  We propose a versatile, parameter-less approach for solving the shape
matching problem, specifically in the context of atomic structures when atomic
assignments are not known a priori. The algorithm Iteratively suggests Rotated
atom-centered reference frames and Assignments (Iterative Rotations and
Assignments, IRA). The frame for which a permutationally invariant set-set
distance, namely the Hausdorff distance, returns minimal value is chosen as the
solution of the matching problem. IRA is able to find rigid rotations,
reflections, translations, and permutations between structures with different
numbers of atoms, for any atomic arrangement and pattern, periodic or not. When
distortions are present between the structures, optimal rotation and
translation are found by further applying a standard Singular Value
Decomposition-based method. To compute the atomic assignments under the
one-to-one assignment constraint, we develop our own algorithm, Constrained
Shortest Distance Assignments (CShDA). The overall approach is extensively
tested on several structures, including distorted structural fragments.
Efficiency of the proposed algorithm is shown as a benchmark comparison against
two other shape matching algorithms. We discuss the use of our approach for the
identification and comparison of structures and structural fragments through
two examples: a replica exchange trajectory of a cyanine molecule, in which we
show how our approach could aid the exploration of relevant collective
coordinates for clustering the data; and an SiO$_2$ amorphous model, in which
we compute distortion scores and compare them with a classical strain-based
potential. The source code and benchmark data are available at
\url{https://github.com/mammasmias/IterativeRotationsAssignments}.
