  This paper addresses the problem of synchronizing orthogonal matrices over
directed graphs. For synchronized transformations (or matrices), composite
transformations over loops equal the identity. We formulate the synchronization
problem as a least-squares optimization problem with nonlinear constraints. The
synchronization problem appears as one of the key components in applications
ranging from 3D-localization to image registration. The main contributions of
this work can be summarized as the introduction of two novel algorithms; one
for symmetric graphs and one for graphs that are possibly asymmetric. Under
general conditions, the former has guaranteed convergence to the solution of a
spectral relaxation to the synchronization problem. The latter is stable for
small step sizes when the graph is quasi-strongly connected. The proposed
methods are verified in numerical simulations.
