  Order parameters based on spherical harmonics and Fourier coefficients
already play a significant role in condensed matter research in the context of
systems of spherical or point particles. Here, we extend these types of order
parameter to more complex shapes, such as those encountered in nanoscale
self-assembly applications. To do so, we build on a powerful set of techniques
that originate in the computer science field of "shape matching." We
demonstrate how shape matching techniques can be applied to identify unknown
structures and create highly-specialized \textit{ad hoc} order parameters.
Additionally, we investigate the special symmetry properties of harmonic
descriptors, and demonstrate how they can be exploited to provide optimal
solutions to certain classes of problems. Our techniques can be applied to
particle systems in general, both simulated and experimental, provided the
particle positions are known.
