  A modulated icosahedral shell with an inclusion is a concise description of
many viruses, including recently-discovered large double-stranded DNA ones.
Many X-ray scattering patterns of such viruses show major polygonal fringes,
which can be reproduced in image reconstruction with a homogeneous icosahedral
shell. A key question regarding a low-resolution reconstruction is how to
introduce further changes to the 3D profile in an efficient way with only a few
parameters. Here, we derive and compile different analytical models of such an
object with consideration of practical optical setups and typical structures of
such viruses. The benefits of such models include 1) inherent filtering and
suppressing different numerical errors of a discrete grid, 2) providing a
concise and meaningful set of descriptors for feature extraction in
high-throughput classification/sorting and higher-resolution cumulative
reconstructions, 3) disentangling (physical) resolution from (numerical)
discretization step and having a vector graphics format for visualization or
further analysis at arbitrary scales, 4) eliminating the phase-retrieval step
and enforcing transparent, relevant, and controlled type/level of a-priori
information in a real-space formulation, and 5) evaluating the reflections and
surface resonances of an icosahedral object, and hence corrections for the
common scattering model.
