstein
distance between the density maps after a rigid transformation. The induced
loss function enjoys a more benign landscape than its Euclidean counterpart and
Bayesian optimization is employed for computation. Numerical experiments show
improved accuracy and efficiency over existing algorithms on the alignment of
real protein molecules. In the context of aligning heterogeneous pairs, we
illustrate a potential need for new distance functions.
